note01
|
|
- たみえ とみもと
- 7 years ago
- Views:
Transcription
1 γ 5
2
3 J, M α J, M α = c JM JM J, M c JM
4 e ipr p / M p = 0 M J(J + 1) / Λ p / M J(J + 1) / Λ ~ 1 / m π
5 m π ~ 138 MeV J P,I = 0,1 π 1, π, π 3 ( ) ( π +, π 0, π ) ( ), π 0 = π 3 π ± = m 1 π1 ± iπ ( ) π ±, π 0 ( + m π )π a F.T. ( x) = 0 q ( + m π )π a q ( ) = 0 J a (π, ρ, N,,etc) ( + m π )π a = J a (π,ρ, N,,etc ) π, ρ,n, π + ~ ud, π 0 ~ uu + dd, π + ~ du q q
6 π = qq + qq qq + qq qq qq +... F π ( q ) γ γ ρ 0 e π ± e π ± ρ ρ π = 6 F π q ( q = 0) = 6 m ρ ρ m ρ = 770 π = 0.63 fm (exp) = 0.65 fm π
7 Space-like egion Pion fom facto F π ( q ) Time-like egion 4m π q [GeV ] q [GeV ] f π = 93
8 q µ, q µ = (ω,q i ), ( ω, q i ) a, b p µ, p µ = (E, p i ), ( E, p i ) α, β ( ) π b ( ω, q ) π a ω, q N α ( ) ( E, p ) N β E, p S-wave P-wave N,, etc
9 S = exp(iδ ) =1 + it = 1 + i q K 1 i q K K = 1 q tanδ a = lim q 0 q (l+1) tanδ = lim q 0 q l K l =1 l = 0 a u Tu = 4π s M χ fχ f u χ s = ( p + q) M q i q i q σ, q σ, q q, σ i q q σ f = B + C q q + id σ q q f q, q τ 1 = , τ = 0 i i 0, τ 3 = i 0 i 0 t 1 = 0 0 i, t = 0 0 0, t 3 = i i 0 i t τ
10 f = b 0 + b 1 t τ + c 0 + c 1 t τ ( ) q q + i d 0 + d 1 t τ σ q q ( ) b 0,1, c 0, 1, d 0,1 S - wave P - wave b 0 b 1 c 0 c 1 d 0 d 1 Spin - - a a f f Isospin a f a f a f a:, f: a I = a 1, a 3 a I,J = a 11, a 13, a 31, a 33 b, c, d a b b c 0 c = 1 d 0 d a 1 a 3 a 11 a 13 a 31 a 33
11
12 0.1 m π 1 10 m π 1 b 0 = m π 1, b 1 = m π 1 a 1 = m π 1, a 3 = m π 1 c 0 = 0.08 m π 3, c 1 = m π 3 a 11 = m π 3, a 13 = m π 3 d 0 = m π 3, d 1 = m π 3 a 31 = m π 3, a 33 = 0.14 m π 3 b 0 a 33 P 33 S-wave ρ P-wave
13 L πnn = ign τ π γ 5 N N g γ 5 γ 5 = N(p) = E + M 1 σ p E + M 1 χ exp(ipx) M σ p χ exp(ipx) M L πnn = ign (p ) τ π g γ 5 N( p 1 ) M M χ σ i τ a χ ( ) ( π i π a ) σ q q σ q p 1 p σ 1 q τ 1 σ q τ
14 V(q) = g M = 1 g 3 M σ 1 q σ q q + m τ 1 τ q q + m σ 1 σ q + q + m S 1 ( q ˆ ) τ 1 τ FT 1 g 3 M 1 m e m 4πδ 3 () σ 4π 1 + m + 3 m + 3 e m S 1 (ˆ ) σ τ 1 τ V στ () = 1 g 3 M 1 e m 4π m σ 1 σ τ 1 τ σ 1 σ τ 1 τ = ( S(S +1) )( I(I +1) ) = S I L odd SO even TE even SE odd TO
15 1 q + m Λ q + Λ 1 q + m Λ Λ q + Λ 1 q + m ~ Λ q + Λ 1 = ~ Λ 4 Λ Λ 1 A q + m + Λ q + Λ 1 q + m (If Λ 1 ~ Λ ) 1 q + Λ 1 1 q + Λ B q + Λ 1 + C q + Λ 1 q + m Λ 1, Λ g 4π ~ 14.9 g ~ 13.7 ; Λ = 1400 MeV a q 1 b q p 1 p T = g u ( p )iγ 5 τ b i / p 1 + q / 1 M τ a iγ 5 u(p 1 ) + g u (p )iγ 5 τ a i p / 1 q / M τ b iγ 5 u( p 1 ) At theshold a=b + i g M u (p )u( p 1 )
16 p 1 ~ p ~ (M,0), q / 1 ~ q / ~ γ 0 m π m π σ = (π) 4 T 4 ( pq) m M dφ m dφ u ( p )u( p 1 ) ~ M dφ = 1 q (π) 5 m + M (angle integated ), T ~ 4g σ ~ g4 4πM ~ 15 fm =150 mb
17 γ 5 m L = ψ ( i / m )ψ, ψ = u d u, d ψ exp(iτ v )ψ ; ψ ψ exp( i τ v ), ( ψ ψ exp( iτ v )) τ v = 3 i=1 τ i v i v i (i =1,,3) ψ / ψ ψ e i τ v / e i τ v ψ = ψ / ψ (invaiant ) ψ ψ ψ e i τ v e i τ v ψ = ψ ψ ( " ) γ 5 γ 5 ψ e i τ a γ 5 ψ ; ψ ψ e i τ a γ 5, ψ ψ e iτ a γ ( 5 ) ψ / ψ ψ e i τ a γ 5 / e i τ a γ 5 ψ = ψ / ψ (invaiant ) ψ ψ ψ e i τ a γ 5 e i τ a γ 5 ψ ψ ψ (noninvaiant )
18 γ 5 ψ = 1+ γ γ 5 ψ ψ R +ψ L P R 1+ γ 5, P L 1 γ 5 P R = P R, P L = P L, P R P L = 0, P R + P L = 1 γ 5 ψ R = γ 5 P R ψ = +ψ R, γ 5 ψ L = γ 5 P L ψ = ψ L ψ R, ψ L P 1+ γ ψ R γ 5 0 ψ = 1 γ 5 γ 0 ψ = 1 γ 5 ψ = ψ L ψ R ψ L c =1 p = (0,0,1) s z = +1 / 1 ψ + = σ p χ = = 1 1 χ
19 χ = ( ) ( 0 1) ψ = 1 σ p ˆ χ = = 1 1 χ P R ψ + = 1 + γ 5 ψ + = P L ψ = 1 γ 5 ψ = P R ψ = P L ψ + = χ = ψ χ = ψ ψ R ψ L exp(iτ v ) ~ 1 + iτ v +... g V exp(iτ a γ 5 ) ~ 1 + iτ a γ g A g V ψ R = (1 + iτ v )ψ R g V ψ L = (1 + iτ v )ψ L g A ψ R = (1 + iτ aγ 5 )ψ R = (1 + iτ a)ψ R g A ψ L = (1 + iτ aγ 5 )ψ L = (1 iτ a)ψ L γ 5 ψ R,L = ±ψ R,L
20 g V + g A ψ R = 1 + iτ v + a ψ R g R ψ R g V g A ψ R = iτ v a ψ R g V + g A ψ L = 1 + iτ v a ψ L g L ψ L g V g A ψ L = iτ v + a ψ L v = a v = a l g R ψ R = (1+ iτ )ψ R, g L ψ R = ψ R g L ψ L = (1 + iτ l )ψ L, g R ψ L = ψ L ψ R, ψ L g R, g L g R, g L g V, g A g R, g L g A, g B g A ψ R, g B ψ L g A ψ L, g B ψ R g A g B g B g A (D A, D B ) g A g B ψ R ~ (D A, D B ), ψ L ~ (D B, D A ) D A A
