I 1 V ( x) = V (x), V ( x) = V ( x ) SO(3) x = R x: R SO(3) Lorentz R t JR = J: J = diag(1, 1, 1, 1) x = x + a Poincarré ( ) 2

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "I 1 V ( x) = V (x), V ( x) = V ( x ) SO(3) x = R x: R SO(3) Lorentz R t JR = J: J = diag(1, 1, 1, 1) x = x + a Poincarré ( ) 2"

Transcription

1 III Jan 30th, 2006 I : II : I : [ I ] (Landau and Lifshitz, Quantum Mechanics chapter 12, 13, 9: Pergamon Pr.) [ ] ( ) (H. Georgi, Lie algebra in particle physics, Perseus Books) [ ] II : H. (H. Flanders, Differential Forms With Applications to the Physical Sciences, Dover) I (M. Nakahara, Geometry, Topology and Physics, Inst of Physics Pub Inc, chap 5,6) M. Spivak, Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (Perseus Books) 1 matsuo( )phys.s.u-tokyo.ac.jp 1

2 I 1 V ( x) = V (x), V ( x) = V ( x ) SO(3) x = R x: R SO(3) Lorentz R t JR = J: J = diag(1, 1, 1, 1) x = x + a Poincarré ( ) 2

3 Maxwell Lorentz Yang-Mills 2 G 1. a, b G a b G 2. a, b, c G (a b) c = a (b c) 3. e G a G a e = e a = a 4. a G a 1 G a a 1 = a 1 a = e 3

4 G (G ) ( ) ( ) a, b G a b = b a ( (Nonabelian group)) ( ) ( ) ( SO(3), (R)) (Z 2 ) x x G = {e, σ} e : x x σ : x x e e = σ σ = e, e σ = σ e = σ (Z n ) z = x + iy σ : z ωz (ω = e 2πi/n ) 360/n G = {e, σ, σ 2,, σ n 1 } σ r σ s = σ r+s r + s mod n Z n = n (S n ) (n ) ( 1 2 n a 1 a 2 a n i a i S n = n! ) GL(n, R) (GL(n, C)): ( ) n n ( ) 4

5 O(n): {a GL(n, R), a t a = E} (E : ) n U(n): { a GL(n, C), a a = E } (a = (a ) t ) n SO(n) (SU(n)) O(n) (U(n)) 1 O(n, m) (U(n, m)) GL(n+m, R) (GL(n+m, C)) a t Ja = J (a Ja = J) J = diag(1,, 1, 1,, 1) (1 n -1 m ) O(3) O(3, 1) ( SU(n)) b.. a a b. Z 2. e σ e e σ σ σ e 3 (group homomorphism) 1. f : G 1 G 2 5

6 2. G 1 2 a 1, a 2 f(a 1 ) f(a 2 ) = f(a 1 a 2 ) Ker f = {g G 1 f(g) = e} G 1, Im f = {f(g) g G 1 } G 2 (isomorphism) G 1 G 2 f (bijection) (subgroup) H G 1. H G H G 2. H G a, b H a b H (coset) H G H Q G g H H g = {h g h H} G g H = {g h h H} G G H H 2 H G g 1 H g 1 H H g 1 = φ h 1 H H g 1 h 2 H h 1 = h 2 g 1 g 1 = h 1 2 h 1 H 3 4 G G = H (H g 1 ) (H g n 1 ) (right coset) H\G H\G {H, H g 1,, H g n 1 } (left coset) G/H {H, g 1 H,, g n 1 H} n = G / H (index) (G : H) 6

7 ( ) (invariant subgroup, normal subgroup) G H G g g H = H g H G H G/H = H\G G/H (coset group), (factor group) (g 1 H) (g 2 H) = (g 1 g 2 ) H H [H H = H H ] (g H) (g 1 H) [ ] (g H) (g 1 H) = g g 1 H H = H 2 a, b (conjugate) G g b = g a g 1 a b a b a b a = b (transitive law) a b b c a c [ ] a = g 1 b g1 1, b = g 2 c g2 1 a = (g 1 ) (g 2 c g 1 2 ) g 1 1 = (g 1 g 2 ) c (g 1 g 2 ) 1 (conjugacy class) G G = C 1 C 2 C n C i G a, b C i a b i j C i C j = φ [ ] e g G g e g 1 = e e e (group ring) G g e g R G = a g e g a g R g G 7

8 : R G R G R G e g1 e g2 = e g1 g 2 a g e g b g e g = a g b g e g g g G g G g,g G R G G C i R G Ĉ i Ĉ i = e a a C i g G e g Ĉi = Ĉi e g [ ] h C i g h g 1 C i g C i = C i g ( ) Ĉ i Ĉj = Ĉj Ĉi Ĉ i Ĉj = k N ij k Ĉ k N k ij S 3 3 3! = 6 ( ) ( ) ( ) e = σ 1 = σ 2 = ( ) ( ) ( ) σ 3 = ω = ω 2 = σ 1 σ 2 = ( ) ( ) ( ) ( = = ( ) = ω )

9 e ω ω 2 σ 1 σ 2 σ 3 e e ω ω 2 σ 1 σ 2 σ 3 ω ω ω 2 e σ 3 σ 1 σ 2 ω 2 ω 2 e ω σ 2 σ 3 σ 1 σ 1 σ 1 σ 2 σ 3 e ω ω 2 σ 2 σ 2 σ 3 σ 1 ω 2 e ω σ 3 σ 3 σ 1 σ 2 ω ω 2 e S 3 {e}, {e, σ 1 }, {e, σ 2 }, {e, σ 3 }, { e, ω, ω 2 }, S 3 S 3 {e}, { e, ω, ω 2 }, S 3 S 3 S 3 / {e} = S 3, S 3 /S 3 = {e}, S 3 / { e, ω, ω 2} = {{ e, ω, ω 2}, {σ 1, σ 2, σ 3 } } = Z 2 S 3 C 1 = {e}, C 2 = { ω, ω 2}, C 3 = {σ 1, σ 2, σ 3 } Ĉ 1 Ĉ1 = Ĉ1, Ĉ 1 Ĉ2 = Ĉ2, Ĉ 1 Ĉ3 = Ĉ3, Ĉ 2 Ĉ2 = 2Ĉ1 + Ĉ2, Ĉ 2 Ĉ3 = 2Ĉ3, Ĉ 3 Ĉ3 = 3Ĉ1 + 3Ĉ2 4 (representation) G GL(n, C) ρ : G GL(n, C) ρ G n ρ g 1 g 2 = g 3 ρ(g 1 ) ρ(g 2 ) = ρ(g 3 ) 9

10 ρ G GL(n, C) ρ(e) = E ( ), ρ(g 1 ) = (ρ(g)) 1 ρ n (representation space) : (trivial ) g G ρ(g) = 1 GL(1, C) (1 ) : g 1 g 2 ρ(g 1 ) ρ(g 2 ) : (unitary ) g G ρ(g) U(n) ρ(g 1 ) = (ρ(g)) 1 = ρ(g) : (direct sum ) ρ 1, ρ 2 n 1, n 2 ρ 1 ρ 2 ρ 1 ρ 2 n 1 + n 2 ρ 1 ρ 2 : g ( ρ1 (g) 0 0 ρ 2 (g) : (direct product ) ρ 1, ρ 2 n 1, n 2 ρ 1 ρ 2 ρ 1 ρ 2 n 1 n 2 ) ρ 1 ρ 2 : g ρ(g) ik,jl = ρ 1 (g) ij ρ 2 (g) kl ρ(g) ik,jl ik jl :(equivalent ) ρ 1, ρ 2 GL(n, C) g G T GL(n, C) ρ 1 (g) = T ρ 2 (g)t 1 ρ 1 ρ 2 (invariant subspace): ρ V ρ(g) ρ(g)v V (irreducible ): : (decomposition into irreducible representations) ρ = ρ 1 ρ 2 ρ n 10

11 (ρ i ) ρ (α) n (α) ρ = α n (α) ρ (α) : (regular ) e g e a = ρ (reg) (a) gg e g g G ρ (reg) 0 1 ρ (reg) (e) (e g e a ) e b = ρ (reg) (a) gg e g e b = g G = e g (e a e b ) = g G ρ (reg) (a b) gg e g. g,g G ρ (reg) (a) gg ρ (reg) (b) g g = ρ(reg) (a b) gg g G ρ (reg) (a) gg ρ (reg) (b) g g e g Shur (Schur s Lemma) 1. ρ i (i = 1, 2) G n i V i M V 1 V 2 g G Mρ 1 (g) = ρ 2 (g)m M ( n 1 = n 2 M 1 ) M = 0 2. ρ, V M V V g G ρ(g)m = Mρ(g) M 11

12 1. KerM = {v V 1 Mv = 0}, ImM = {Mv V 2 v V 1 } V 1, V 2 v Ker M Mv = 0 M Mρ 1 (g)v = ρ 2 (g)mv = 0 ρ 1 (g)v Ker M Ker M ρ 1 0 Ker M = 0 Ker M = V 1 M = 0 Im M 0 V 2 Ker M = 0 Im M = V 2 M 2. M v V 1 Mv = λv (λ C ) g G (M λe)ρ(g) = ρ(g)(m λe) M λe M λe v M λe = 0 { } ρ (α) (α = 1,, #( )) ρ (α) ji (g 1 )ρ (β) kl (g) = G δ ik δ jl δ αβ d α g G ρ (α), ρ (β) V (α), V (β) B V (β) V (α) M = g G ρ (α) (g 1 )Bρ (β) (g) M V (β) V (α) g G ρ (α) (g)m = Mρ (β) (g) ρ (α) (g)m = ρ (α) (g) ρ (α) (g 1 )Bρ (β) (g ) g G = ρ (α) (gg 1 )Bρ (β) (g ) = ρ (α) (g 1 )Bρ (β) (g g) g G g = g G ρ (α) (g 1 )Bρ (β) (g ) ρ (β) (g) = Mρ (β) (g) Schur 1 α β 12

13 2 M = 0 B rs = δ ri δ sk M jl = g r,s ρ (α) jr (g 1 )δ ri δ sk ρ (β) sl (g) = g ρ (α) ji (g 1 )ρ (β) kl (g) = 0 α = β Shur 2 M = ce B rs = δ ri δ sk ρ (α) ji (g 1 )ρ (α) kl (g) = c ik δ jl g c ik j, l d α c ik = ρ (α) ji (g 1 )ρ (α) kj (g) = ρ (α) kj (g)ρ(α) ji (g 1 ) = ρ (α) ki (g g 1 ) = G δ ki g g g j j (d α ρ (α) G G ρ (α) (e) ki = δ ki ) c ik = G d α δ ik ρ g G ρ (α) ij (g)ρ (β) kl (g) = G δ ik δ jl δ αβ d α (character) ρ χ(g) = Tr(ρ(g)) ρ, ρ ρ (g) = T ρ(g)t 1 χ (g) = χ(g) g, g χ(g) = χ(g ) [ ] g g g = k 1 gk (k G) χ(g ) = Trρ(k 1 gk) = Trρ(k 1 )ρ(g)ρ(k) = χ(g) ρ (α) ρ (β) χ(ρ (α) ρ (β) ) = χ (α) + χ (β) χ(ρ (α) ρ (β) ) = χ (α) χ (β) 2 α β d α = d β M 0 Mρ 1 (g) = ρ 2 (g)m ρ 2 (g) = Mρ 1 (g)m 1 ρ 1 ρ 2 M = 0 13

14 (I) ρ (α), ρ (β) χ α, χ (β) χ (α) (g)χ (β) (g) = G δ αβ g G [ ] i = j, k = l χ {C i } (i I) G χ (α) = χ (α) (g) g Ci i I C i χ (α) i χ (β) i = G δ αβ i (I) ρ(g) ρ(g) = α q α ρ (α) (g) ( α ) q α χ(g) = α q α χ α (g) χ α(g) g χ α(g)χ(g) = g β q α = 1 G g q β χ α(g)χ β (g) = g β χ (α ) (g)χ (α) (g) = 1 G i I q β G δ αβ = q α G C i χ (α) i χ (α) i ρ (reg) (g) g = e χ (reg) (e) = E χ (reg) (g) = { G (for g = e) 0 (otherwize) q α = 1 G g χ (reg) (g)χ (α) (g) = 1 G χ(reg) (e)χ (α) (e) = d α 14

