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

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

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

Transcription

1 Hanbury-Brown Twiss (ver. 1.) van Cittert - Zernike mutual coherence Hanbury-Brown Twiss ( ) HBT Mandel A Hanbury-Brown Twiss Mark Fox Quantum Optics An Introduction 1 1

2 kawa (Spin-Orbit Tomography: Kawahara and Fujii 21,Kawahara and Fujii 211,Fujii & Kawahara submitted) 2 van Cittert-Zernike Appendix A V 2.1 van Cittert - Zernike (mutual coherence function) Γ(Q 1, Q 2, τ) V (Q 1, t)v (Q 2, t + τ) (1) (complex degree of coherence) γ(q 1, Q 2, τ) V (Q 1, t)v (Q 2, t + τ) I(Q1 )I(Q 2 ) (2) Sm Rm2 Rm1 P2 P1 1: S m P 1, P 2 1 S m A m P 1 P 2 V m1 (t) = A m (t R m1 /c) e iω(t Rm1/c) R m1 (3) V m2 (t) = A m (t R m2 /c) e iω(t Rm2/c) R m2 (4) 2

3 P 1 P 2 Γ(P 1, P 2, ) V (P 1 )V (P 2 ) (5) = m A m(t R m1 /c)a m (t R m2 /c) e iω(r m2 R m1 )/c R m1 R m2 (6) R m1 R m2 2πc/ ω A m Γ(P 1, P 2, ) = m (7) A m(t)a m (t) e iω(r m2 R m1 )/c R m1 R m2 (8) I(S) e iω(r2 R1)/c R 1 R 2 ds (9) R m1 R 1 R m2 R 2 1 γ(p 1, P 2, ) = I(S) e iω(r 2 R 1 )/c ds (1) I(P1 )I(P 2 ) R 1 R 2 I(S) I(P j ) Γ(P j, P j, ) = ds (11) van Cittert-Zernike (α, β) P 1,P 2 (X 1, Y 1 ), (X 2, Y 2 ) R 2 j γ(p 1, P 2, ) = eiψ dαdβi(α, β)e ik(αx+βy) dαdβi(α, β) (12) ψ k[(x2 2 + Y 2 2 ) (X Y 2 1 )] 2R x X 2 X 1, y Y 2 Y 1 P 1, P 2 ( ) mutual coherence ( ) P 1 P 2 (13) 2.2 mutual coherence mutual coherence 12 ρ b γ(p 1, P 2, ) = 2J 1(ν) e iψ (14) ν ν kρb (15) 3

4 γ(p 1, P 2, ) = 2J 1(ν) ν mutual coherence (16) b =.61 λ ρ (17) 6.3 mas ( ).5 µ m 1 m m (18).63π/(36 18) P 1 P 2 Q V (Q, t) = k 1 V (P 1, t t 1 ) + k 2 V (P 2, t t 2 ) (19) I(Q) = V (Q, t)v (Q, t) (2) = k 1 2 V (P 1, t t 1 )V (P 1, t t 1 ) + k 2 2 V (P 2, t t 2 )V (P 2, t t 2 ) + 2Re[ k 1 k 2 V (P 1, t t 1 )V (P 2, t t 2 ) ] (21) = k 1 2 I(P 1 ) + k 2 2 I(P 2 ) + 2 k 1 k 2 Re[Γ(P 1, P 2, t 1 t 2 )] (22) = I (1) (Q) + I (2) (Q) + 2 I (1) (Q)I (2) (Q)Re[γ(P 1, P 2, t 1 t 2 )] (23) (I (j) (Q) ) γ(p 1, P 2, t 1 t 2 ) A(t) Φ(t) δ = ντ = 2π(R 2 R 1 )/λ 2I (1) (Q)(1 ± γ(p 1, P 2, t 1 t 2 ) ) (24) (I (1) = I (2) ), mutual coherence visibility γ(p 1, P 2, τ) = I max(p ) I min (P ) I max (P ) + I min (P ) τ = visibility γ(p 1, P 2, ) (25) 3 Hanbury-Brown Twiss ( ) visibility mutual coherence ( ) mutual coherence phase 4

5 P2 Q P1 2: P2 P1 3: Hanbury-Brown ( ) I(r j, t) I(r j, t) I(r j, t) (26) I(r 1, t) I(r 2, t + τ) = (I(r 1, t) I(r 1, t) )(I(r 2, t + τ) I(r 2, t + τ) ) (27) = I(r 1, t)i(r 2, t + τ) I(r 1, t) I(r 2, t + τ) (28) = V (r 1, t)v (r 1, t)v (r 2, t + τ)v (r 2, t + τ) V (r 1, t)v (r 1, t) V (r 2, t + τ)v (r 2, t + τ) (29) V x j Lsserlis x 1x 2 x 3x 4 = x 1x 2 x 3x 4 + x 1x 4 x 2 x 3 I(r 1, t) I(r 2, t + τ) = V (r 1, t)v (r 2, t + τ) V (r 2, t + τ)v (r 1, t) (3) = Γ(r 1, r 2, τ)γ(r 1, r 2, τ) (31) = Γ(r 1, r 2, τ) 2 (32) 2 2 Γ(r 1, r 2, τ) = Γ (r 1, r 2, τ) 5