21 g L(φ, µ φ) φ = ( φ 1,φ,L,φ n) ( ) G φ = φ 1,φ,L,φ n φ a g ( D(g)φ ) a D(g) ab φ b b ~ 1 + i ε m T m φ φ + δφ m D(g) T m φ n n ε m 0 = δl(φ, µ φ) L(φ + δφ, µ φ + µ δφ) L(φ, µ φ) ~ L φ δφ + L ( µ φ) µ δφ L = i µ ( µ φ) T m φ ε m µ J m m m ( ) µ ε m ( J m ) µ = i L ( µ φ) T m φ ε m m µ ( J m ) µ = 0 V µ a = ψ γ µ τ a ψ, A µ a = ψ γ µ γ 5 τ a ψ
22 V µ a = ψ R γ µ τ a ψ R +ψ L γ µ τ a ψ L R µ a + L µ a A µ a =ψ R γ µ τ a ψ R ψ L γ µ τ a ψ L R µ a L µ a V µ a ψ (1 iτ v)γ µ τ a (1 + iτ v)ψ A µ a ~ ψ γ µ τ a ψ + i ψ γ µ [ τ a, τ v]ψ a c = V µ ε abc v b V µ A µ a c ε abc v b A µ V µ a V µ a ε abc a b A µ c A µ a A µ a ε abc a b V µ c a a c R µ R µ ε abc b R µ a a L µ L µ a a c L µ L µ ε abc l b R µ R µ a R µ a φ π a L (x) = ( 0 φ a )
23 [ π a (x),φ b (y)] = iδ ab δ 3 (x y) ( J m ) µ = i L ( µ φ) T m φ µ =0 i πt m φ Q m d3 x J m 0 (x) = i d 3 x πt m φ [ ] = d 3 y [ π(y)t m φ(y),φ(x) ] i Q m, φ(x) = i T m φ(x) e i Qm ε m φ(x)e i Q m ε m = D(g)φ(x) Q a b [ V, Q V ] = iε abc Q c V, Q a b [ V, Q A ] = iε abc Q c A, Q a b c [ A, Q A ] = iε abc Q V Q R a = 1 Q a a a ( V + Q A ), Q L = 1 Q a a ( V Q A ) Q a b [ R, Q R ] = iε abc Q c R, Q a b [ L, Q L ] = iε abc Q c L, Q a b [ R, Q L ] = 0
24 SU() SU() SU() SU() J(J +1) J H ghg 1 = H
25 g g A = a 0, B = b 0 B g A E A = A H A = B ghg B = B H B = E B GeV Meson spectum σ(~ 600) 0 + a 1 (160) ρ(770) f 1 (185) ω (780) π (1300) η(195) a 0 (980) f 0 (975) K 1 (1400) K 1 (170) K * (890) 0 π(139) GeV 1.5 Bayon spectum 1 N (1535) Λ (1670) Λ (110) Σ (1750) Σ (1190) (1700) (13) N (939)
26 A g A = ga 0 = ga g g 0 = b g 0 g 0 = 0 g SU() SU() SU() Isospin SU() Isospin SU() SU() SU() SU() SU() Isospin ~ U = exp(iτ π / f π )
27 ν π + u µ d 0 A µ π d A µ u W + ν µ L int ~ J µ J µ J µ ~ J µ (h) + J µ (l) J µ (l) J µ (l) = l γ µ (1 γ 5 )l v µ a µ µν µ L int (x) π(q) ~ µν µ J µ (x)j µ (x) π(q) ~ µν µ J µ(l) (x) 0 0 A µ (x) π(q) q µ A µ A a µ = σ µ π a π a µ σ f π f π µ π a f π
28 µν µ J µ (l) 0 0 A µ π A µ a (x) 0 A µ a (x) π b (q) = iq µ δ ab f π exp( iqx) f π q f π exp( iqx) 0 µ A µ a (x) π b (q) = δ ab m π f π exp( iqx) 0 π a (x) π b (q) = δ ab exp( iqx) 0 µ A µ a (x) π b (q) = m π f π 0 π a (x) π b (q) µ A µ a (x) = m π f π π a (x) m π m π = 0
29 p(p f )eν e L int n( p i ) = µν e J µ (l) 0 p(p f ) (V µ A µ ) n(p i ) p(p f ) A µ a n( p i ) p(p f ) A a µ (x) n(p i ) = u p ( p f ) γ µ g A (q ) + q µ h A (q τ [ a )]γ 5 u n ( p i ) exp(iqx) q = p f p i g A (q ) h A (q ) ν e f π = + g π NN n p g A tem h A tem µ A µ a (x) = 0 m π = 0 0 = p( p f ) µ A a µ n( p i ) = iu p ( p f ) q / g A (q ) + q h A (q τ [ a )]γ 5 u n (p i ) h A (q ) = M N g A (q ) q h A (q ) 1 / q A µ a = f π µ π a +...
30 x = 0 p(p f ) A µ a n( p i ) 1π = if π q µ p(p f ) π a n( p i ) m π = 0 p(p f ) π a n(p i ) i q g πnn (q ) u p ( p f )γ 5 τ a u n ( p i ) p(p f ) π a n(p i ) = 1 q p( p f ) J a n(p i ) i q g πnn (q ) u p ( p f )γ 5 τ a u n ( p i ) 1 / q J a A µ a p(p f ) A µ a n( p i ) 1π = f π q µ q g πnn (q ) u p (p f )γ 5 τ a u n (p i ) p(p f ) A µ a n( p i ) 1π = u p (p f )q µ M N g A (q ) q γ 5 τ a u n ( p i ) g πnn M N = g A f π g πnn = 13.7, g A =1.5, M N = 938 MeV, f π = 93 MeV g πnn f π
31 a q 1 b q p 1 p π b (q )N(p ) π a (q 1 )N(p 1 ) I = π b (q )N(p ) π a (q 1 )N(p 1 ) ~ d 4 xd 4 y e iq 1 x e +iq x N(p ) Tπ b (x)π a (y) N( p 1 ) I ~ d 4 xd 4 y e iq 1 x e +iq x N(p ) T µ A b µ (x) ν A a ν (y) N(p 1 ) I = I 1 + I + I 3 I 1 = d 4 xd 4 y e iq 1 x e +iq y δ(x 0 y 0 ) N(p ) A b 0 (x), ν A a ν (y) I = d 4 xd 4 y e iq 1 x e +iq y µ x ν y N(p ) A b 0 (x),a a ν (y) I 3 = i d 4 xd 4 y e iq 1 x e +iq y q µ δ(x 0 y 0 ) N(p ) A b 0 (x), A a µ (y) [ ] N(p 1 ) [ ] N(p 1 ) [ ] N(p 1 )
32 I I 1 I 3 O(m π ) O(m π ) O(m π ) I 1 I 3 I m π I 1 ν A a ν (y) ~ m π O(m π ) O(m π ) O(1) I 3 O(m π ) T = i f π I π I N, I N = τ, (I π a ) bc = iε abc a = m π 8πf π = m π 8πf π 1 + m π M 1+ m π M 1 I π I N 1 [ I(I +1) I N (I N +1) ] a 1 = 0. m π 1, a 3 = 0.1 m π 1 I N = 1 / I = I π + I N =1 / o 3 / a 1 = m 1 1 π, a 3 = m π
33 ψ,σ,π L SU() R SU() L 1 (, 0 ) ( 0, 1 ) 1 ( ), 1 1 (, 0 ) SU() R SU() L SU() R ~ Isospin 1 /, SU() L ~ Isospin 0 ( ) SU() R ~ Isospin 0, SU() L ~ Isospin 1 / 0, 1 1 ( ) ~ ψ R 0, 1 B α, 0 ( ) ~ ψ L B α ψ R τ α ψ L