15 χ (reg) (e) = G, χ (α) (e) = d α ρ (reg) (g) = α d α ρ (α) (g) g = e G = α (d α ) 2 (II) n n α=1 χ (α) i χ (α) j = G C i δ ij [ ] Ĉ i = g C i e g ρ (α) (Ĉi) = [ ] g C i ρ (α) (g) ( ) eg, Ĉi = 0 g G [ ρ (α) (g), ρ (α) (Ĉi) ] = 0 Schur ρ (α) (Ĉi) = λe trace d α ρ (α) λ C i χ (α) i = λd α λ = C i χ (α) i d α ρ (α) (Ĉi)ρ (α) (Ĉj) = k N ij k ρ (α) (Ĉk) ρ (α) (Ĉi) C i C j d 2 α χ (α) i χ (α) j = k C i C j χ (α) i χ (α) j = k k N C k χ (α) k ij d α N ij k d α C k χ (α) k C i C j α χ (α) i χ (α) j = C k N k ij d α χ (α) k = α,k k C k N ij k χ (reg) k = N ij 1 χ (reg) 1 3 d α χ (α) k 15

16 C 1 N 1 ij = δîj C j, χ (reg) 3 1 = G α χ (α) i χ (β) G j = δ ij C j U (α) = C i #class i U (α) i U (β) i = δ αβ, #irreps α U (α) i U (α) j = δ ij. i G χ(α) i U Trρ (α) (g) ρ (β) (g) = Trρ (α) (g)trρ (β) (g) ρ (α) ρ (β) = irreps γ C αβ γ ρ (γ) (g) C αβ γ = 1 G χ (γ) (g)χ (α) (g)χ (β) (g) = g i C i G χ(γ) i χ (α) i χ (β) i (II) C i C j χ (α) i χ (α) j = k N ij k d α C k χ (α) k d α l α k α χ (α) N k ij C k χ (α) k χ(α) l = N k ij C k G C k δ kl = N k ij G 3 î (complex conjugate class) C i g g 1 ρ(g 1 ) = ρ(g) χî = χ i 16

17 N ij k = α C i C j χ (α) i d α G χ (α) j χ (α) l 5 (point group and its representation) 5.1 (point group) (symmetry transformation of point group) (rotation) 2π/n C n (C n ) n = e (reflection) σ σ 2 = e σ h σ v, rotation-reflection S n S n = C n σ h (inversion) x x I I = S 2 = σ h C 2 (classification of point group) 5 C n : e, C n, (C n ) 2,, (C n ) n 1 n Z n 2 D n : n 2 n 2 n 2 n 2n (C n n C 2 n ) D 3 = V 17

18 4 T: T = 12 8 O: O = I: C 5 6 C 3 10 C 2 15 I = 60 ( ) S 2n : ( S n ) 2n S 2n n = 2p + 1 (S 4p+2 ) 2p+1 = I S 4p+2 = C 2p+1 C i C i {e, I} C nh : n (C n ) p, (C n ) p σ h (p = 0,, n 1) 2n C nv : n n C nv = 2n D nh : D n n 2 D nd : n 2 2 T d : T T h : T T h = T C i O h : O O h = O C i I h : I I h = I C i H 2 O : C 2v NH 3 : C 3v CH 3 Cl : C 3v CH 4 : T d 18

19 OsF 8 : O h UF 6 : O h C 2 H 6 : D 3d C 2 H 4 : D 2h 5.2 (Representation of point group) C n (=Z n ) g 1, g 2 G [g 1, g 2 ] = 0 [ρ(g 1 ), ρ(g 2 )] = 0 ρ(g) (g G) v ρ(g) v = λ(g) v, λ(g) C v 1 λ(g) 1 C n = Z n {e, C n, (C n ) 2,, (C n ) n 1 } (C n ) n = e ρ(c n ) = λ C 1 (C n ) n = e λ n = 1 n 1 ρ (α) ((C n ) p ) = e 2πiαp/n p = 0, 1,, n 1, α = 0, 1,, n 1 n χ (α) ((C n ) p ) = e 2πiαp/n ( ρ (0) ) discrete Fourier n 1 p=0 (χ (α) ((C n ) p )) χ (β) ((C n ) p ) = n 1 p=0 e 2πipα/n e 2πipβ/n = nδ α,β 19

20 α irrep. (χ (α) ((C n ) p ) χ (α) ((C n ) q ) = n 1 α=0 e 2πipα/n e 2πiqα/n = nδ p,q x p χ (α) (g) C nh C 3v (= S 3 3 ) {e}, {ω, ω 2 }, {σ 1, σ 2, σ 3 } ρ (1) ρ (2), ρ (1) (g) = 1 ρ (2) (e) = ρ (2) (ω) = ρ (2) (ω 2 ) = 1, g C 3v ρ (2) (σ 1 ) = ρ (2) (σ 2 ) = ρ (2) (σ 3 ) = 1 3 d d 2 = 6 d = 2 ( ) ( ) ( ) 1 0 c s c s ρ (3) (e) =, ρ (3) (ω) =, ρ (3) (ω 2 ) = 0 1 s c s c ( ) ( ) ( ) 1 0 c s c s ρ (3) (σ 1 ) =, ρ (3) (σ 2 ) =, ρ (3) (σ 3 ) =. 0 1 s c s c c = cos(2π/3) = 1/2, s = sin(2π/3) = 3/2 2 ρ (α) (g) = χ (α) (g) ρ (3) χ (3) (e) = 2, χ (3) (ω) = χ (3) (ω 2 ) = 1, χ (3) (σ 1 ) = χ (3) (σ 2 ) = χ (3) (σ 3 ) = ρ(e) = 0 1 0, ρ(ω) = ρ(σ 1 ) = , ρ(σ 2) = , ρ(ω2 ) =, ρ(σ 3) =

21 D 2n a (2π/n ) b (2π/2 ) a n = b 2 = e, b 1 a b = a 1 D 2n = {e, a,, a n 1, b, ba,, ba n 1 } a p a l a p = a l, a r (a l b)a r = a l+2r b, a l a n l, (a p b)a l (a p b) 1 = a n l (a r b)(a l b)(a r b) 1 = a 2r l b a l b a l+2 b a n l b n: : {e}, { a i, a n i} { (1 i n/2), } { ba 2i, } ba 2i 1 (1 i n/2) n: : {e}, { a i, a n i} { (1 i (n 1)/2), } ba i (1 i n) n n/2 + 3, (n 1)/ G 1 (G : G ) G G 4 D2n a 2 n 1 4 n 2 n ρ (1) (a) = ρ (1) (b) = 1 ρ (2) (a) = 1, ρ (2) (b) = 1 ρ (3) (a) = 1, ρ (2) (a) = 1, ρ (2) (b) = 1 ρ (1) (a) = ρ (1) (b) = 1 ρ (2) (a) = 1, [ ] 2 n 2 ( ρ (k) ω k 0 2 (a) = 0 ω k ) ρ (2) (b) = 1 ρ (2) (b) = 1 (, ρ (k) (b) = G 2 g 1, g 2 g 1 g 2 g1 1 g )

22 (k = 1, 2,, [ ] n 2 ) ω = e 2πi/n : x i (i = 1,, N) H = 1 M ij ẋ i ẋ j + 1 K ij x i x j 2 i,j 2 i,j x i (x q = Rx, R t R = M) H = 1 2 i ( ) 2 dqi + 1 dt 2 L ij q i q j i,j q (q Q = Sq, S t LS = diag(ω 2 i )) ( S O(N) ) H = 1 2 i ( ) 2 dqi + 1 dt 2 Ω 2 i Q 2 i i G Ω G q G q i = ρ ij (g)q i H H(ρ(g)q) = H(q) ρ ρ ρ = α n α ρ (α) 22

23 ρ 1 (g)... ρ(g) = S ρ2(g) S t Sρ (diag) (g)s t S q Q = Sq L Ω = S t LS Hamiltonian q ρ(g)q ρ t (g)lρ(g) = L S t LSρ (diag) (g) = ρ (diag) S t LS S t LS g G ρ (diag) (g) Schur Schur I Ω Schur II Ω Hamiltonian H = 1 2 n α d α α i=1 s=1 ( ( Q (α,i) s ) 2 + (Ω (α) i ) 2 (Q (α,i) s ) 2) i s Ω (α) (NH 3 ) C 3v = S 3 N x 1, H x 2,3,

24 =6 6 C 3v 3 {e}, {ω, ω 2 }, σ 1, σ 2, σ 3 e, ω, σ 3 e χ(e) = 6 ω x 1 R x 1 x R 0 x 2 = x R x 3 x 4 0 R 0 0 x 4 R = c s 0 s c (c = cos 2π 3 = 1 2, s = sin 2π 3 3 = 2 ) χ(ω) = 0 σ 3 x 1 Σ x 1 x 2 0 Σ 0 0 x 2 = x Σ x 3 x Σ 0 x 4 Σ = χ(σ 3 ) = 2trΣ = 2 C 3v C 1 = {e}, C 2 = {ω, ω 2 }, C 3 = {σ 1, σ 2, σ 3 } C 1 C 2 C 3 ρ (1) ρ (2) ρ (3) n α = 1 6 g G χ (α) (g) χ(g) n 1 = 2, n 2 = 0, n 3 = 2 ρ (1) 2 ρ (2) 0 ρ (3) = 6 24

25 6 ( ) (Symmetry (Permutation) group) Young 6.1 S n 1,, n σ ( 1 n ) σ(1) σ(n) S n σ σ τ = ( ) ( ) 1 n 1 n σ(1) σ(n) τ(1) τ(n) = ( ) ( ) τ(1) τ(n) 1 n σ τ(1) σ τ(n) τ(1) τ(n) = ( 1 n ) σ τ(1) σ τ(n) σ, τ ( ) ( ) σ(1) σ(n) σ 1 1 n = = 1 n σ 1 (1) σ 1 (n) (Transposition) ( 1 i j ) n 1 j i n (ij) ( ) :

26 2 (ij) (ij) = e p i σ p n p 1 σ 1 = p 1 p n σ ( ) Van der Monde (x 1,, x n ) = 1 1 x 1 x n = (x.. i x j ) i<j x n 1 1 xn n 1 S n σ (x 1,, x n ) = (x σ(1),, x σ(n) ) σ = ± σ σ = σ = ± ( +, ) (cycle) l ( ( ) a1 a 2 a l 1 a l (a 1,, a l ) a 2 a 3 a l a 1 (cycle) S n ) = (134)(25)(6) 3, 2, 1 S n n = λ λ n λ i 0, λ i λ i+1, n i=1 λ i = n [λ 1,, λ n ] n (partition) n p(n) n p(n) 1 1 [1] 2 2 [2], [1, 1] 3 3 [3], [2, 1], [1, 1, 1] 4 5 [4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1] 26

27 (p(0) = 1 ) p(n)q n 1 = n=0 n=1 1 q n : σ 1 σ 2 ( S n ) σ 1 σ 2 [λ 1,, λ n ] σ = (σ(1) σ(λ 1 )) (σ(λ 1 + 1) σ(λ 1 + λ 2 )) (σ(λ λ n 1 + 1),, σ(λ λ n )) τ = (τ(1) τ(λ 1 )) (τ(λ 1 + 1) τ(λ 1 + λ 2 )) (τ(λ λ n 1 + 1),, τ(λ λ n )) µ µ = ( σ(1) σ(n) τ(1) τ(n) σ = µ 1 τµ ) Young diagram (Young ) [λ 1, λ 2,, λ n ] λ 2 λ 1 λ 3 1: Young diagram Young Young 27

28 ( ) S 3 ρ 1 [3], ρ 2 [1, 1, 1], ρ 3 [2, 1] 6.2 S n S n n Young Young λ = [λ 1,, λ n ] d λ = f! s 1 s 2 s f f = i λ i ( ) s i i (hook length)= + +1 [ ] S 5 5! = λ d 2 λ 1,4,5,6,5,4,1 [ ] d [n] = d [1,1,,1] = 1 ρ(σ) = 1 ρ(σ) = ( 1) σ Young ( n ) 1,, n (board) S 5 Young [2, 2, 1] 2: (board) B H B ( (2, 4), (1, 5) ) S n 28