34 B 0 B B B 0 B 0 B + v B B 0 + i a B B + i a B 0 B = τ α ( ψ R τ α ψ L ) = ψ L ψ R U B U ψ R τ α ψ L ψ L ψ R Paity ψ R τ α ψ L ψ L τ α ψ R S α = ψ R τ α ψ L + ψ L τ α ψ R =ψ τ α ψ ( ) = iψ τ α γ 5 ψ P α = i ψ R τ α ψ L ψ L τ α ψ R i S 0 S 0 S S + v S, P P 0 P 0 P + v P S 0 S 0 + a P P 0 P 0 a S, S S + a P 0 P P a S 0 S 0 + P P 0 + S S 0 + P, P 0 + S : Invaiant
35 S 0 + P (S 0, P ) (σ, π ) L σ = 1 ( µ σ) + ( µ π ) ( ) V (φ ) V(φ ) φ = σ + π V(φ ) = µ φ + λ 4 φ 4 λ µ H = 1 p α + 1 i φ α ( ) + V (φ ), p α = L φ = φ α α µ µ m π 10 1 mπ
36 φ φ π π σ 0 σ µ µ µ > (σ, π ) = (0, 0 ) µ < 0 φ = µ λ
37 0 µ λ, 0 f π, 0 ( ) SU() SU() SU() SU() R SU() L SU() V χ = ( χ 0, χ 1, χ, χ 3 ) φ = φ vac + χ χ V(φ vac + χ) = V(φ vac ) + χ α α V(φ vac ) + 1 χ α χ β α β V (φ vac ) +... V(φ), φ = φ 0 + φ 1 +φ + φ 3
38 α V(φ) = φ V (φ ) = φ α V (φ) φ ˆ φ α φ α V (φ ) ( ) = P αβ α β V(φ) = ˆ β φ α V (φ) P αβ δ αβ ˆ φ α ˆ φ β, φ V (φ) + φ ˆ ˆ α φ β V (φ ) P αβ ˆ φ α = P αβ ˆ φ β = 0 φ vac = ( f π, 0,0,0) L σ ~ 1 ( µ χ ) 1 V''(φ vac ) χ 0 ( ψ ψ, iψ τ γ 5 ψ ) σ, π ( ) σψ ψ + π iψ τ γ 5 ψ = ψ ( σ + iτ π γ 5 )ψ L = i ψ / ψ gψ ( σ +iτ π γ 5 )ψ + 1 µ σ ( ) + ( µ π) V(φ) ψ i ψ L / ψ L + i ψ R / ψ R gψ L ( σ + iτ π )ψ R gψ R ( σ iτ π )ψ L gσψ ψ gf π ψ ψ
39 M = gf π g A g A g A σ ψ ψ = ψ R ψ L +ψ L ψ R ψψ ψψ = ψ R ψ R +ψ L ψ L + 1 µ σ L = i ψ / ψ gψ ( σ + iτ π γ 5 )ψ [( ) + ( µ π) ] λ 4 σ + π ( f π ) m σ = λf π M = gf π
40 σσσ, σππ σσσσ, σσππ, ππσσ a q 1 b q π π p 1 p σ (3d tem ) = g u u i q m λf π i g σ M k 0 L PS = iψ γ 5 τ π ψ ψ ψ (x) = u n ( x )exp( ie n t)b n + v m ( x ) exp(+ie m t)d m E n >0 E m <0 L PS n L PS n v ~ p / E m L PS n
41 n Bon tem n ~ n iψ γ 5 τ π ψ iψ γ 5 τ π ψ n ~ u n γ 5 0 T(ψψ ) 0 γ 5 u n ~ u n (x)γ 5 ~ dω π dω π u n (x)u n (y) ω E n + iε + v m (x)v m (y) γ ω + E m iε 5 u n (y) u n (x)γ 5 u n (x)u n (y)γ 5 u n (y) ω E n + iε + u n (x)γ 5 v m (x)v m (y)γ 5 u n (y) ω + E m iε γ 5 O(1) L PV = 1 f π ψ γ µ γ 5 τ µ π ψ O(q) µ = 1,,3 n L PV n n L PV n n L PV n ~ O(q) ( n L PV n ) / (Enegy denominato ) O(q) O(q ) σ q
42 1 (, 1 ) f 0 (600) o σ ( σ,π 1,π,π 3 ) O(4) f = σ +π 1 +π + π 3 P Q P(x,y, z) Q( x, y, z ) δx δy = δz 0 ε 3 ε ε 3 0 ε 1 ε ε 1 0 x y z (x, y, z) (ε 1,ε,ε 3 )
43 x = sinθ cosϕ, y = sinθ sinϕ, z = cosθ δθ = δϕ sinϕ ε 1 cosϕ ε cot θ cosϕ ε 1 + cot θ sinϕ ε ε 3 (θ,ϕ) sinϕ φ µ ~ ( σ,π 1,π,π 3 ) σ = f cosθ 1, π 1 = f sinθ 1 sinθ cosθ 3, π = f sinθ 1 sinθ sinθ 3, π 3 = f sinθ 1 cosθ. ( σ,π 1,π,π 3 ) (θ 1,θ,θ 3 ) L = 1 ( µ ) φα = 1 m θ m x µ φ α θ m = 1 m,n g mn (θ ) θ m x µ θ n x µ g mn (θ ) (θ 1,θ,θ 3 ) g mn (θ ) = δ mn + θ m θ n f π θ L = f π ( ) ( µ σ) + µ θ, σ = 1 θ
44 g mn (θ) = θ m θ n θ + δ mn θ θ m θ n θ 4 sin θ L = f π 4 t µ UU µ UU = f π 4 t µ U µ U ( ) U = exp i τ θ U U U U g L U g R g V U g V g A U g A V µ a A µ a = i f 4 tτ a U, µ U = i f 4 tτ a U, µ U [ ] ~ ε abc φ b µ φ c { } ~ f µ φ a L = i ψ / ψ gψ ( σ +iτ π γ 5 )ψ 1 + µ σ ( ) + ( µ π ) V(φ) σ + iτ π f U, U = exp( iτ φ / f π ) σ + iτ π γ 5 f U 5, U 5 = exp iτ φ γ 5 / f π ( ), ψ ( σ +iτ πγ 5 )ψ = f ψ U 5 ψ = f ψ U 5 U 5 ψ f N N
45 ξ 5 ψ = N, ψ = ξ 1 5 N U 5 = exp iτ φ γ 5 / f π ( τ ) ( ) ξ 5 exp( i φ / f ) ξ π ψ N N L = i N ξ 1 5 / ( ξ 1 5 N) gf N N + f 4 t µ U µ U + 1 ( µ f ) + V ( f ) N ξ 5 1 L K = i N ξ 1 5 / ( ξ 1 5 N) = i N / N +i N ξ 1 5 / 1 ( ξ 5 )N i N ξ 1 5 ( µ ξ 5 )γ µ N = i N 1 ξ ( ( µ ξ 5 ) + ξ 5 ( µ ξ 5 ))γ N + i N 1 ξ ( ( µ ξ 5 ) ξ 5 ( µ ξ 5 ))γ N φγ 5 γ 5 φγ 5 1 ( ( ) + ξ 5 ( µ ξ 5 ) v µ = 1 ξ 5 1 µ ξ 5 a µ = 1 ξ ( ( µ ξ 5 ) ξ 5 ( µ ξ 5 ))γ 5 τ a L = i N D / N + i N a / γ 5 N gf N N + f 4 t µ U µ U + 1 ( µ f ) + V( f ) 1 ξ 5 / 1 ξ 5 ( ) = ξ 1 5 γ µ µ ξ 1 5 = ξ 1 5 µ ξ 5 γ µ γ 1 µ ξ 5 1 ξ 5 ξ 5
46 D µ = µ iv µ ξ U U U = ξ g L Ug R = ( g L ξ) ( ξg R ) U U = ξ ( ) h ξ = g L ξh = hξg R h ξ (g R, g L ) ξ g R = g L = g V h h = g V ε δφ = ε φ h h ~ 1+ iε φ τ + L
47 SU() R SU() L SU() R SU() L SU() R SU() L SU() V π ~ SU() R SU() L / SU() V SU() V U SU() R SU() L SU() R SU() L τ R a ~ SU() R, τ L a ~ SU() L τ R a τ L a τ a (R) 1+ γ 5 τ a = P R τ a, τ a (L) 1 γ 5 τ a = P L τ a τ a γ 5 P R,L τ V a = τ R a + τ L a = τ a τ A a = τ R a τ L a = γ 5 τ a τ V a = τ a τ A a = γ 5 τ a γ 5 H G G G
48 g Π = G / H a H h g = ha a i i a h ha i i { ha 1,ha,ha 3,L} G Π = G / H SU() SU() / SU() ( ) ξ 5 (φ) = exp iφ τ γ 5 / a i ξ 5 (φ) φ γ 5 τ a = τ R a τ L a = τ A a ~ SU() SU() / SU() ξ 5 ( ) γ 5 ξ 5 ξ = exp iτ φ / γ 5 ξ 5 (φ) G ξ 5 (φ) g G ξ 5 ( φ ) h ξ 5 (φ) g = h(g,φ ) ξ 5 ( φ ) ha i h h (g,φ) h ξ 5 (φ) φ g ξ 5 ( φ ) = h(g,φ)ξ 5 (φ )g φ φ
49 N h N = h(g,φ)n i N / N i N h / (hn) = i N h ( / h)n +i N / N v µ h v µ h h ( / h) a µ h a µ h i N D / N i N ( / + v /)N i N D / N, N a / γ 5 N N a / γ 5 N L = i N D / N + i g A N a / γ 5 N gf N N + f 4 t µ U µ U + 1 ( µ f ) + V( f ) g A =1 g A g A N = ξ 5 ψ g A
50 λ 1 = , λ = 0 i 0 i 0 0, λ 3 = λ 4 = , λ 5 = 0 0 i i 0 0 λ 6 = , λ 7 = i, λ 8 = i U = exp( iθ a λ a ), θ a = θ 1, θ,lθ 8 I 3 Y U + Y V + I I + I 3 V U I ± = 1 ( λ 1 ± iλ ), V ± = 1 ( λ 4 ± iλ 5), U ± = 1 ( λ 6 ± iλ 7) 3 3 = 1 + 8, =
51 π + I + + π I + π 0 I 0 + K + V + + K V + K 0 U + + K 0 U + η λ 8 π 0 + η π + K + = π π 0 + η K 0 = λ a φ a φ K K 0 η 6 exp( iφ / f π ) U U Peudoscala meson octet K 0 ( ds) K + (us) Vecto meson octet K *0 (ds) K *+ (us) π ( d u ) π 0 (uu dd) π + (ud ) ρ ( du) ρ 0 (uu dd) ρ + (ud) η(uu + dd ) ω ( uu + dd ) K (su) K 0 (sd) K * (su) K *0 (sd ) n(udd) Bayon octet p(uud) (ddd) Bayon decuplet 0 (udd) + (uud) ++ (uuu ) Σ (dds) Σ 0 (uds) Λ (uds) Σ + (uus) Σ * ( dds) Σ *0 (uds) Σ *+ (uus) Ξ (ssd ) Ξ 0 (ssu) Ξ (dss) Ξ *0 (uss ) Ω (sss)
52 Σ 0 + η Σ + p B = Σ Σ0 + Λ n Ξ Ξ 0 Λ 6 p n 8 8 = (8 8) 8 = ( ) L = f π ( ) + t B i / 4 t µ U µ U ( D B) + m 0 tb B + F tb γ 5 [ a /, B] + D tγ 5 { a /, B}
53 R 3 SU() ~ S 3 x = (x 1, x,x 3 ) R 3 R 3 π = (π 1,π, π 3 ) S 3 S 3 N
54 N = 1 V dπ = 1 V (π ) (x) dx V S 3 dπ x (π) / (x) V N π 3 (S 3 ) = Z (intege ) x π ( x ) ( ) U( x ) = exp iτ π ( x ) / f π f π π µ = ( µ U)U N = ε ijk 4π t ( ) U = exp iτ π / f π d3 x i j k L = f π 4 t µ µ E = f π d 3 x 4 t i i U( x ) x ax a
55 E E ae a U( x ) 1 E 4 E 4 / a E = E + E 4 a L 4 = 1 3e t [ µ, ] ν e ρ π L Skyme = f π [ ] 4 t µ µ + 1 3e t µ, ν E = E + E 4 E E 4 a
56 ( ) U H exp( iτ ˆ F() ) U = exp i τ π / f π F() K = I + J K I = τ / U H U H Isospin : U H AU H A A SU() J = i Spin : ( ) R O(3) U H exp iτ Rˆ F() B R R ab = i t [ τ a B τ b B] A = B K U H E E = kq ˆ
57 1 / B = 1 E ~ Y 1, e [ ] 0 0 Y 1 e E = d 3 x f π 4 t i i + 1 3e t i, j [ ] = 4π d f π F + sin F + sin F e F + sin F F + sin F F + e f π sin F F + sinf F sin F sin F = 0 B = 1 π d 3 x sin F F = 1 π ( F(0) F( ) ) F() F(0) = π F( ) = 0 F( ) =1 / 3 F( ) Radial distance [fm].0
58 e f π = 93 MeV e ~ 5 E ~ 1.4 GeV ρ π R 0 e ~ 4.5 E ~ 1.4 GeV I = J K = 0 (I, I z ) = (J, J z ) I = J U H φ c ϕ φ = φ c + ϕ φ c L( φ c + ϕ) = L(φ c ) + L φ (φ c ) ϕ + 1 L φ (φ c ) ϕ + L L / φ ϕ ϕ ϕ L φ (φ c ) = 0 φ c ϕ L φ (φ c ) ϕ = 0 ϕ t ϕ = Hϕ
59 H H ϕ(t, x ) = n a n e iω n t ϕ n ( x ) φ Gφ φ Gφ G L φ (Gφ c ) = L φ (φ c ) = 0 ε T G ~ 1 + iεt 0 = L φ (Gφ c ) ~ L φ ((1 + iεt)φ c ) ~ L φ (φ c ) + i L φ (φ c ) εtφ c L φ (φ c ) Tφ c = 0 Tφ c 0 Tφ c = 0 Tφ c Tφ c ω = 0 Tφ c 0 φ c φ c
60 U H AU H A A SU() A A A(t) A A SU() ~ S 3 (α,β,γ ) ψ ~ exp(ip α α)exp(ip β β)exp(ip γ γ ) ψ ~ D t, s (α,β, γ ) J J K = I + J = 0 I = J J J = SU(N f > ) N c N c N c 1/ p ~ D 1/, 1/ (α,β,γ ) = e i α cos β i e γ A / A U(x) U(x) AU H A d 3 x A A A / A
61 H = d 3 x L = M H + Λ π Ω Λ π Ω = (i / )t[ τ A A ] (~ A ) A F() Λ π = 8π 3 f π R d sin F e F + sin F f π A A / A H = M H + 1 Λ π J(J +1) J(J +1) M N M f π 1/ I =0 1/ M,I =1 µ p µ n µ p / µ n g A g πnn g πn µ N
62 1 (,0 ) ( 0, 1 ) 1 (, 1 ) 1 (,0 ) ( 0, 1 ) g A =1
63
7 9 7..................................... 9 7................................ 3 7.3...................................... 3 A A. ω ν = ω/π E = hω. E
B 8.9.4, : : MIT I,II A.P. E.F.,, 993 I,,, 999, 7 I,II, 95 A A........................... A........................... 3.3 A.............................. 4.4....................................... 5 6..............................
More information7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
More information24.15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 information5 36 5................................................... 36 5................................................... 36 5.3..............................
9 8 3............................................. 3.......................................... 4.3............................................ 4 5 3 6 3..................................................