29 R B ( (2, 1, 3), (4, 5) ) a B = b B = 1 H B 1 R B e σ σ H B σ R B ( 1) σ e σ, a B a B = a B, b B b B = b B e B = r n! a B b B (r = d λ, n = λ ) e B e B = e B tre B = r e B V B = e B C S n C Sn r S n λ ( 1, 2,, n) ψ(1, 2,, n) (1,, n ) S n σ ψ(1, 2,, n) = ψ(σ 1,, σ n ) S n n! S n Young e B [n] ψ [n] (1, 2,, n) = σ S n ψ(σ 1,, σ n ) 1 [1, 1,, 1] ψ [1,1,,1] (1, 2,, n) = σ S n ( 1) σ ψ(σ 1,, σ n ) 29

30 1 Young λ B n! σ ψ Young e B d λ (= ) ρ (reg) (g) = i d α ρ (α) (g) d λ ρ ρ (α) π α = d α χ G α(g)ρ(g) g G ρ (α) π α π β = d αd β G 2 g,g G g g G χ α(g)χ α(g )ρ(g g ) = d αd β χ (α) (g g 1 )χ (β) (g ) ρ(g) G 2 = δ αβ d α G g G χ α(g)ρ(g) = δ αβ π β 2 3 g G χ (α) (g 1 )χ (β) (g g) = δ αβ G χ (α) (g) d α ρ(g) = α n (α) ρ (α) (g) Tr (π α ) = d α χ G α(g)trρ(g) = n α d α g G π α n α d α ρ (α) d α n α 30

31 6.3 SU(n) GL(n, C) n ( V = C n ) M GL(n, C) (i, j = 1,, n) M j i m V V v 1 v m M M v 1 M v m GL(n, C) S m σ ( v 1 v m ) = v σ1 v σm S m GL(n, C) σ (M( v 1 v m )) = M(σ( v 1 v m )) = M v σ1 M v σm S m (Young ) e B e B (V V ) GL(n, C) GL(n, C) GL(n, C) Young V e i (i = 1,, n) 1 m = 2 λ = [2] 2 ( e 2 i e j + e j e i ) n(n + 1)/2 1 λ = [1, 1] 2 ( e 2 i e j e j e i ) n(n 1)/2 m = 3 : [3] 3 n(n+1)(n+2)/6 [1, 1, 1] 3 n(n 1)(n 2)/6 λ = [2, 1] Young n(n 2 1)/3 2 n n + 1 Young n 31

32 SU(n) SU(n) n GL(n, C) M SU(n) det(m) = 1 n e 1 e n det(m) e 1 e n = e 1 e n e 1 e n = 1 ( 1) σ e σ1 e σn n! σ S n 5 SU(n) Young 1. n 2. n Young n Young [λ 1,, λ n 1, λ n ] n 1 Young [λ 1 λ n,, λ n 1 λ n ] ( n 1 ) SU(2) SU(2) Young 2 1 = 1 λ Young [λ] σ S λ s σ1 s σλ s 2 Up Down) λ + 1 λ λ/2 SU(3) Young 2 λ 1 λ 2 SU(3) Young [1], 3; [1, 1], 3; [2], 6; 5 32

33 [2, 2], 6; [2, 1], 8; [3], 10 [1] (3 ) u, d, s [3], [2, 1] (baryon) [2, 1] (meson) (color) SU(3) [1] (3 ) [1 2 ] (3 ) [2, 1] (8 ) SU(m) Young λ ( n ) F/H, F = f 1 f n, H = s 1 s n s i i hook length (S n ) (factor)f i i f = m +1, 1 f 3 λ = [2, 2, 1] hook length 3: hook length m 2 (m + 1)(m 1)(m 2) = m2 (m 2 1)(m 2) 24 SU(2) SU(3) 33

34 7 (Representation theory of Lie group and Lie algebra) SU(n) 7.1 GL(n, C), GL(n, R): (general linear group) (det g 0) SL(n, C), SL(n, R): (special linear group) (det g = 1) U(n): (unitary group) g GL(n, C), g g = E O(n): (orthogonal group) g GL(n, R), g t g = E. SU(n) = U(n) SL(n, C) SO(n) = O(n) SL(n, R) Sp(n, K): (symplectic group) (K = R, C). ω = n i=1 (ξ i η i+n η i ξ i+n ) g GL(2n, K) ( 0 g t En Jg = J J = E n 0 ) Sp(n, C) U(2n) Sp(n) (classical Lie group) U(n), SU(n), O(n), SO(n), Sp(n) G 2, F 4, E 6, E 7, E 8 (exceptional Lie group) 34

35 ( ) (Lie algebra, ring) g 1. (linearity) X, Y g ax + by g (a, b C) 2. (commutator) X, Y g [X, Y ] g ( [X, Y ] = XY Y X) 1. [X, ay + bz] = a [X, Y ] + b [X, Z] [ax + by, Z] = a [X, Y ] + b [Y, Z] 2. [X, Y ] = [Y, X] 3. Jacobi [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0 g T 1,, T d X g X = d i=1 a i T i d [T A, T B ] = i f C AB T C C=1 f C AB f AB C = f BA C ( ) f AB D f CD E + f BC D f AD E + f CA D f BD E = 0 (Jacobi ) ɛ g = e + iɛx ɛ2 2 X2 + O(ɛ 3 ) = exp(iɛx), X g. 35

36 1. U(n) u(n) E = g g = E + iɛ(x X ) + O(ɛ 2 ) T = T Hermite 2. SU(n) su(n): det(g) = 1 + iɛtr(x) + O(ɛ 2 ) = 1 trx = 0. Hermite 3. O(n) o(n): X t = X 4. SO(n) so(n): X t = X tr(x) = 0 X tr(x) = 0 O(n) SO(n) o(n) = so(n) 5. Sp(n, C) sp(n, C): ( E + iɛa iɛb g = iɛc E + iɛd ) + O(ɛ 2 ) d = a t, b t = b, c t = c 6. Sp(n) = Sp(n, C) U(2n) sp(n): a = a, b = c Campbell-Hausdorff g 1 = e X 1, g 2 = e X 2 g 1 g 2 e X 3 Campbell-Haussdorff X 3 = X 1 + X [X 1, X 2 ] [X 1 X 2, [X 1, X 2 ]] + = 1 m=1 m {Z m(x 1, X 2 ) + ( 1) m Z m (X 2, X 1 )} Z m (X, Y ) = ( 1) n+1 [ad X] p 1 [ad Y ] q1 [ad X] p n 1 [ad Y ] q n 1 X n=1 n (p i,q i p ) 1!q 1! p n 1!q n 1! n 1 (p i + q i ) = m 1, p i + q i > 0 i=1 36

37 M. Reinsch: arxiv:mathphys/ (Homotopy ) 7.2 (Global structure of Lie group) (π 0 (G)) G G 2 g 1, g 2 G g(t) G, t = [0, 1] g(0) = g 1, g(1) = g 2 O(3) = {g GL(3, R) g g t = E} g t g = E det(g) 2 = 1 det(g) = ±1 det(g 1 ) = 1 ( g 1 = E) det(g 2 ) = 1 ( g 2 = E) O(3) det(g) +1 1 G 0 G G G 0 G g G g G 0 g 1 = G 0 G/G 0 G = O(3) G/G 0 = Z 2 π 0 (G) (homotopy group) (π 1 (G)) e e G 0 G e e g(t) G (t [0, 1]) g(0) = g(1) = e g 0 (t) = e (t [0, 1]) g 1 g 2 (t) g 12 (t, s), t, s [0, 1] g 12 (t, 0) = g 1 (t), g 12 (t, 1) = g 2 (t) g 1 g 2 e e g 1 g 2 g 1 g 2 (t) = { g2 (2t) t [0, 1] 2 g 1 (2t 1) t [ 1, 1] 2 37

38 g g 1 (t) = g(1 t) t [0, 1] e e G (π 1 (G 0 )) G π 1 (G 0 ) 1. U(1) = {a C a 2 = 1}: n g n (t) = e 2πint (t [0, 1]) g n g m g n+m π 1 (U(1)) = Z 2. SO(3): SU(2) SO(3) g SU(2) g SO(3)) : 3 g σ i g = σ j g ji j=1 σ i Pauli SU(2) ( ) e πit 0 g(t) =, t [0, 1] 0 e πit SU(2) E E SO(3) g(t) E E SU(2) 2 g g SU(2) SO(3) g g π 1 (SO(3)) = Z 2 SU(2) S 3 (3 ) SO(3) SU(2)/ {E, E} = RP 3 (3 ) 38

39 Lie Lie g Lie G Lie Lie (universal covering group) UG Lie Lie UG/D D G g UG, d D g 1 dg D Lie Schur D {λe} ( λ C) SU(n) Lie D g g = E λ 2 = 1, det(g) = 1 λ n = 1 D = { ω l E }, ω = e 2πi/n, l = 0, 1,, n 1 SU(n) SU(n)/Z n Lie n = 2 SU(2)/Z 2 = SO(3) Lie Lie Lie g ρ : g GL(n, C) X 1, X 2 g [ρ(x 1 ), ρ(x 2 )] = ρ([x 1, X 2 ]) ρ e X Lie Lie ρ(e X ) = e ρ(x) Lie Lie UG/D D d ρ(d) = E Lie Lie UG/D 39

40 SU(2) j (j = 0, 1, 1, 3, ) 2j SO(3) Lie j j ρ( E) = E SO(3) 7.3 su(2) su(3) su(2) Lie su(2) [J i, J j ] = i k ɛ ijk J k J ± = J 1 ± ij 2 [J 3, J ± ] = ±J ±, [J +, J ] = 2J 3 1. J 3 J 2 = i Ji 2 2. J + j, j = 0, J 3 j, j = j j, j 2 j, j J 2 j(j + 1) 3. j, j J J 3 J 3 (J ) p j, j = (j p)(j ) p j, j J 2 J J 2 (J ) p j, j j, j p 4. j, j j, m j, m j, m = 1 J j, m = N j,m j, m 1 N j,m 2 = j, m J + J j, m = j, m (J 2 + J J 3 ) j, m = (j m + 1)(j + m) 40

41 j + m > 0 N j,m = (j m + 1)(j + m) 5. j (J ) p j, j (p = 0, 1, 2, ) j +m = j +(j p) < 0 p > 2j J j, p j = 0 j 1/2 l/2 2j + 1 j, j, j, j 1,, j, j + 1, j, j su(3) Lie su(3) su(2) su(3) 3 3 X X = X, tr(x) = 0. 8 Pauli Gellmann i λ 1 = 1 0 0, λ 2 = i 0 0, λ 3 = 0 1 0, λ 4 = λ 7 = i 0 i , λ 5 =, λ 8 = i i 0 0, λ 6 = T a λ a , tr(t a T b ) = 1 2 δ ab 41

42 su(2) J 3 T 3 T 8 2 H 1 = T 3, H 2 = T 8 Cartan (Cartan subalgebra) su(2) J ± 1 2 (T 1 ± it 2 ) = E ± α1, 1 2 (T 4 ± it 5 ) = E ± α2, 1 2 (T 6 it 7 ) = E ± α3 α i R 2 (i = 1, 2, 3) ( ) ( ) ( 1 1/2 1/2 α 1 =, α 2 =, α 3 = 0 3/2 3/2 H i (i = 1, 2) [ ] Hi, E αj = ( αj ) i E α ) ( α j ) i α j i Cartan 2 (2 ) 2 su(3) 4 2 H 1,2 Cartan (root system) ± α i (root vector) SU(2) Cartan 1 (±1), (0) ( J ±, J 3 ) SU(3) 2 su(3) (Young [1]) e 1 = 0, e 2 = 1, e 3 = Cartan H 1,2 H 1 e 1 = 1 2 e 1, H 1 e 2 = 1 2 e 2, H 1 e 3 = 0, H 2 e 1 = e 1, H 2 e 2 = e 2, H 2 e 3 = 1 3 e 3 42