More information0.,,., 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 informationhttp://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 informationNote5.dvi
12 2011 7 4 2.2.2 Feynman ( ) S M N S M + N S Ai Ao t ij (i Ai, j Ao) N M G = 2e2 t ij 2 (8.28) h i μ 1 μ 2 J 12 J 12 / μ 2 μ 1 (8.28) S S (8.28) (8.28) 2 ( ) (collapse) j 12-1 2.3 2.3.1 Onsager S B S(B)
More information基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
More information第85 回日本感染症学会総会学術集会後抄録(III)
β β α α α µ µ µ µ α α α α γ αβ α γ α α γ α γ µ µ β β β β β β β β β µ β α µ µ µ β β µ µ µ µ µ µ γ γ γ γ γ γ µ α β γ β β µ µ µ µ µ β β µ β β µ α β β µ µµ β µ µ µ µ µ µ λ µ µ β µ µ µ µ µ µ µ µ
More informationPart. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3. 39. 4.. 4.. 43. 46.. 46..
Cotets 6 6 : 6 6 6 6 6 6 7 7 7 Part. 8. 8.. 8.. 9..... 3. 3 3.. 3 3.. 7 3.3. 8 Part. 4. () 4.. () 4.. 3 5. 5 5.. 5 5.. 6 5.3. 7 Part 3. 8 6. 8 6.. 8 6.. 8 7. 8 7.. 8 7.. 3 8. 3 9., 34 9.. 34 9.. 37 9.3.
More information1 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 informationI.2 z x, y i z = x + iy. x, y z (real part), (imaginary part), x = Re(z), y = Im(z). () i. (2) 2 z = x + iy, z 2 = x 2 + iy 2,, z ± z 2 = (x ± x 2 ) +
I..... z 2 x, y z = x + iy (i ). 2 (x, y). 2.,,.,,. (), ( 2 ),,. II ( ).. z, w = f(z). z f(z), w. z = x + iy, f(z) 2 x, y. f(z) u(x, y), v(x, y), w = f(x + iy) = u(x, y) + iv(x, y).,. 2. z z, w w. D, D.
More information1 913 10301200 A B C D E F G H J K L M 1A1030 10 : 45 1A1045 11 : 00 1A1100 11 : 15 1A1115 11 : 30 1A1130 11 : 45 1A1145 12 : 00 1B1030 1B1045 1C1030
1 913 9001030 A B C D E F G H J K L M 9:00 1A0900 9:15 1A0915 9:30 1A0930 9:45 1A0945 10 : 00 1A1000 10 : 15 1B0900 1B0915 1B0930 1B0945 1B1000 1C0900 1C0915 1D0915 1C0930 1C0945 1C1000 1D0930 1D0945 1D1000
More informationuntitled
10 log 10 W W 10 L W = 10 log 10 W 10 12 10 log 10 I I 0 I 0 =10 12 I = P2 ρc = ρcv2 L p = 10 log 10 p 2 p 0 2 = 20 log 10 p p = 20 log p 10 0 2 10 5 L 3 = 10 log 10 10 L 1 /10 +10 L 2 ( /10 ) L 1 =10
More information「数列の和としての積分 入門」
7 I = 5. introduction.......................................... 5........................................... 7............................................. 9................................................................................................
More informationE B m e ( ) γma = F = e E + v B a m = 0.5MeV γ = E e m =957 E e GeV v β = v SPring-8 γ β γ E e [GeV] [ ] NewSUBARU.0 957 0.999999869 SPring-8 8.0 5656
SPring-8 PF( ) ( ) UVSOR( HiSOR( SPring-8.. 3. 4. 5. 6. 7. E B m e ( ) γma = F = e E + v B a m = 0.5MeV γ = E e m =957 E e GeV v β = v SPring-8 γ β γ E e [GeV] [ ] NewSUBARU.0 957 0.999999869 SPring-8
More informationuntitled
3,,, 2 3.1 3.1.1,, A4 1mm 10 1, 21.06cm, 21.06cm?, 10 1,,,, i),, ),, ),, x best ± δx 1) ii), x best ), δx, e,, e =1.602176462 ± 0.000000063) 10 19 [C] 2) i) ii), 1) x best δx
More information第86回日本感染症学会総会学術集会後抄録(II)
χ μ μ μ μ β β μ μ μ μ β μ μ μ β β β α β β β λ Ι β μ μ β Δ Δ Δ Δ Δ μ μ α φ φ φ α γ φ φ γ φ φ γ γδ φ γδ γ φ φ φ φ φ φ φ φ φ φ φ φ φ α γ γ γ α α α α α γ γ γ γ γ γ γ α γ α γ γ μ μ κ κ α α α β α
More information,..,,.,,.,.,..,,.,,..,,,. 2
A.A. (1906) (1907). 2008.7.4 1.,.,.,,.,,,.,..,,,.,,.,, R.J.,.,.,,,..,.,. 1 ,..,,.,,.,.,..,,.,,..,,,. 2 1, 2, 2., 1,,,.,, 2, n, n 2 (, n 2 0 ).,,.,, n ( 2, ), 2 n.,,,,.,,,,..,,. 3 x 1, x 2,..., x n,...,,
More information2 1 17 1.1 1.1.1 1650
1 3 5 1 1 2 0 0 1 2 I II III J. 2 1 17 1.1 1.1.1 1650 1.1 3 3 6 10 3 5 1 3/5 1 2 + 1 10 ( = 6 ) 10 1/10 2000 19 17 60 2 1 1 3 10 25 33221 73 13111 0. 31 11 11 60 11/60 2 111111 3 60 + 3 332221 27 x y xy
More information,,..,. 1
016 9 3 6 0 016 1 0 1 10 1 1 17 1..,,..,. 1 1 c = h = G = ε 0 = 1. 1.1 L L T V 1.1. T, V. d dt L q i L q i = 0 1.. q i t L q i, q i, t L ϕ, ϕ, x µ x µ 1.3. ϕ x µ, L. S, L, L S = Ld 4 x 1.4 = Ld 3 xdt 1.5
More information467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 B =(1+R ) B +G τ C C G τ R B C = a R +a W W ρ W =(1+R ) B +(1+R +δ ) (1 ρ) L B L δ B = λ B + μ (W C λ B )
More information3
- { } / f ( ) e nπ + f( ) = Cne n= nπ / Eucld r e (= N) j = j e e = δj, δj = 0 j r e ( =, < N) r r r { } ε ε = r r r = Ce = r r r e ε = = C = r C r e + CC e j e j e = = ε = r ( r e ) + r e C C 0 r e =
More informationEndoPaper.pdf
Research on Nonlinear Oscillation in the Field of Electrical, Electronics, and Communication Engineering Tetsuro ENDO.,.,, (NLP), 1. 3. (1973 ),. (, ),..., 191, 1970,. 191 1967,,, 196 1967,,. 1967 1. 1988
More informationチュートリアル:ノンパラメトリックベイズ
{ x,x, L, xn} 2 p( θ, θ, θ, θ, θ, } { 2 3 4 5 θ6 p( p( { x,x, L, N} 2 x { θ, θ2, θ3, θ4, θ5, θ6} K n p( θ θ n N n θ x N + { x,x, L, N} 2 x { θ, θ2, θ3, θ4, θ5, θ6} log p( 6 n logθ F 6 log p( + λ θ F θ
More informationhirameki_09.dvi
2009 July 31 1 2009 1 1 e-mail: mtakahas@auecc.aichi-edu.ac.jp 2 SF 2009 7 31 3 1 5 1.1....................... 5 1.2.................................. 6 1.3..................................... 7 1.4...............................