43 4: SU(3) H i e j = ( ω j ) i e j ( ω j ) i 2 ω j i 3 ω j (weight vector) H i (fundamental weight) SU(3) Young e j1 e jl Cartan H i H i e j1 e jl = ( ω j1 + + ω jl ) i e j1 e jl Young 2 [2] (6 ) [1, 1](3 ) [2] e i e i, (i = 1, 2, 3), 1 2 ( e i e j + e j e i ), (i < j) 6 [1, 1] 1 2 ( e i e j e j e i ), (i < j)

44 Η 2 Η 2 Η 2 2ω 2 2ω 1 ω 2 ω 1 ω 3 Η 1 Η 1 Η 1 2ω 3 5: SU(3) [ ] 3 [3], [2, 1], [1 3 ] [2, 1] ( 4) ω ω E α ω ω + α : H i E α ω = [H i, E α ] ω + E α H i ω = (α i + ω i ) ω SU(3) ω 1 ω 2 = α 1, ω 1 ω 3 = α 2, ω 3 ω 2 = α 3 α i SU(2) j, m j, m ± 1 J ± 7.4 SU(3) quark SU(3) quark quark 2 quark, lepton, Higgs (gauge ) ( U(1)) photon γ, (SU(2)) weak boson Z, W ±, (SU(3)) gluon 44

45 quark 6 u (up), d(down), s(strange), c(charm), b(bottom), t(top) quark SU(3) u, d, s 3 3 quark 2 SU(3) SU(3) 3 quark u i, d i, s i (i = 1, 2, 3) SU(3) u d s (flavor) SU(3) 2 SU(3) quark quark baryon meson color color color SU(3) singlet (confinement) (QCD) SU(3) singlet 3 q i quark ɛ ijk q i q j q k quark (baryon) [1, 1] (3 ) 3 ([1]) 3 ([1, 1]) = 8 ([2, 1]) 1 ([1, 1, 1]) quark ( q) [1, 1] q q (meson) quark H i (i = 1, 2) quark (Q), (B), strangeness (S), (Y), (T 3 ) Y = B + S, Q = T 3 + Y 2 u,d,s 45

46 Q B S Y T 3 u 2/3 1/3 0 1/3 1/2 d 1/3 1/3 0 1/3 1/2 s 1/3 1/3 1 2/3 0 T 3 Y Y d u s Τ3 6: quark T 3 H 1 3Y/2 H 2 (u ω 1, d ω 2, s ω 3 ) SU(3) quark quark fermion baryon flavor SU(3) flavor SU(3) Young = 10 ([3]) 2 8 ([2, 1]) 1 ([1, 1, 1]) SU(2) = 4 ([3]) 2 2 ([2, 1]) Young (Young Young 46

47 ) SU(2) SU(3) 1 SU(2) 4 3/2, 2 1/2 10 3/2 8 1/2 baryon - 0 Y + ++ N Y P Σ Ξ *- *- Σ Ω *0 - Ξ Σ *0 *+ T Σ - Ξ - 0 Σ Λ Ξ 0 Σ + T 7: Baryon [ ] 7.5 Lie Cartan Cartan H i (i = 1,, m) r m E α r m (rank) α m root [H i, H j ] = 0, [H i, E α ] = α i E α weight m ω H i ω = ω i ω H i E α ω = (α i + ω i )E α ω weight ω root E α ω ω + α 47

48 λ Tr (T a T b ) = λδ ab E α E β := λ 1 Tr ( E α E β ) = δ α, β, H i H j := λ 1 Tr (H i H j ) = δ i,j λ su(3) 1/2 m [E α, E α ] = α i H i, i=1 E α E α weight E α E α = mi=1 β i H i β i β i = H i E α E α = λ 1 Tr (H i [E α, E α ]) = λ 1 Tr (E α [H i, E α ]) = α i λ Tr (E αe α ) = α i root su(2) J + j, j = 0 j, j J j, j, j, j 1,, j, j E α root α root α α i i = 1 m α i α α E α su(2) E α ω = 0 ( α > 0) su(3) (1, 0), (1/2, ± 3/2) su(3) (1/2, ± 3/2) (1, 0) = (1/2, 3/2) + (1/2, 3/2) (1, 0) (Cartan ) weight 48

49 E α ω = N α, ω ω + α, N α, ω 1. N α, ω α 2 N α, ω 2 = α ω ( ) ( )[E α, E α ] = m i=1 α i H i ω [E α, E α ] ω = m i=1 α i ω H i ω = mi=1 α i ω i ω E α E α ω ω E α E α ω = N α, ω 2 N α, ω 2, N α, ω = ω α E α ω = ω α E α ω = (N α, ω α) 2. ω p, q E α ω+p α = E α ω q α = 0 α ω α = 1 (p q). ( ) 2 2 ( ) (*) ω N α, ω+(p 1) α 2 0 = α ( ω + p α) N α, ω+(p 2) α 2 N α, ω+(p 1) α 2 = α ( ω + (p 1) α) 0 N α, ω q α 2 = α ( ω q α). ( ) p(p + 1) (p + q + 1) α ω + α 2 q(q + 1) 2 2 { = (p + q + 1) α ω + 1 } 2 α 2 (p q) p + q root (**) ω root α β (m ) α β β = q p := m β α = q p := m α α β 2 α 2 β = mm 2 4 = cos 2 θ. 49

50 θ root mm 4 mm = 0 (θ = π/2), mm = 1 (θ = π/3, 2π/3), mm = 2 (θ = π/4, 3π/4), mm = 3 (θ = π/6, 5π/6) Dynkin α, β 6 E α E β = 0 q = q = 0 = p/2 0, α 2 p /2 0 α β π/2 θ < π θ = π 2, 2π 3, 3π 4, 5π 6. β α β 2 = θ = π (p, p ) cos θ = 1 2 pp, β 2 / α 2 = p/p θ = 2π/3 p = p = 1 θ = 3π/4, 5π/6 (p, p ) = (1, 2), (1, 3) 2 3 SU(3) (1/2, ± 3/2) 2π/3 Dynkin su(n + 1) ( A n so(2n + 1) (B n ), sp(n) (C n ), so(2n) (D n ), G 2, F 4, E 6, E 7, E 8 Dynkin A 3 = D 3, B 2 = C 2, D 2 = A 1 A 1 6 [E α, E β ] = 0 β α root β α = γ γ root β = α + γ β γ root γ root 50

51 θ=5π/6 θ=2π/3 θ=3π/4 θ=π/2 8: (Dynkin ) su(4) = so(6), so(5) = sp(2), so(4) = su(2) su(2) A D E simply laced Lie algebra A B C D G F E 9: Dynkin (fundamental weight) E α ω = 0 (**) 2 α i ω α 2 = q i 0, (i = 1, 2,, m) m q i ω m ω = q i ω i i=1 51

52 ω i (fundamental weight) 2 α i ω j α i 2 = δ ij q i su(2) j 52

53 II 8 wedge V n (R n ) wedge ( ) α 1 α p = 1 ( 1) σ α σ(1) α σ(p), p! σ S p α i V p p V p wedge p- dim( p V ) = ( n p ) = n(n 1) (n p + 1) p! wedge 1. α 1 α i α j α p = α 1 α j α i α p 2. α 1 (a 1 α i + a 2 α i) α p = a 1 α 1 α i α p + a 2 α 1 α i α p a 1, a 2 R, α i V V = R 3 v = 3 i=1 v i e i u = 3 i=1 u i e i v u = i<j(v i u j u i v j ) e i e j = w 1 e 2 e 3 + w 2 e 3 e 1 + w 3 e 1 e 2 w i v u v u i 53

54 p V 1. V e i (i = 1,, n) p V 1 i 1 < i 2 < < i p n e i1 e ip 2. p = n dim( n V ) = 1 e 1 e n v i = nj=1 R ij e j v 1 v n = det(r ij ) e 1 e n 3. p > n dim( p V ) = 0 n n p V p V λ, µ λ = v 1 v p, µ = u 1 u p V (λ, µ) = det i,j=1,,p ( v i, u j ) p V λ = λ i1 i p e i1 e ip, µ = µ i1 i p e i1 e ip i 1 < <i p i 1 < <i p V ( e i, e j ) = G ij (λ, µ) = i 1 < <i p G i1 j 1 G ip j 1 λ i1 i p µ j1 j p.. j 1 < <j p G i1 j p G ip j p e i V e i ( e i, e j ) = δ j i V p V (p = 0, 1,, n) 2 1. ψi : λ p V ψ i (λ) = e i λ p+1 V 54

55 2. ψ i : λ p V ψ i (λ) = i( e i )λ p 1 V i( v) λ = v 1 v p p i( u)λ = ( 1) j 1 ( u, v j ) v 1 v j 1 v j+1 v p j=1 { ψ i, ψ j } = δ i j, { ψ i, ψ j} = 0, { ψi, ψ j } = 0. V e i e 1 e p = ψ 1 ψ p (1) (1) p Hodge dim( p V ) = dim( n p V ) Hodge e i V e i = e i λ = λ i1 i p e i1 e ip = λ i1 i p ψi1 ψ ip (1) p V i 1 < <i p i 1 < <i p Hodge λ n p V λ = λ i1 i p ψ ip ψ i1 σ i 1 < <i p σ = e 1 e n Hodge λ p V λ = ( 1) p(n p) λ : V = R 3 v u = ( v u) 55

56 9 n M (R n ) (x 1,, x n ) p- (differential p-form) ω = 0 i 1 < <i p n ω i1 i p (x) dx i 1 dx i p p- Ω p (M) ω = p dx i x i n 7 ( e i ) wedge dx i dx j = dx j dx i ω i1 i p (x) x i ω i1 i r i s i p (x) = ω i1 i s i r i p (x) Ω p (M) ω i1 i ( ) p n p p q ω η = ω = η = 0 i 1 < <i p n 0 j 1 < <j q n 0 i 1 < <i p n 0 j 1 < <j q n ω i1 i p (x) dx i 1 dx ip η j1 j q (x) dx j 1 dx jq ω i1 i p (x)η j1 j q (x) dx i 1 dx ip dx j 1 dx jq ω η = ( 1) pq η ω 7 (cotangent bundle) (fiber) 56

57 M x i y i dx i dx i = x i = φ i (y) n j=1 φ i (y) dy j y j p ω(x) = ω i1 i p (x)dx i 1 dx i p i 1 < i p = ω i1 i p (φ(y)) φi1 i 1 < i p j 1,,j p dy j 1 (φ ω)(y) φip dy j p dyj 1 dy j p y φ:n M M ω N φ ω φ ω(y) M ω (pullback) (exterior derivative) d Ω p (M) Ω p+1 (M) ω = 0 i 1 < <i p n ω i1 i p (x) dx i 1 dx i p dω = 0 i 1 < <i p n j ω i1 i p x j dx j dx i 1 dx i p 1. d(ω + η) = dω + dη 2. d(ω η) = dω η + ( 1) ω ω η 3. d 2 = 0 4. d(φ ω) = φ (dω) 57

58 3 d 2 ω = j,k 2 ω i1 ip i 1 < <i p x j x k dxk dx j dx i 1 dx i p dx j dx k j k 2 ω i1 ip x j x k 4 1 ω = ω i (x)dx i φ ω = ω i (φ(y)) φi y j dyj d(φ ω) = ( y k = ω i (φ(y)) φi y j ) dy k dy j ( ωi x l φ l y k φ i y j + ω i(φ(y)) 2 φ i y k y j = ω i x l φ l y k φ i y j dyk dy j = φ (dω)(y) ) dy k dy j Hodge Hodge dx i x i M ds 2 = ij g ij (x)dx i dx j dx i n e a = Ei a dx i i=1 n ds 2 = e a e a a=1 e a (vierbein) dx i 58

59 Hodge e a ω ω = ω i1 i p (x) dx i 1 dx ip 0 i 1 < <i p n e a = n i=1 Ei a dx i dx i = n a=1 Eae i a ω = ω a1 a p e a 1 e a p a 1 < <a p ω Hodge ω = ω a1 a p ψ ap ψ a 1 σ a 1 < <a p ψ σ σ = e 1 e p Hodge Ω p (M) Ω n p (M) Ω p (M) ω ω = ( 1) p(n p) ω δ Hodge d Ω p (M) Ω p 1 (M) δ δ = ( 1) np n+1 d 8 d, φ,, δ δ d δ 2 = 0 f(x) Ω 0 (M) ( M ) δf(x) = 0 ( 1 ) Laplacian Laplacian Ω p (M) Ω p (M) 2 = (d + δ) 2 = dδ + δd f Ω 0 (M) f(x) = δdf(x) = d df(x) 8 n 1 ( 1) p 59

60 10 M = R 3 x 1, x 2, x 3 ds 2 = (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 dx i : (dx i, dx j ) = δ ij 0 : f(x) 1 : V 1 (x)dx 1 + V 2 (x)dx 2 + V 3 (x)dx 3 2 : A 1 (x)dx 2 dx 3 + A 2 (x)dx 3 dx 1 + A 3 (x)dx 1 dx 2 3 : g(x)dx 1 dx 2 dx 3 x i x i f f, A i A i φ(x), E i (x), B i (x) (1 ) (grad), (div), (rot), Laplacian 1. (gradient) (0 ) (1 ) df = f x 1 dx1 + f x 2 dx2 + f x 3 dx3 0 d 2. (divergence) (1 ) (0 ) 1 ω = i V i (x)dx i δ δω = d (V 1 dx 1 + V 2 dx 2 + V 3 dx 3 ) = d(v 1 dx 2 dx 3 + V 2 dx 3 dx 1 + V 3 dx 1 dx 2 ) = ( x V x V x V 3)dx 1 dx 2 dx 3 = div V 3 δ 60

61 3. (rotation) Hodge (rot) dω = d(v 1 dx 1 + V 2 dx 2 + V 3 dx 3 ) (( V3 = x V ) ) 2 dx 2 dx x 3 = ( ( V ) 1 dx 1 + ( V ) 2 dx 2 + ( V ) 3 dx 3) rot = d 4. Laplacian ( ) (d + δ) 2 ω = ± ( ωi1 i p (x) ) dx i 1 dx i p i 1 < <i p = ( ) 2 ( ) 2 ( ) 2 x + 1 x + 2 x 3 ( ) 1. :(r, θ, ϕ) dx 2 = (dr) 2 + r 2 ( (dθ) 2 + sin 2 θ(dϕ) 2) 2. : (r, θ, z) dx 2 = (dr) 2 + r 2 (dθ) 2 + dz 2 ds 2 = (h 1 (y)) 2 (dy 1 ) 2 + (h 2 (y)) 2 (dy 2 ) 2 + (h 3 (y)) 2 (dy 3 ) 2 (h i (x) > 0 ) e i = h i (y)dy i e i (y) 3 3 ω(y) = V i (y)dy i = (V i /h i )e i i=1 i=1 V i /h i i grad, div, rot 61

62 1. grad: df = i f y i dyi = i (grad f) i = 1 h i f y i 1 f h i y i ei 2. div: Hodge e i dx i δv = d (V 1 e 1 + V 2 e 2 + V 3 e 3 ) 3. rot: = d (V 1 e 2 e 3 + V 2 e 3 e 1 + V 3 e 1 e 2 ) = d ( V 1 h 2 h 3 dx 2 dx 3 + V 2 h 3 h 1 dx 3 dx 1 + V 3 h 1 h 2 dx 1 dx 2) ( = x (V 1h 1 2 h 3 ) + x (V 2h 2 3 h 1 ) + ) x (V 3h 3 1 h 2 ) dx 1 dx 2 dx 3 ( 1 = h 1 h 2 h 3 x (V 1h 1 2 h 3 ) + x (V 2h 2 3 h 1 ) + ) x (V 3h 3 1 h 2 ) divv dv = d (V 1 e 1 + V 2 e 2 + V 3 e 3 ) = d ( h 1 V 1 dx 1 + h 2 V 2 dx 2 + h 3 V 3 dx 3) (( (h3 V 3 ) = (h ) ) 2V 2 ) dy 2 dy 3 + cyclic perm. x 2 x 3 ( ( 1 (h3 V 3 ) = (h ) ) 2V 2 ) e 2 e 3 + cyclic perm. h 2 h 3 x 2 x 3 ( 1 (h3 V 3 ) = (h ) 2V 2 ) e 1 + cyclic perm. h 2 h 3 x 2 x 3 (rot V ) 1 = 1 h 2 h 3 ( (h3 V 3 ) (h ) 2V 2 ), x 2 x 3 4. Laplacian: (d + δ) 2 f = δdf = d df ( f = d y 1 dy1 + f y 2 dy2 + f ) y 3 dy3 ( 1 f = d h 1 y 1 e1 + 1 f h 2 y 2 e2 + 1 ) f h 3 y 3 e3 62

63 ( 1 f = d h 1 y 1 e2 e f h 2 y 2 e3 e f h 3 y 3 e1 e 2 ( h2 h 3 f = d = = ( y 1 1 h 1 h 2 h 3 h 1 ( h2 h 3 ( y 1 ) f y 2 dy3 dy 1 + h 1h 2 h 3 y 1 dy2 dy 3 + h 3h 1 h 2 ) ) f + dy 1 dy 2 dy 3 h 1 y 1 ( ) h2 h 3 f + ( h3 h 1 h 1 y 1 y 2 h 2 ) f y 3 dy1 y 2 ) f + ( h1 h 2 y 2 y 3 h 3 )) f y 3 11 Maxwell Maxwell ( c = 1 ) E = ρ, B = 0 B E t = J, E + B t = 0 (t = x 0 ) 4 4 Minkowski ds 2 = (dx 0 ) 2 + (dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 e 0 = idx 0, e j = dx j, (j = 1, 2, 3) Hodge 4 2 F = 3 F µν dx µ dx ν = E i (x)dx 0 dx i + 1 µ<ν i=1 2 3 i,j,k=1 ɛ ijk B i dx j dx k F µν 0 E 1 E 2 E 3 E 1 0 B 3 B 2 F = E 2 B 3 0 B 1 E 3 B 2 B

64 1 3 j = ρ(x)dx 0 + J k (x)dx k k=1 Maxwell δf = j, df = 0 df = 1 ( ɛ ijk ( 2 E) k + B ) k dx 0 dx i dx j +( x B)dx 1 dx 2 dx 3 0 i,j,k df = 0 Maxwell 2 4 Hodge F = i j B j dx 0 dx j + i ɛ jkl E j dx j dx l 2 j,k,l i E B Hodge δf = d F δf = j Maxwell 1 3 φ A B = A, E A = φ x 0 F = da 1 A 3 A = φ(x)dx 0 + A j (x)dx j j=1 p ω(x) dω(x) = 0 p 1 µ ω = dµ d 2 = 0 64

65 Maxwell df = 0 F = da 1 F = da A = A + dχ F ( da = da + d(dχ) = F ) χ M p M p 9 p f = f(p ) [P ] p = 1 2 R 2 1 x = x(t), y = y(t) 1 t P 0 P 1 t P 0 = (x(0), y(0)), P 1 = (x(1), y(1)) A = A x dx + A y dy P1 [ 1 A = dt A x (x(t)) dx P 0 0 dt + A y(x(t)) dy ] dt t 1 t t dt dx dt dx dx = d t = d t dt d t dt d t 9 M 65

66 1 A = df (F 0 ) P1 [ 1 df dx df = dt P 0 0 dx dt + df ] dy dy dt 1 df (x(t)) = dt = F (x(1)) F (x(0)) 0 dt = F (P 1 ) F (P 0 ) = F [P 1 ] [P 0 ] Green Green Green ( Vy dxdy D x V ) x = (V x dx + V y dy) y D D R 2 D D D ω = V x dx+v y dy dω = ω D D 12.2 R n+1 n + 1 P 0, P 1,, P n 3 n n n P = t i P i, (t i 0, t i = 1) i=0 i=0 n = 0 P 0, n = 1 2 P 0, P 1 n = 2 3 P 0, P 1, P 2 3 n + 1 P 0, P 1,, P n n (P 0, P 1,, P n ) 66

67 t 1,, t n 0 t i 1, 0 n i=1 t i 1 c = i a i i ( i a i ) (chain) n n 3 (P 1, P 2, P 3 ) 3 3 (P 0, P 1 ) + (P 1, P 2 ) + (P 2, P 0 ) R 2 4 P 1, P 2, P 3, P (2 ) (P 1, P 2, P 3 ) (P 2, P 3, P 4 ) (orientation) 1 (P 0, P 1 ) (P 1, P 0 ) (P 1, P 0 ) = (P 0, P 1 ) (P σ(1),, P σ(n) ) = ( 1) σ (P 1,, P n ) n n 1 n (P 0,, P n ) = ( 1) i (P 0,, P i 1, P i+1,, P n ) i=0 i c = l a l l ( l a l ) c = l a l l (P 0 P 1 ) = (P 1 ) (P 0 ) (P 0 P 1 P 2 ) = (P 1 P 2 ) (P 0 P 2 ) + (P 0 P 1 ) 4 (P 1, P 2, P 3 ) (P 2, P 3, P 4 ) 4 1 (P 1 P 2 ) + (P 2 P 4 ) + (P 4 P 3 ) + (P 3 P 1 ) 67

68 2 2 = 0 ( (P 0 P n )) ( n ) = ( 1) i (P 0 PX i P n ) i=0 n = ( 1) i i 1 ( 1) j (P 0 PX j PX i P n ) + i=0 = 0 j=0 n j=i+1 ( 1) j 1 (P 0 X P i X P j P n ) p ω(x) = 1 ω µ1 µ p! p (x)dx µ 1 dx µp µ 1 µ p p X X p X = X 1 X L X α (α = 1,, L) ϕ α (t) α x µ = ϕ µ α(t), x X α ω X X ω = L α=1 X α ω ω = 1 X α p! µ 1 µ p = (ϕ αω)(t) α t i 0, t i 1 d p t ω µ1 µ p (ϕ α (t)) ϕµ 1 α t 1 ϕµp α t p ϕ αω ω ϕ α α t i (i = 1,, p) 68

69 Stokes Green p p ω (p 1) dω (p 1) = p p ω (p 1) p = dt df(t) dt = f(1) f(0). p = 2 Green X d dω (p 1) = ω (p 1) X X ( )Stokes p M n (ω, µ) = ω µ M (ω, µ Ω p (M)) (ω, ν) = (ν, ω) M ( M = 0) Stokes ω Ω p 1 (M), ν Ω p (M) (dω, ν) = (ω, δν) (dω, ν) = dω ν = d(ω ν) + ( 1) p dω ν M M = ω ν + ( 1) np n+1 ω ( d ν) = ω δν = (ω, δν) M M M Maxwell df = 0, δf = j S[A] = 1 (da, da) (j, A) 2 69

70 F = da df = 0 A (A = A + ɛa 1 ) S[A ] S[A] = ɛ ((da, da 1 ) (j, A 1 )) = ɛ ((δda j, A 1 )) O(ɛ 2 ) A δda j = 0 F A Homology Cohomology M Homology Cohomology d d 2 = 0, 2 = 0 M X X = 0 X = Y Z p (M) = {c c p, c = 0} B p (M) = {c there exists b p 1, c = b} H p (M) = Z p (M)/B p (M) 10 H 0 (T 2 ) = Z, H 1 (T 2 ) = Z Z, H 2 (T 2 ) = Z, H 0 (S 2 ) = Z, H 1 (S 2 ) = 0, H 2 (S 2 ) = Z. p- dω = 0 ω = dµ Z p (M) = {ω Ω p (M) dω = 0}, B p = {ω Ω p (M) there exists µ Ω p 1 (M), ω = dµ} H p (M) = Z p (M)/B p (M) c H p (M), ω H p (M) ω 10 T 2 2 S 2 2 c 70

71 ω = ω + dω = ω, (ω + dµ) = ω + µ = ω c+ b c b c c Stokes dω = 0, c = 0 c c c 71

『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

120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2

120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2 9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0

More information

I-2 (100 ) (1) y(x) y dy dx y d2 y dx 2 (a) y + 2y 3y = 9e 2x (b) x 2 y 6y = 5x 4 (2) Bernoulli B n (n = 0, 1, 2,...) x e x 1 = n=0 B 0 B 1 B 2 (3) co

I-2 (100 ) (1) y(x) y dy dx y d2 y dx 2 (a) y + 2y 3y = 9e 2x (b) x 2 y 6y = 5x 4 (2) Bernoulli B n (n = 0, 1, 2,...) x e x 1 = n=0 B 0 B 1 B 2 (3) co 16 I ( ) (1) I-1 I-2 I-3 (2) I-1 ( ) (100 ) 2l x x = 0 y t y(x, t) y(±l, t) = 0 m T g y(x, t) l y(x, t) c = 2 y(x, t) c 2 2 y(x, t) = g (A) t 2 x 2 T/m (1) y 0 (x) y 0 (x) = g c 2 (l2 x 2 ) (B) (2) (1)

More information

* 1 2014 7 8 *1 iii 1. Newton 1 1.1 Newton........................... 1 1.2............................. 4 1.3................................. 5 2. 9 2.1......................... 9 2.2........................