More informationMicrosoft Word - Wordで楽に数式を作る.docx
Ver. 3.1 2015/1/11 門 馬 英 一 郎 Word 1 する必要がある Alt+=の後に Ctrl+i とセットで覚えておく 1.4. 変換が出来ない場合 ごく稀に以下で説明する変換機能が無効になる場合がある その際は Word を再起動するとまた使えるようになる 1.5. 独立数式と文中数式 数式のスタイルは独立数式 文中数式(2 次元)と文中数式(線形)の 3 種類があ り 数式モードの右端の矢印を選ぶとメニューが出てくる
More information46 Y 5.1.1 Y Y Y 3.1 R Y Figures 5-1 5-3 3.2mm Nylon Glass Y (X > X ) X Y X Figure 5-1 X min Y Y d Figure 5-3 X =X min Y X =10 Y Y Y 5.1.2 Y Figure 5-
45 5 5.1 Y 3.2 Eq. (3) 1 R [s -1 ] ideal [s -1 ] Y [-] Y [-] ideal * [-] S [-] 3 R * ( ω S ) = ω Y = ω 3-1a ideal ideal X X R X R (X > X ) ideal * X S Eq. (3-1a) ( X X ) = Y ( X ) R > > θ ω ideal X θ =
More information木オートマトン•トランスデューサによる 自然言語処理
木オートマトン トランスデューサによる 自然言語処理 林 克彦 NTTコミュニケーション科学基礎研究所 hayashi.katsuhiko@lab.ntt.co.jp n I T 1 T 2 I T 1 Pro j(i T 1 T 2 ) (Σ,rk) Σ rk : Σ N {0} nσ (n) rk(σ) = n σ Σ n Σ (n) Σ (n)(σ,rk)σ Σ T Σ (A) A
More information168 13 Maxwell ( H ds = C S rot H = j + D j + D ) ds (13.5) (13.6) Maxwell Ampère-Maxwell (3) Gauss S B 0 B ds = 0 (13.7) S div B = 0 (13.8) (4) Farad
13 Maxwell Maxwell Ampère Maxwell 13.1 Maxwell Maxwell E D H B ε 0 µ 0 (1) Gauss D = ε 0 E (13.1) B = µ 0 H. (13.2) S D = εe S S D ds = ρ(r)dr (13.3) S V div D = ρ (13.4) ρ S V Coulomb (2) Ampère C H =
More informationexample2_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 information3 3.3. I 3.3.2. [ ] N(µ, σ 2 ) σ 2 (X 1,..., X n ) X := 1 n (X 1 + + X n ): µ X N(µ, σ 2 /n) 1.8.4 Z = X µ σ/ n N(, 1) 1.8.2 < α < 1/2 Φ(z) =.5 α z α
2 2.1. : : 2 : ( ): : ( ): : : : ( ) ( ) ( ) : ( pp.53 6 2.3 2.4 ) : 2.2. ( ). i X i (i = 1, 2,..., n) X 1, X 2,..., X n X i (X 1, X 2,..., X n ) ( ) n (x 1, x 2,..., x n ) (X 1, X 2,..., X n ) : X 1,
More informationFourier (a) C, (b) C, (c) f 2 (a), (b) (c) (L 2 ) (a) C x : f(x) = a 0 2 + (a n cos nx + b n sin nx). ( N ) a 0 f(x) = lim N 2 + (a n cos nx + b n sin
( ) 205 6 Fourier f : R C () (2) f(x) = a 0 2 + (a n cos nx + b n sin nx), n= a n = f(x) cos nx dx, b n = π π f(x) sin nx dx a n, b n f Fourier, (3) f Fourier or No. ) 5, Fourier (3) (4) f(x) = c n = n=
More information(w) F (3) (4) (5)??? p8 p1w Aさんの 背 中 が 壁 を 押 す 力 垂 直 抗 力 重 力 静 止 摩 擦 力 p8 p
F 1-1................................... p38 p1w A A A 1-................................... p38 p1w 1-3................................... p38 p1w () (1) ()?? (w) F (3) (4) (5)??? -1...................................
More information0 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 informationf (x) f (x) f (x) f (x) f (x) 2 f (x) f (x) f (x) f (x) 2 n f (x) n f (n) (x) dn f f (x) dx n dn dx n D n f (x) n C n C f (x) x = a 1 f (x) x = a x >
5.1 1. x = a f (x) a x h f (a + h) f (a) h (5.1) h 0 f (x) x = a f +(a) f (a + h) f (a) = lim h +0 h (5.2) x h h 0 f (a) f (a + h) f (a) f (a h) f (a) = lim = lim h 0 h h 0 h (5.3) f (x) x = a f (a) =
More information330
330 331 332 333 334 t t P 335 t R t t i R +(P P ) P =i t P = R + P 1+i t 336 uc R=uc P 337 338 339 340 341 342 343 π π β τ τ (1+π ) (1 βτ )(1 τ ) (1+π ) (1 βτ ) (1 τ ) (1+π ) (1 τ ) (1 τ ) 344 (1 βτ )(1
More informationJune 2016 i (statistics) F Excel Numbers, OpenOffice/LibreOffice Calc ii *1 VAR STDEV 1 SPSS SAS R *2 R R R R *1 Excel, Numbers, Microsoft Office, Apple iwork, *2 R GNU GNU R iii URL http://ruby.kyoto-wu.ac.jp/statistics/training/
More information受賞講演要旨2012cs3
アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート アハ ート α β α α α α α
More information70 : 20 : A B (20 ) (30 ) 50 1
70 : 0 : A B (0 ) (30 ) 50 1 1 4 1.1................................................ 5 1. A............................................... 6 1.3 B............................................... 7 8.1 A...............................................
More information4 2 4.1: =, >, < π dθ = dφ = 0 3 4 K = 1/R 2 rdr + udu = 0 dr 2 + du 2 = dr 2 + r2 1 R 2 r 2 dr2 = 1 r 2 /R 2 = 1 1 Kr 2 (4.3) u iu,r ir K = 1/R 2 r R
1 4 4.1 1922 1929 1947 1965 2.726 K WMAP 2003 1. > 100Mpc 2. 10 5 3. 1. : v = ȧ(t) = Ha [ ] dr 2. : ds 2 = c 2 dt 2 a(t) 2 2 1 kr 2 + r2 (dθ 2 + sin 2 θdφ 2 ) a(t) H k = +1 k *1) k = 0 k = 1 dl 2 = dx
More informationi 0 1 0.1 I................................................ 1 0.2.................................................. 2 0.2.1...........................
2008 II 21 1 31 i 0 1 0.1 I................................................ 1 0.2.................................................. 2 0.2.1............................................. 2 0.2.2.............................................
More informationNumRu::GPhys::EP Flux 2 2 NumRu::GPhys::EP Flux 3 2.................................. 3 2.2 EP............................. 4 2.3.....................
NumRu::GPhys::EP Flux 7 2 9 NumRu::GPhys::EP Flux 2 2 NumRu::GPhys::EP Flux 3 2.................................. 3 2.2 EP............................. 4 2.3................................. 5 2.4.............................
More informationM ω f ω = df ω = i ω idx i f x i = ω i, i = 1,..., n f ω i f 2 f 2 f x i x j x j x i = ω i x j = ω j x i, 1 i, j n (3) (3) ω 1.4. R 2 ω(x, y) = a(x, y
1 1.1 M n p M T p M Tp M p (x 1,..., x n ) x 1,..., x n T p M dx 1,..., dx n Tp M dx i dx i ( ) = δj i x j Tp M Tp M i a idx i 1.1. M x M ω(x) Tx M ω(x) = n ω i (x)dx i i=1 ω i C r ω M C r C ω( x i ) C
More information25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
More information…K…E…X„^…x…C…W…A…fi…l…b…g…‘†[…N‡Ì“‚¢−w‘K‡Ì‹ê™v’«‡É‡Â‡¢‡Ä
2009 8 26 1 2 3 ARMA 4 BN 5 BN 6 (Ω, F, µ) Ω: F Ω σ 1 Ω, ϕ F 2 A, B F = A B, A B, A\B F F µ F 1 µ(ϕ) = 0 2 A F = µ(a) 0 3 A, B F, A B = ϕ = µ(a B) = µ(a) + µ(b) µ(ω) = 1 X : µ X : X x 1,, x n X (Ω) x 1,,
More information2011de.dvi
211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
More information平成18年度弁理士試験本試験問題とその傾向
CBA CBA CBA CBA CBA CBA Vol. No. CBA CBA CBA CBA a b a bm m swkmsms kgm NmPa WWmK σ x σ y τ xy θ σ θ τ θ m b t p A-A' σ τ A-A' θ B-B' σ τ B-B' A-A' B-B' B-B' pσ σ B-B' pτ τ l x x I E Vol. No. w x xl/ 3
More information: , 2.0, 3.0, 2.0, (%) ( 2.