More information

2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a

More information

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e

Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e 7 -a 7 -a February 4, 2007 1. 2. 3. 4. 1. 2. 3. 1 Gauss Gauss ɛ 0 E ds = Q (1) xy σ (x, y, z) (2) a ρ(x, y, z) = x 2 + y 2 (r, θ, φ) (1) xy A Gauss ɛ 0 E ds = ɛ 0 EA Q = ρa ɛ 0 EA = ρea E = (ρ/ɛ 0 )e z

More information

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,,

0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ). f ( ). x i : M R.,, 2012 10 13 1,,,.,,.,.,,. 2?.,,. 1,, 1. (θ, φ), θ, φ (0, π),, (0, 2π). 1 0.,,., m Euclid m m. 2.., M., M R 2 ψ. ψ,, R 2 M.,, (x 1 (),, x m ()) R m. 2 M, R f. M (x 1,, x m ), f (x 1,, x m ) f(x 1,, x m ).

More information

Z: Q: R: C: 3. Green Cauchy

Z: Q: R: C: 3. Green Cauchy 7 Z: Q: R: C: 3. Green.............................. 3.............................. 5.3................................. 6.4 Cauchy..................... 6.5 Taylor..........................6...............................

More information

( ) 2002 1 1 1 1.1....................................... 1 1.1.1................................. 1 1.1.2................................. 1 1.1.3................... 3 1.1.4......................................

More information

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n

http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 2 N(ε 1 ) N(ε 2 ) ε 1 ε 2 α ε ε 2 1 n N(ɛ) N ɛ ɛ- (1.1.3) n > N(ɛ) a n α < ɛ n N(ɛ) a n http://www2.math.kyushu-u.ac.jp/~hara/lectures/lectures-j.html 1 1 1.1 ɛ-n 1 ɛ-n lim n a n = α n a n α 2 lim a n = 1 n a k n n k=1 1.1.7 ɛ-n 1.1.1 a n α a n n α lim n a n = α ɛ N(ɛ) n > N(ɛ) a n α < ɛ

More information

量子力学A

量子力学A c 1 1 1.1....................................... 1 1............................................ 4 1.3.............................. 6 10.1.................................. 10......................................

More information

p.2/76

p.2/76 kino@info.kanagawa-u.ac.jp p.1/76 p.2/76 ( ) (2001). (2006). (2002). p.3/76 N n, n {1, 2,...N} 0 K k, k {1, 2,...,K} M M, m {1, 2,...,M} p.4/76 R =(r ij ), r ij = i j ( ): k s r(k, s) r(k, 1),r(k, 2),...,r(k,

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

= hυ = h c λ υ λ (ev) = 1240 λ W=NE = Nhc λ W= N 2 10-16 λ / / Φe = dqe dt J/s Φ = km Φe(λ)v(λ)dλ THBV3_0101JA Qe = Φedt (W s) Q = Φdt lm s Ee = dφe ds E = dφ ds Φ Φ THBV3_0102JA Me = dφe ds M = dφ ds

More information

q π =0 Ez,t =ε σ {e ikz ωt e ikz ωt } i/ = ε σ sinkz ωt 5.6 x σ σ *105 q π =1 Ez,t = 1 ε σ + ε π {e ikz ωt e ikz ωt } i/ = 1 ε σ + ε π sinkz ωt 5.7 σ

q π =0 Ez,t =ε σ {e ikz ωt e ikz ωt } i/ = ε σ sinkz ωt 5.6 x σ σ *105 q π =1 Ez,t = 1 ε σ + ε π {e ikz ωt e ikz ωt } i/ = 1 ε σ + ε π sinkz ωt 5.7 σ H k r,t= η 5 Stokes X k, k, ε, ε σ π X Stokes 5.1 5.1.1 Maxwell H = A A *10 A = 1 c A t 5.1 A kη r,t=ε η e ik r ωt 5. k ω ε η k η = σ, π ε σ, ε π σ π A k r,t= q η A kη r,t+qηa kηr,t 5.3 η q η E = 1 c A

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

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t 6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]

More information

Γ Ec Γ V BIAS THBV3_0401JA THBV3_0402JAa THBV3_0402JAb 1000 800 600 400 50 % 25 % 200 100 80 60 40 20 10 8 6 4 10 % 2.5 % 0.5 % 0.25 % 2 1.0 0.8 0.6 0.4 0.2 0.1 200 300 400 500 600 700 800 1000 1200 14001600

More information

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1.

1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N m 1. 1.1 1. 1.3.1..3.4 3.1 3. 3.3 4.1 4. 4.3 5.1 5. 5.3 6.1 6. 6.3 7.1 7. 7.3 1 1 variation 1.1 imension unit L m M kg T s Q C QT 1 A = C s 1 MKSA F = ma N N = kg m s 1.1 J E = 1 mv W = F x J = kg m s 1 = N

More information

kawa (Spin-Orbit Tomography: Kawahara and Fujii 21,Kawahara and Fujii 211,Fujii & Kawahara submitted) 2 van Cittert-Zernike Appendix A V 2

kawa (Spin-Orbit Tomography: Kawahara and Fujii 21,Kawahara and Fujii 211,Fujii & Kawahara submitted) 2 van Cittert-Zernike Appendix A V 2 Hanbury-Brown Twiss (ver. 1.) 24 2 1 1 1 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 3 3 Hanbury-Brown Twiss ( ) 4 3.1............................................

More information

(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law

(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law I (Radom Walks ad Percolatios) 3 4 7 ( -2 ) (Preface),.,,,...,,.,,,,.,.,,.,,. (,.) (Basic of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios, Meas).............................

More information

1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1

1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1 1 1.1......... 1............. 1.3... 1.4......... 1.5.............. 1.6................ Bownian Motion.1.......... Einstein.............. 3.3 Einstein........ 3.4..... 3.5 Langevin Eq.... 3.6................

More information

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si

I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin. sin. sin + π si I 8 No. : No. : No. : No.4 : No.5 : No.6 : No.7 : No.8 : No.9 : No. : I No. sin cos sine, cosine : trigonometric function π : π =.4 : n =, ±, ±, sin + nπ = sin cos + nπ = cos sin = sin : cos = cos :. sin.

More information

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1)

n Y 1 (x),..., Y n (x) 1 W (Y 1 (x),..., Y n (x)) 0 W (Y 1 (x),..., Y n (x)) = Y 1 (x)... Y n (x) Y 1(x)... Y n(x) (x)... Y n (n 1) (x) Y (n 1) D d dx 1 1.1 n d n y a 0 dx n + a d n 1 y 1 dx n 1 +... + a dy n 1 dx + a ny = f(x)...(1) dk y dx k = y (k) a 0 y (n) + a 1 y (n 1) +... + a n 1 y + a n y = f(x)...(2) (2) (2) f(x) 0 a 0 y (n) + a 1 y

More information

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B 9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A

More information

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C

1990 IMO 1990/1/15 1:00-4:00 1 N N N 1, N 1 N 2, N 2 N 3 N 3 2 x x + 52 = 3 x x , A, B, C 3,, A B, C 2,,,, 7, A, B, C 0 9 (1990 1999 ) 10 (2000 ) 1900 1994 1995 1999 2 SAT ACT 1 1990 IMO 1990/1/15 1:00-4:00 1 N 1990 9 N N 1, N 1 N 2, N 2 N 3 N 3 2 x 2 + 25x + 52 = 3 x 2 + 25x + 80 3 2, 3 0 4 A, B, C 3,, A B, C 2,,,, 7,

More information

Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib

Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib ( ) Donaldson Seiberg-Witten Witten Göttsche [GNY] L. Göttsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki s formula and instanton counting, Publ. of RIMS, to appear Donaldson

More information

10:30 12:00 P.G. vs vs vs 2

10:30 12:00 P.G. vs vs vs 2 1 10:30 12:00 P.G. vs vs vs 2 LOGIT PROBIT TOBIT mean median mode CV 3 4 5 0.5 1000 6 45 7 P(A B) = P(A) + P(B) - P(A B) P(B A)=P(A B)/P(A) P(A B)=P(B A) P(A) P(A B) P(A) P(B A) P(B) P(A B) P(A) P(B) P(B

More information

13 0 1 1 4 11 4 12 5 13 6 2 10 21 10 22 14 3 20 31 20 32 25 33 28 4 31 41 32 42 34 43 38 5 41 51 41 52 43 53 54 6 57 61 57 62 60 70 0 Gauss a, b, c x, y f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 b/2 c x

More information

CKY CKY CKY 4 Kerr CKY

CKY CKY CKY 4 Kerr CKY ( ) 1. (I) Hidden Symmetry and Exact Solutions in Einstein Gravity Houri-Y.Y: Progress Supplement (2011) (II) Generalized Hidden Symmetries and Kerr-Sen Black Hole Houri-Kubiznak-Warnick-Y.Y: JHEP (2010)

More information

R R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q Riemann 15

R R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q Riemann 15 (Gen KUROKI) 1 1 : Riemann Spec Z 2? 3 : 4 2 Riemann Riemann Riemann 1 C 5 Riemann Riemann R compact R K C ( C(x) ) K C(R) Riemann R 6 (E-mail address: kuroki@math.tohoku.ac.jp) 1 1 ( 5 ) 2 ( Q ) Spec

More information

1

1 016 017 6 16 1 1 5 1.1............................................... 5 1................................................... 5 1.3................................................ 5 1.4...............................................

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

4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t

4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t 1 1.1 sin 2π [rad] 3 ft 3 sin 2t π 4 3.1 2 1.1: sin θ 2.2 sin θ ft t t [sec] t sin 2t π 4 [rad] sin 3.1 3 sin θ θ t θ 2t π 4 3.2 3.1 3.4 3.4: 2.2: sin θ θ θ [rad] 2.3 0 [rad] 4 sin θ sin 2t π 4 sin 1 1

More information

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi I (Basics of Probability Theory ad Radom Walks) 25 4 5 ( 4 ) (Preface),.,,,.,,,...,,.,.,,.,,. (,.) (Basics of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios,

More information

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi

(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi II (Basics of Probability Theory ad Radom Walks) (Preface),.,,,.,,,...,,.,.,,.,,. (Basics of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios, Meas).............................

More information

PDF

PDF 1 1 1 1-1 1 1-9 1-3 1-1 13-17 -3 6-4 6 3 3-1 35 3-37 3-3 38 4 4-1 39 4- Fe C TEM 41 4-3 C TEM 44 4-4 Fe TEM 46 4-5 5 4-6 5 5 51 6 5 1 1-1 1991 1,1 multiwall nanotube 1993 singlewall nanotube ( 1,) sp 7.4eV

More information

Big Bang Planck Big Bang 1 43 Planck Planck quantum gravity Planck Grand Unified Theories: GUTs X X W X 1 15 ev 197 Glashow Georgi 1 14 GeV 1 2

Big Bang Planck Big Bang 1 43 Planck Planck quantum gravity Planck Grand Unified Theories: GUTs X X W X 1 15 ev 197 Glashow Georgi 1 14 GeV 1 2 12 Big Bang 12.1 Big Bang Big Bang 12.1 1-5 1 32 K 1 19 GeV 1-4 time after the Big Bang [ s ] 1-3 1-2 1-1 1 1 1 1 2 inflationary epoch gravity strong electromagnetic weak 1 27 K 1 14 GeV 1 15 K 1 2 GeV

More information

統計学のポイント整理

統計学のポイント整理 .. September 17, 2012 1 / 55 n! = n (n 1) (n 2) 1 0! = 1 10! = 10 9 8 1 = 3628800 n k np k np k = n! (n k)! (1) 5 3 5 P 3 = 5! = 5 4 3 = 60 (5 3)! n k n C k nc k = npk k! = n! k!(n k)! (2) 5 3 5C 3 = 5!