2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................
More information34 2 2 h = h/2π 3 V (x) E 4 2 1 ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2 ガウス 型 関 数 1.2 1 関 数 値 0.8 0.6 0.4 0.2 0 15 10 5 0 5 10
33 2 2.1 2.1.1 x 1 T x T 0 F = ma T ψ) 1 x ψ(x) 2.1.2 1 1 h2 d 2 ψ(x) + V (x)ψ(x) = Eψ(x) (2.1) 2m dx 2 1 34 2 2 h = h/2π 3 V (x) E 4 2 1 ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2
More information0406_total.pdf
59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ
More informationb3e2003.dvi
15 II 5 5.1 (1) p, q p = (x + 2y, xy, 1), q = (x 2 + 3y 2, xyz, ) (i) p rotq (ii) p gradq D (2) a, b rot(a b) div [11, p.75] (3) (i) f f grad f = 1 2 grad( f 2) (ii) f f gradf 1 2 grad ( f 2) rotf 5.2
More informationgenron-3
" ( K p( pasals! ( kg / m 3 " ( K! v M V! M / V v V / M! 3 ( kg / m v ( v "! v p v # v v pd v ( J / kg p ( $ 3! % S $ ( pv" 3 ( ( 5 pv" pv R" p R!" R " ( K ( 6 ( 7 " pv pv % p % w ' p% S & $ p% v ( J /
More informationA B 5 C 9 3.4 7 mm, 89 mm 7/89 = 3.4. π 3 6 π 6 6 = 6 π > 6, π > 3 : π > 3
π 9 3 7 4. π 3................................................. 3.3........................ 3.4 π.................... 4.5..................... 4 7...................... 7..................... 9 3 3. p
More information1 180m g 10m/s 2 2 6 1 3 v 0 (t=0) z max t max t z = z max 1 2 g(t t max) 2 (6) 1.3 2 3 3 r = (x, y, z) e x, e y, e z r = xe x + ye y + ze z. (7) v =
1. 2. 3 3. 4. 5. 6. 7. 8. 9. I http://risu.lowtem.hokudai.ac.jp/ hidekazu/class.html 1 1.1 1 a = g, (1) v = g t + v 0, (2) z = 1 2 g t2 + v 0 t + z 0. (3) 1.2 v-t. z-t. z 1 z 0 = dz = v, t1 dv v(t), v
More information入試の軌跡
4 y O x 4 Typed by L A TEX ε ) ) ) 6 4 ) 4 75 ) http://kumamoto.s.xrea.com/plan/.. PDF) Ctrl +L) Ctrl +) Ctrl + Ctrl + ) ) Alt + ) Alt + ) ESC. http://kumamoto.s.xrea.com/nyusi/kumadai kiseki ri i.pdf
More information2 (f4eki) ρ H A a g. v ( ) 2. H(t) ( ) Chapter 5 (f5meanfp) ( ( )? N [] σ e = 8π ( ) e mc 2 = cm 2 e m c (, Thomson cross secion). Cha
http://astr-www.kj.yamagata-u.ac.jp/~shibata P a θ T P M Chapter 4 (f4a). 2.. 2. (f4cone) ( θ) () g M θ (f4b) T M L 2 (f4eki) ρ H A a g. v ( ) 2. H(t) ( ) Chapter 5 (f5meanfp) ( ( )? N [] σ e = 8π ( )
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 informationuntitled
B2 3 2005 (10:30 12:00) 201 2005/10/04 10/04 10/11 9, 15 10/18 10/25 11/01 17 20 11/08 11/15 22 11/22 11/29 ( ) 12/06 12/13 L p L p Hölder 12/20 1/10 1/17 ( ) URL: http://www.math.tohoku.ac.jp/ hattori/hattori.htm
More information662/04-直立.indd
l l q= / D s HTqq /L T L T l l ε s ε = D + s 3 K = αγk R 4 3 K αγk + ( α + β ) K 4 = 0 γ L L + K R K αβγ () ㅧ ర ㅧ ర (4) (5) ()ᑼ (6) (8) (9) (0) () () (3) (3) (7) Ƚˎȁ Ȇ ၑა FYDFM වႁ ޙ 䊶䊶 䊶 䊶䊶 䊶 Ƚˏȁζ υ ίυέρθ
More information第 1 章 書 類 の 作 成 倍 角 文 字 SGML 系 書 類 のみ 使 用 できます 文 字 修 飾 改 行 XML 系 書 類 では 文 字 修 飾 ( 半 角 / 下 線 / 上 付 / 下 付 )と 改 行 が 使 用 できます SGML 系 書 類 では 文 字 修 飾 ( 半 角
1.2 HTML 文 書 の 作 成 基 準 1.2.2 手 続 書 類 で 使 用 できる 文 字 全 角 文 字 手 続 書 類 で 使 用 できる 文 字 種 類 文 字 修 飾 について 説 明 します 参 考 JIS コードについては 付 録 J JIS-X0208-1997 コード 表 をご 覧 ください XML 系 SGML 系 共 通 JIS-X0208-1997 情 報 交 換 用
More informationL A TEX ver L A TEX LATEX 1.1 L A TEX L A TEX tex 1.1 1) latex mkdir latex 2) latex sample1 sample2 mkdir latex/sample1 mkdir latex/sampl
L A TEX ver.2004.11.18 1 L A TEX LATEX 1.1 L A TEX L A TEX tex 1.1 1) latex mkdir latex 2) latex sample1 sample2 mkdir latex/sample1 mkdir latex/sample2 3) /staff/kaede work/www/math/takase sample1.tex
More informationz z x = y = /x lim y = + x + lim y = x (x a ) a (x a+) lim z z f(z) = A, lim z z g(z) = B () lim z z {f(z) ± g(z)} = A ± B (2) lim {f(z) g(z)} = AB z
Tips KENZOU 28 6 29 sin 2 x + cos 2 x = cos 2 z + sin 2 z = OK... z < z z < R w = f(z) z z w w f(z) w lim z z f(z) = w x x 2 2 f(x) x = a lim f(x) = lim f(x) x a+ x a z z x = y = /x lim y = + x + lim y
More informationA (1) = 4 A( 1, 4) 1 A 4 () = tan A(0, 0) π A π
4 4.1 4.1.1 A = f() = f() = a f (a) = f() (a, f(a)) = f() (a, f(a)) f(a) = f 0 (a)( a) 4.1 (4, ) = f() = f () = 1 = f (4) = 1 4 4 (4, ) = 1 ( 4) 4 = 1 4 + 1 17 18 4 4.1 A (1) = 4 A( 1, 4) 1 A 4 () = tan
More informationSO(2)
TOP URL http://amonphys.web.fc2.com/ 1 12 3 12.1.................................. 3 12.2.......................... 4 12.3............................. 5 12.4 SO(2).................................. 6
More informationesba.dvi
Ehrenberg-Siday-Bohm-Aharonov 1. Aharonov Bohm 1) 0 A 0 A A = 0 Z ϕ = e A(r) dr C R C e I ϕ 1 ϕ 2 = e A dr = eφ H Φ Φ 1 Aharonov-Bohm Aharonov Bohm 10 Ehrenberg Siday 2) Ehrenberg-Siday-Bohm-Aharonov ESBA(
More informationA. Fresnel) 19 1900 (M. Planck) 1905 (A. Einstein) X (A. Ampère) (M. Faraday) 1864 (C. Maxwell) 1871 (H. R. Hertz) 1888 2.2 1 7 (G. Galilei) 1638 2
1 2012.8 e-mail: tatekawa (at) akane.waseda.jp 1 2005-2006 2 2009 1-2 3 x t x t 2 2.1 17 (I. Newton) C. Huygens) 19 (T. Young) 1 A. Fresnel) 19 1900 (M. Planck) 1905 (A. Einstein) X (A. Ampère) (M. Faraday)
More information1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3.....................................
1 1 3 1.1 (Frequecy Tabulatios)................................ 3 1........................................ 8 1.3........................................... 1 17.1................................................
More information5.. 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ボールねじ
A A 506J A15-6 A15-8 A15-8 A15-11 A15-11 A15-14 A15-19 A15-20 A15-24 A15-24 A15-26 A15-27 A15-28 A15-30 A15-32 A15-35 A15-35 A15-38 A15-38 A15-39 A15-40 A15-43 A15-43 A15-47 A15-47 A15-47 A15-47 A15-49
More informationA 2008 10 (2010 4 ) 1 1 1.1................................. 1 1.2..................................... 1 1.3............................ 3 1.3.1............................. 3 1.3.2..................................
More information1 1 ( ) ( 1.1 1.1.1 60% mm 100 100 60 60% 1.1.2 A B A B A 1
1 21 10 5 1 E-mail: qliu@res.otaru-uc.ac.jp 1 1 ( ) ( 1.1 1.1.1 60% mm 100 100 60 60% 1.1.2 A B A B A 1 B 1.1.3 boy W ID 1 2 3 DI DII DIII OL OL 1.1.4 2 1.1.5 1.1.6 1.1.7 1.1.8 1.2 1.2.1 1. 2. 3 1.2.2
More information1.1 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 information1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
More information) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
More information( ) 2.1. C. (1) x 4 dx = 1 5 x5 + C 1 (2) x dx = x 2 dx = x 1 + C = 1 2 x + C xdx (3) = x dx = 3 x C (4) (x + 1) 3 dx = (x 3 + 3x 2 + 3x +
(.. C. ( d 5 5 + C ( d d + C + C d ( d + C ( ( + d ( + + + d + + + + C (5 9 + d + d tan + C cos (sin (6 sin d d log sin + C sin + (7 + + d ( + + + + d log( + + + C ( (8 d 7 6 d + 6 + C ( (9 ( d 6 + 8 d
More informationa (a + ), a + a > (a + ), a + 4 a < a 4 a,,, y y = + a y = + a, y = a y = ( + a) ( x) + ( a) x, x y,y a y y y ( + a : a ) ( a : a > ) y = (a + ) y = a
[] a x f(x) = ( + a)( x) + ( a)x f(x) = ( a + ) x + a + () x f(x) a a + a > a + () x f(x) a (a + ) a x 4 f (x) = ( + a) ( x) + ( a) x = ( a + a) x + a + = ( a + ) x + a +, () a + a f(x) f(x) = f() = a
More information7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6
26 11 5 1 ( 2 2 2 3 5 3.1...................................... 5 3.2....................................... 5 3.3....................................... 6 3.4....................................... 7
More information2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =
More informationα = 2 2 α 2 = ( 2) 2 = 2 x = α, y = 2 x, y X 0, X 1.X 2,... x 0 X 0, x 1 X 1, x 2 X 2.. Zorn A, B A B A B A B A B B A A B N 2
1. 2. 3. 4. 5. 6. 7. 8. N Z 9. Z Q 10. Q R 2 1. 2. 3. 4. Zorn 5. 6. 7. 8. 9. x x x y x, y α = 2 2 α x = y = 2 1 α = 2 2 α 2 = ( 2) 2 = 2 x = α, y = 2 x, y X 0, X 1.X 2,... x 0 X 0, x 1 X 1, x 2 X 2.. Zorn
More information24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x
24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),
More informationx (0, 6, N x 2 (4 + 2(4 + 3 < 6 2 3, a 2 + a 2+ > 0. x (0, 6 si x > 0. (2 cos [0, 6] (0, 6 (cos si < 0. ( 5.4.6 (2 (3 cos 0, cos 3 < 0. cos 0 cos cos
6 II 3 6. π 3.459... ( /( π 33 π 00 π 34 6.. ( (a cos π 2 0 π (0, 2 3 π (b z C, m, Z ( ( cos z + π 2 (, si z + π 2 (cos z, si z, 4m, ( si z, cos z, 4m +, (cos z, si z, 4m + 2, (si z, cos z, 4m + 3. (6.
More information応用数学III-4.ppt
III f x ( ) = 1 f x ( ) = P( X = x) = f ( x) = P( X = x) =! x ( ) b! a, X! U a,b f ( x) =! " e #!x, X! Ex (!) n! ( n! x)!x! " x 1! " x! e"!, X! Po! ( ) n! x, X! B( n;" ) ( ) ! xf ( x) = = n n!! ( n
More informationB. 41 II: 2 ;; 4 B [ ] S 1 S 2 S 1 S O S 1 S P 2 3 P P : 2.13:
B. 41 II: ;; 4 B [] S 1 S S 1 S.1 O S 1 S 1.13 P 3 P 5 7 P.1:.13: 4 4.14 C d A B x l l d C B 1 l.14: AB A 1 B 0 AB 0 O OP = x P l AP BP AB AP BP 1 (.4)(.5) x l x sin = p l + x x l (.4)(.5) m d A x P O
More information136 pp p µl µl µl
135 2006 PCB C 12 H 10-n Cl n n 1 10 CAS No. 42 PCB: 53469-21-9, 54 PCB: 11097-69-1 0.01 mg/m 3 PCB PCB 25 µg/l 136 pp p µl µl µl 137 1 γ 138 1 γ γ γ µl µl µl µl µl µl µl l µl µl µl µl µl l 139 µl µl µl
More information2 p T, Q
270 C, 6000 C, 2 p T, Q p: : p = N/ m 2 N/ m 2 Pa : pdv p S F Q 1 g 1 1 g 1 14.5 C 15.5 1 1 cal = 4.1855 J du = Q pdv U ( ) Q pdv 2 : z = f(x, y). z = f(x, y) (x 0, y 0 ) y y = y 0 z = f(x, y 0 ) x x =
More information基礎から学ぶトラヒック理論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. 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.
基礎から学ぶトラヒック理論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/085221 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 +α 3 1 2 4 5 1 2 ii 3 4 5 6 7 8 9 9.3 2014 6 iii 1 1 2 5 2.1 5 2.2 7
More information2 T(x - v τ) i ix T(x + v τ) i ix x T = ((dt/dx),, ) ( q = c T (x i ) v i ( ) ) dt v ix τ v i dx i i ( (dt = cτ ) ) v 2 dx ix,, () i x = const. FIG. 2
Y. Kondo Department of Physics, Kinki University, Higashi-Osaka, Japan (Dated: September 3, 27) [] PACS numbers: I. m cm 3 24 e =.62 9 As m = 9.7 3 kg A. Drude-orentz Drude orentz N. i v i j = N q i v
More informationDecember 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
More informationE F = q b E (2) E q a r q a q b N/C q a (electric flux line) q a E r r r E 4πr 2 E 4πr 2 = k q a r 2 4πr2 = 4πkq a (3) 4πkq a 1835 4πk 1 ɛ 0 ɛ 0 (perm
1 1.1 18 (static electricity) 20 (electric charge) A,B q a, q b r F F = k q aq b r 2 (1) k q b F F q a r?? 18 (Coulomb) 1 N C r 1m 9 10 9 N 1C k 9 10 9 Nm 2 /C 2 1 k q a r 2 (Electric Field) 1 E F = q
More information