More information

MUFFIN3

MUFFIN3 MUFFIN - MUltiFarious FIeld simulator for Non-equilibrium system - ( ) MUFFIN WG3 - - JCII, - ( ) - ( ) - ( ) - (JSR) - - MUFFIN sec -3 msec -6 sec GOURMET SUSHI MUFFIN -9 nsec PASTA -1 psec -15 fsec COGNAC

More information

< qq > (Quark Gluon Plasma,QGP) QGP (< qq >= ) < qq > π - π K + Nambu-Goldstone K + S = + S = K K + K + - K + t free ρ K + N K + N next-to-leading ord

< qq > (Quark Gluon Plasma,QGP) QGP (< qq >= ) < qq > π - π K + Nambu-Goldstone K + S = + S = K K + K + - K + t free ρ K + N K + N next-to-leading ord -K + < qq > (Quark Gluon Plasma,QGP) QGP (< qq >= ) < qq > π - π K + Nambu-Goldstone K + S = + S = K K + K + - K + t free ρ K + N K + N next-to-leading order (NLO) NLO (low energy constant,lec) χ I = I

More information

1. 2. C2

1. 2. C2 2000 7 6 (I) (II) ( 47, 1999) C1 1. 2. C2 1 ˆk AIC T C3 1.1 ( : 3 ) Y N ( µ(x a,x b,x c ),σ 2) µ(x a,x b,x c )=β 0 + β a x a + β b x b + β c x c x a,x b,x c α α {a, b, c} Θ α = {(σ, β) σ >0,β i =0,i α

More information

2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := 0

2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := 0 1473 : de nouvelles perspectives 2006 2 pp 102 119 VI q 1 Tetsuya Kikuchi Sabro Kakei Drinfel d-sokolov Painlevé [KK1] [KK2] [KK3] [KIK] [ ] [ ] [KK3] three-wave equation Painlevé VI q q Drinfel d-sokolov

More information

untitled

untitled V. 8 9 9 8.. SI 5 6 7 8 9. - - SI 6 6 6 6 6 6 6 SI -- l -- 6 -- -- 6 6 u 6cod5 6 h5 -oo ch 79 79 85 875 99 79 58 886 9 89 9 959 966 - - NM /6 Nucl Ml SI NM/6/685 85co /./ /h / /6/.6 / /.6 /h o NM o.85

More information

C p (.2 C p [[T ]] Bernoull B n,χ C p p q p 2 q = p p = 2 q = 4 ω Techmüller a Z p ω(a a ( mod q φ(q ω(a Z p a pz p ω(a = 0 Z p φ Euler Techmüller ω Q

C p (.2 C p [[T ]] Bernoull B n,χ C p p q p 2 q = p p = 2 q = 4 ω Techmüller a Z p ω(a a ( mod q φ(q ω(a Z p a pz p ω(a = 0 Z p φ Euler Techmüller ω Q p- L- [Iwa] [Iwa2] -Leopoldt [KL] p- L-. Kummer Remann ζ(s Bernoull B n (. ζ( n = B n n, ( n Z p a = Kummer [Kum] ( Kummer p m n 0 ( mod p m n a m n ( mod (p p a ( p m B m m ( pn B n n ( mod pa Z p Kummer

More information

untitled

untitled . 96. 99. ( 000 SIC SIC N88 SIC for Windows95 6 6 3 0 . amano No.008 6. 6.. z σ v σ v γ z (6. σ 0 (a (b 6. (b 0 0 0 6. σ σ v σ σ 0 / v σ v γ z σ σ 0 σ v 0γ z σ / σ ν /( ν, ν ( 0 0.5 0.0 0 v sinφ, φ 0 (6.

More information

untitled

untitled (a) (b) (c) (d) Wunderlich 2.5.1 = = =90 2 1 (hkl) {hkl} [hkl] L tan 2θ = r L nλ = 2dsinθ dhkl ( ) = 1 2 2 2 h k l + + a b c c l=2 l=1 l=0 Polanyi nλ = I sinφ I: B A a 110 B c 110 b b 110 µ a 110

More information

86 6 r (6) y y d y = y 3 (64) y r y r y r ϕ(x, y, y,, y r ) n dy = f(x, y) (6) 6 Lipschitz 6 dy = y x c R y(x) y(x) = c exp(x) x x = x y(x ) = y (init

86 6 r (6) y y d y = y 3 (64) y r y r y r ϕ(x, y, y,, y r ) n dy = f(x, y) (6) 6 Lipschitz 6 dy = y x c R y(x) y(x) = c exp(x) x x = x y(x ) = y (init 8 6 ( ) ( ) 6 ( ϕ x, y, dy ), d y,, dr y r = (x R, y R n ) (6) n r y(x) (explicit) d r ( y r = ϕ x, y, dy ), d y,, dr y r y y y r (6) dy = f (x, y) (63) = y dy/ d r y/ r 86 6 r (6) y y d y = y 3 (64) y

More information

1. Introduction Palatini formalism vierbein e a µ spin connection ω ab µ Lgrav = e (R + Λ). 16πG R µνab µ ω νab ν ω µab ω µac ω νcb + ω νac ω µcb, e =

1. Introduction Palatini formalism vierbein e a µ spin connection ω ab µ Lgrav = e (R + Λ). 16πG R µνab µ ω νab ν ω µab ω µac ω νcb + ω νac ω µcb, e = Chiral Fermion in AdS(dS) Gravity Fermions in (Anti) de Sitter Gravity in Four Dimensions, N.I, Takeshi Fukuyama, arxiv:0904.1936. Prog. Theor. Phys. 122 (2009) 339-353. 1. Introduction Palatini formalism

More information

LHC ALICE (QGP) QGP QGP QGP QGP ω ϕ J/ψ ALICE s = ev + J/ψ

LHC ALICE (QGP) QGP QGP QGP QGP ω ϕ J/ψ ALICE s = ev + J/ψ 8 + J/ψ ALICE B597 : : : 9 LHC ALICE (QGP) QGP QGP QGP QGP ω ϕ J/ψ ALICE s = ev + J/ψ 6..................................... 6. (QGP)..................... 6.................................... 6.4..............................

More information

example2_time.eps

example2_time.eps Google (20/08/2 ) ( ) Random Walk & Google Page Rank Agora on Aug. 20 / 67 Introduction ( ) Random Walk & Google Page Rank Agora on Aug. 20 2 / 67 Introduction Google ( ) Random Walk & Google Page Rank

More information

( 9 1 ) 1 2 1.1................................... 2 1.2................................................. 3 1.3............................................... 4 1.4...........................................

More information

body.dvi

body.dvi ..1 f(x) n = 1 b n = 1 f f(x) cos nx dx, n =, 1,,... f(x) sin nx dx, n =1,, 3,... f(x) = + ( n cos nx + b n sin nx) n=1 1 1 5 1.1........................... 5 1.......................... 14 1.3...........................

More information

[Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2

[Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2 On the action of the Weil group on the l-adic cohomology of rigid spaces over local fields (Yoichi Mieda) Graduate School of Mathematical Sciences, The University of Tokyo 0 l Galois K F F q l q K, F K,

More information

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................

5.. z = f(x, y) y y = b f x x g(x) f(x, b) g x ( ) A = lim h g(a + h) g(a) h g(x) a A = g (a) = f x (a, b)............................................ 5 partial differentiation (total) differentiation 5. z = f(x, y) (a, b) A = lim h f(a + h, b) f(a, b) h........................................................... ( ) f(x, y) (a, b) x A (a, b) x (a, b)

More information

Confinement dual Meissener effect dual Meissener effect

Confinement dual Meissener effect dual Meissener effect BASED ON WORK WITH KENICHI KONISHI (UNIV. OF PISA) [0909.3781 TO APPEAR IN NPB] Confinement dual Meissener effect dual Meissener effect 1) Perturbed SU(N) Seiberg WiRen theory : 2) SU(N) with Flavors at

More information

2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP

2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP 1. 1 213 1 6 1 3 1: ( ) 2: 3: SF 1 2 3 1: 3 2 A m 2. 2 P M A 2 F = mmg AP AP 2 AP (G > : ) AP/ AP A P P j M j F = n j=1 mm j G AP j AP j 2 AP j 3 P ψ(p) j ψ(p j ) j (P j j ) A F = n j=1 mgψ(p j ) j AP

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

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63> 通信方式第 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/072662 このサンプルページの内容は, 第 2 版発行当時のものです. i 2 2 2 2012 5 ii,.,,,,,,.,.,,,,,.,,.,,..,,,,.,,.,.,,.,,.. 1990 5 iii 1 1

More information

CVMに基づくNi-Al合金の

CVMに基づくNi-Al合金の CV N-A (-' by T.Koyama ennard-jones fcc α, β, γ, δ β α γ δ = or α, β. γ, δ α β γ ( αβγ w = = k k k ( αβγ w = ( αβγ ( αβγ w = w = ( αβγ w = ( αβγ w = ( αβγ w = ( αβγ w = ( αβγ w = ( βγδ w = = k k k ( αγδ

More information

縺 縺8 縺, [ 縺 チ : () () () 4 チ93799; () "64": ィャ 9997ィ

縺 縺8 縺, [ 縺 チ : () () () 4 チ93799; () 64: ィャ 9997ィ 34978 998 3. 73 68, 86 タ7 9 9989769 438 縺48 縺 378364 タ 縺473 399-4 8 637744739 683 6744939 3.9. 378,.. 68 ィ 349 889 3349947 89893 683447 4 334999897447 (9489) 67449, 6377447 683, 74984 7849799 34789 83747

More information

42 3 u = (37) MeV/c 2 (3.4) [1] u amu m p m n [1] m H [2] m p = (4) MeV/c 2 = (13) u m n = (4) MeV/c 2 =

42 3 u = (37) MeV/c 2 (3.4) [1] u amu m p m n [1] m H [2] m p = (4) MeV/c 2 = (13) u m n = (4) MeV/c 2 = 3 3.1 3.1.1 kg m s J = kg m 2 s 2 MeV MeV [1] 1MeV=1 6 ev = 1.62 176 462 (63) 1 13 J (3.1) [1] 1MeV/c 2 =1.782 661 731 (7) 1 3 kg (3.2) c =1 MeV (atomic mass unit) 12 C u = 1 12 M(12 C) (3.3) 41 42 3 u

More information

学習内容と日常生活との関連性の研究-第2部-第6章

学習内容と日常生活との関連性の研究-第2部-第6章 378 379 10% 10%10% 10% 100% 380 381 2000 BSE CJD 5700 18 1996 2001 100 CJD 1 310-7 10-12 10-6 CJD 100 1 10 100 100 1 1 100 1 10-6 1 1 10-6 382 2002 14 5 1014 10 10.4 1014 100 110-6 1 383 384 385 2002 4

More information

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y

No2 4 y =sinx (5) y = p sin(2x +3) (6) y = 1 tan(3x 2) (7) y =cos 2 (4x +5) (8) y = cos x 1+sinx 5 (1) y =sinx cos x 6 f(x) = sin(sin x) f 0 (π) (2) y No1 1 (1) 2 f(x) =1+x + x 2 + + x n, g(x) = 1 (n +1)xn + nx n+1 (1 x) 2 x 6= 1 f 0 (x) =g(x) y = f(x)g(x) y 0 = f 0 (x)g(x)+f(x)g 0 (x) 3 (1) y = x2 x +1 x (2) y = 1 g(x) y0 = g0 (x) {g(x)} 2 (2) y = µ

More information

5 36 5................................................... 36 5................................................... 36 5.3..............................

5 36 5................................................... 36 5................................................... 36 5.3.............................. 9 8 3............................................. 3.......................................... 4.3............................................ 4 5 3 6 3..................................................

More information

1 1 1 1 1 1 2 f z 2 C 1, C 2 f 2 C 1, C 2 f(c 2 ) C 2 f(c 1 ) z C 1 f f(z) xy uv ( u v ) = ( a b c d ) ( x y ) + ( p q ) (p + b, q + d) 1 (p + a, q + c) 1 (p, q) 1 1 (b, d) (a, c) 2 3 2 3 a = d, c = b

More information

2 σ γ l σ ο 4..5 cos 5 D c D u U b { } l + b σ l r l + r { r m+ m } b + l + + l l + 4..0 D b0 + r l r m + m + r 4..7 4..0 998 ble4.. ble4.. 8 0Z Fig.4.. 0Z 0Z Fig.4.. ble4.. 00Z 4 00 0Z Fig.4.. MO S 999

More information

213 2 katurada AT meiji.ac.jp http://nalab.mind.meiji.ac.jp/~mk/pde/ 213 9, 216 11 3 6.1....................................... 6.2............................. 8.3................................... 9.4.....................................

More information

[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)

[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization) . D............................................... : E = κ ............................................ 3.................................................

More information

untitled

untitled 0 ( L CONTENTS 0 . sin(-x-sinx, (-x(x, sin(90-xx,(90-xsinx sin(80-xsinx,(80-x-x ( sin{90-(ωφ}(ωφ. :n :m.0 m.0 n tn. 0 n.0 tn ω m :n.0n tn n.0 tn.0 m c ω sinω c ω c tnω ecω sin ω ω sin c ω c ω tn c tn ω

More information

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63> 単純適応制御 SAC 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/091961 このサンプルページの内容は, 初版 1 刷発行当時のものです. 1 2 3 4 5 9 10 12 14 15 A B F 6 8 11 13 E 7 C D URL http://www.morikita.co.jp/support

More information

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0

1.1 ft t 2 ft = t 2 ft+ t = t+ t 2 1.1 d t 2 t + t 2 t 2 = lim t 0 t = lim t 0 = lim t 0 t 2 + 2t t + t 2 t 2 t + t 2 t 2t t + t 2 t 2t + t = lim t 0 A c 2008 by Kuniaki Nakamitsu 1 1.1 t 2 sin t, cos t t ft t t vt t xt t + t xt + t xt + t xt t vt = xt + t xt t t t vt xt + t xt vt = lim t 0 t lim t 0 t 0 vt = dxt ft dft dft ft + t ft = lim t 0 t 1.1

More information

17 3 31 1 1 3 2 5 3 9 4 10 5 15 6 21 7 29 8 31 9 35 10 38 11 41 12 43 13 46 14 48 2 15 Radon CT 49 16 50 17 53 A 55 1 (oscillation phenomena) e iθ = cos θ + i sin θ, cos θ = eiθ + e iθ 2, sin θ = eiθ e

More information

B

B B YES NO 5 7 6 1 4 3 2 BB BB BB AA AA BB 510J B B A 510J B A A A A A A 510J B A 510J B A A A A A 510J M = σ Z Z = M σ AAA π T T = a ZP ZP = a AAA π B M + M 2 +T 2 M T Me = = 1 + 1 + 2 2 M σ Te = M 2 +T

More information

The painter of the Lascaux Cave (B.C.15,000) knew the geometry of apparent contours. http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 2 / 5

The painter of the Lascaux Cave (B.C.15,000) knew the geometry of apparent contours. http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html 25 ( ) 2 / 5 1 / 52 25 http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html The painter of the Lascaux Cave (B.C.15,000) knew the geometry of apparent contours. http://www.math.sci.hokudai.ac.jp/ ohmoto/class.html

More information

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9

0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c), (6) ( b) c = (b c), (7) (b + c) = b + c, (8) ( + b)c = c + bc (9 1-1. 1, 2, 3, 4, 5, 6, 7,, 100,, 1000, n, m m m n n 0 n, m m n 1-2. 0 m n m n 0 2 = 1.41421356 π = 3.141516 1-3. 1 0 1-4. 1-5. (1) + b = b +, (2) b = b, (3) + 0 =, (4) 1 =, (5) ( + b) + c = + (b + c),

More information

第86回日本感染症学会総会学術集会後抄録(I)

第86回日本感染症学会総会学術集会後抄録(I) κ κ κ κ κ κ μ μ β β β γ α α β β γ α β α α α γ α β β γ μ β β μ μ α ββ β β β β β β β β β β β β β β β β β β γ β μ μ μ μμ μ μ μ μ β β μ μ μ μ μ μ μ μ μ μ μ μ μ μ β

More information

離散研究会2013

離散研究会2013 2013 9 27-30, S. Carlip, Challenges for Emergent Gravity, arxiv:1207.2504 [gr-qc] Wheeler-DeWitt N= M abc,p abc a, b, c =1, 2,...,N M abc = M bca = M bac O(N) N.Sasakura, Quantum canonical tensor model

More information

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3

> > <., vs. > x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D > 0 x (2) D = 0 x (3 13 2 13.0 2 ( ) ( ) 2 13.1 ( ) ax 2 + bx + c > 0 ( a, b, c ) ( ) 275 > > 2 2 13.3 x 2 x y = ax 2 + bx + c y = 0 2 ax 2 + bx + c = 0 y = 0 x ( x ) y = ax 2 + bx + c D = b 2 4ac (1) D >

More information

H = H 1 (Jac(R); Z) Sp 1 H (Jac(R); Z) = Λ Z H, H (Jac(R); Z) = Λ Z H = Λ Z H Poincaré duality canonical ( ) canonical symplectic form foliation (2) F

H = H 1 (Jac(R); Z) Sp 1 H (Jac(R); Z) = Λ Z H, H (Jac(R); Z) = Λ Z H = Λ Z H Poincaré duality canonical ( ) canonical symplectic form foliation (2) F 6 11 5 1 Sp-modules symplectic Sp-module low dimensional., 0 1, 2, 3, 4 (n),. foliation n. Sp-modules, intersection form H = H 1 (Σ; Z), µ : H H Z H rank 2g free module Q C Q foliation R Q, R H Q = H Q,

More information

On the Limited Sample Effect of the Optimum Classifier by Bayesian Approach he Case of Independent Sample Size for Each Class Xuexian HA, etsushi WAKA

On the Limited Sample Effect of the Optimum Classifier by Bayesian Approach he Case of Independent Sample Size for Each Class Xuexian HA, etsushi WAKA Journal Article / 学術雑誌論文 ベイズアプローチによる最適識別系の有限 標本効果に関する考察 : 学習標本の大きさ がクラス間で異なる場合 (< 論文小特集 > パ ターン認識のための学習 : 基礎と応用 On the limited sample effect of bayesian approach : the case of each class 韓, 雪仙 ; 若林, 哲史

More information

,, Mellor 1973),, Mellor and Yamada 1974) Mellor 1973), Mellor and Yamada 1974) 4 2 3, 2 4,

,, Mellor 1973),, Mellor and Yamada 1974) Mellor 1973), Mellor and Yamada 1974) 4 2 3, 2 4, Mellor and Yamada1974) The Turbulence Closure Model of Mellor and Yamada 1974) Kitamori Taichi 2004/01/30 ,, Mellor 1973),, Mellor and Yamada 1974) Mellor 1973), 4 1 4 Mellor and Yamada 1974) 4 2 3, 2

More information

( ) 24 1 ( 26 8 19 ) i 0.1 1 (2012 05 30 ) 1 (), 2 () 1,,, III, C III, C, 1, 2,,, ( III, C ),, 1,,, http://ryuiki.agbi.tsukuba.ac.jp/lec/12-physics/ E104),,,,,, 75 3,,,, 0.2, 1,,,,,,,,,,, 2,,, 1000 ii,

More information

nsg04-28/ky208684356100043077

nsg04-28/ky208684356100043077 δ!!! μ μ μ γ UBE3A Ube3a Ube3a δ !!!! α α α α α α α α α α μ μ α β α β β !!!!!!!! μ! Suncus murinus μ Ω! π μ Ω in vivo! μ μ μ!!! ! in situ! in vivo δ δ !!!!!!!!!! ! in vivo Orexin-Arch Orexin-Arch !!

More information

橡博論表紙.PDF

橡博論表紙.PDF Study on Retaining Wall Design For Circular Deep Shaft Undergoing Lateral Pressure During Construction 2003 3 Study on Retaining Wall Design For Circular Deep Shaft Undergoing Lateral Pressure During Construction

More information

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1

II 2014 2 (1) log(1 + r/100) n = log 2 n log(1 + r/100) = log 2 n = log 2 log(1 + r/100) (2) y = f(x) = log(1 + x) x = 0 1 f (x) = 1/(1 + x) f (0) = 1 II 2014 1 1 I 1.1 72 r 2 72 8 72/8 = 9 9 2 a 0 1 a 1 a 1 = a 0 (1+r/100) 2 a 2 a 2 = a 1 (1 + r/100) = a 0 (1 + r/100) 2 n a n = a 0 (1 + r/100) n a n a 0 2 n a 0 (1 + r/100) n = 2a 0 (1 + r/100) n = 2

More information

On a branched Zp-cover of Q-homology 3-spheres

On a branched Zp-cover of Q-homology 3-spheres Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 On a branched Zp -cover of Q-homology 3-spheres 植木 潤 九州大学大学院数理学府 D2 December 23, 2014 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres

More information

mains.dvi

mains.dvi 8 Λ MRI.COM 8.1 Mellor and Yamada (198) level.5 8. Noh and Kim (1999) 8.3 Large et al. (1994) K-profile parameterization 8.1 8.1: (MRI.COM ) Mellor and Yamada Noh and Kim KPP (avdsl) K H K B K x (avm)

More information

(I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 Typeset by Akio Namba usig Powerdot. 2 / 47

(I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 Typeset by Akio Namba usig Powerdot. 2 / 47 4 Typeset by Akio Namba usig Powerdot. / 47 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 Typeset by Akio Namba usig Powerdot. 2 / 47 (I) (II) 2 (I) 2 (II) 2 (III) (I) (II) (II) : 2 (radom variable):

More information

(2) N elec = D p,q p,q χ q χ p dr = p,q D p,q S q,p Mulliken PA D Mull p = p = group A D p,p 1 + D p,q S q,p p q p [ r A D Mull p ] group χ p G Mull A

(2) N elec = D p,q p,q χ q χ p dr = p,q D p,q S q,p Mulliken PA D Mull p = p = group A D p,p 1 + D p,q S q,p p q p [ r A D Mull p ] group χ p G Mull A 7 - (Electron-Donor Acceptor) : Charge-Transfer ( CT) ( (Charge-Transfer) - (electron donor-electron acceptor) [1][2][3][4] Van der Waals CT [5] Population Analysis population analysis ( ), observable

More information

O E ( ) A a A A(a) O ( ) (1) O O () 467

O E ( ) A a A A(a) O ( ) (1) O O () 467 1 1.0 16 1 ( 1 1 ) 1 466 1.1 1.1.1 4 O E ( ) A a A A(a) O ( ) (1) O O () 467 ( ) A(a) O A 0 a x ( ) A(3), B( ), C 1, D( 5) DB C A x 5 4 3 1 0 1 3 4 5 16 A(1), B( 3) A(a) B(b) d ( ) A(a) B(b) d AB d = d(a,

More information

nsg02-13/ky045059301600033210

nsg02-13/ky045059301600033210 φ φ φ φ κ κ α α μ μ α α μ χ et al Neurosci. Res. Trpv J Physiol μ μ α α α β in vivo β β β β β β β β in vitro β γ μ δ μδ δ δ α θ α θ α In Biomechanics at Micro- and Nanoscale Levels, Volume I W W v W

More information

24.15章.微分方程式

24.15章.微分方程式 m d y dt = F m d y = mg dt V y = dy dt d y dt = d dy dt dt = dv y dt dv y dt = g dv y dt = g dt dt dv y = g dt V y ( t) = gt + C V y ( ) = V y ( ) = C = V y t ( ) = gt V y ( t) = dy dt = gt dy = g t dt

More information