6 3.2 HBT Hanbury-Brown Twiss Hanbury-Brown Narrabri Stellar Intensity Interferometer 32 (Hanbury-Brown, Davis, Allen 1974) 32 mas ζp up.41 ±.3 mas (1969 ) 4 Bigot et al mas beam spliter 2 PhotoMultiplier 1 PhotoMultiplier correlator 4: HBT (33) g 2 ( ) g 2 1 g 2 (τ) = γ(r 1, r 2, τ) (33) g 2 (τ) I(r 1, t)i(r 2, t + τ) (34) I(r 1, t) I(r 2, t + τ) 3.3 Mandel Mandel HBT Mandel t-t + t I(t) = V (t)v (t) P (t) = αi(t) t (35) α t t + T n p(n, t, T ) T T/ t 6

7 t r1,..., t rn ( ) 1 T/ t T/ t T/ t rn T/ t p(n, t, T ) = lim... (α t) n I(t t n! r ) i= [1 αi(t i)δt] n j=1 [1 αi(t (36) rj)δt] r1= r2= 1 3 = lim { 2} t n! 1 rn= r=r1 (37) 1 1 no(δt) 1 (38) n [ T/δt ] n t+t 2 αi(t r1 )δt α I(t )dt (39) 3 exp r 1= [ α t+t t I(t )dt ] t (4) p(n, t, T ) = 1 n! [αw (t, T )]n e αw (t,t ) (41) W (t, T ) t+t t I(t )dt (42) I W p(w ) { } 1 P (n, t, T ) = p(n, t, T ) = dw p(w ) n! [αw (t, T )]n αw (t,t ) e (43) p(n, t, T ) = dw p(w )P p (n, W ) (44) ( dw p(w )f(w ) f(w ) ) Mandek {} P p (n, W ) t t + T n = = np(n, t, T ) = n= dw p(w ) np p (n, W ) (45) n= dw p(w )αw (46) = αw (47) n 2 = np(n, t, T ) = dw p(w ) n 2 P p (n, W ) (48) = n= n= dw p(w )(αw + α 2 W 2 ) (49) = αw + α 2 W 2 (5) ( n) 2 = n 2 n 2 = αw + α 2 W 2 α 2 W 2 (51) = n + α 2 [ W ] 2 (52) 7

8 Intensity ( n) 2 > n ( ) ( n) 2 = n ( n) 2 < n ( ) HBT Mandel n 1 n 2 = = n 1 n 2 p 1 (n 1, t, T )p 2 (n 2, t, T ) (53) n 1 = n 2 = n 1 p 1 (n 1, t, T ) n 2 p 2 (n 2, t, T ) = α 1 α 2 W 1 W 2 (54) n 1= n 2= n 1 n 2 = n 1 n 2 n 1 n 2 = α 1 α 2 W 1 W 2 (55) W j W j W j (56) W 3.4 E n = (n + 1/2)ħω (57) n : P ω (n) = exp ( E n /kt ) n= exp ( E n/kt ) (58) = x n n= xn (59) = x n (1 x) (6) x exp ( ħω/kt ) (61) 8

9 n= xn = 1/(1 x) (x < 1) n = = np ω (n) (62) n= nx n (1 x) (63) n= = (1 x)x d dx = (1 x)x d dx x = 1 x ( ) x n n= ( 1 1 x ) (64) (65) (66) (67) n = 1 exp (ħω/kt ) 1 (68) P ω (n) n Bose-Einstein ( n) 2 = P ω (n) = 1 ( ) n n (69) n + 1 n + 1 (n n) 2 P ω (n) = n + n 2 (7) n= N m (Mandel & Wolf 95) ( n) 2 = n + n 2 /N m (71) 68 HBT 3.5 HBT HBT 3 3 Jam session 9

10 1: creation and annihilation operators : â n = n + 1 n + 1 â n = n n 1 â = [â, â ] = 1 number operator : ˆn = â â ˆn n = n n Hamiltonian : Ĥ = ħω (ˆn ) Ĥ ψ = ħω ( n + 2) 1 ψ HBT 4 g 2 (τ) g 2 (τ) = n 1(t)n 2 (t + τ) n 1 (t) n 2 (t + τ) (72) g 2 (τ) = â 1 (t)â 2 (t + τ)â 2(t + τ)â 1 (t) â 1 (t)â 1(t) â 2 (t + τ)â 2(t + τ) (73) normal ordering (Mandel & Wolf 95) â 1 = â / 2 (74) â 2 = â / 2 (75) â 1â1 = ψ â â ψ /2 = ψ ˆn ψ /2 (76) â 2â2 = ψ â â ψ /2 = ψ ˆn ψ /2 (77) â 1â 2â2â 1 = ψ â â ââ ψ /4 (78) = ψ â (â â 1)â ψ /4 (79) = ψ ˆn (ˆn 1) ψ /4 (8) (81) g 2 (τ) = ˆn(ˆn 1) ˆn 2 (82) 1

11 ( ) photon number state: ψ photon number state n coherent state: g 2 (τ) = coherent state α â α = α α : n(n 1) n 2 < 1 (83) ( n) 2 = n (ˆn n) 2 n (84) = n ˆn 2 n n 2 = (85) α â â ââ α = α 4 (86) α â â α = α 2 (87) g 2 (τ) = 1 (88) ( n) 2 = α (ˆn n) 2 α (89) = α ˆn 2 α n 2 (9) = α â ââ â α n 2 (91) = α â â + â â ââ α n 2 (92) = (n + n 2 ) n 2 = n (93) 4 (?) A V (r) (t) a(ν) cos (ϕ(ν) 2πνt) V (r) (t) = 4 TeX dν a(ν) cos (ϕ(ν) 2πνt) (94) 11

12 2: g 2 (τ) > 1 ( n) 2 > n g 2 (τ) = 1 ( n) 2 = n g 2 (τ) < 1 ( n) 2 < n 5 V (t) = V (r) (t) + iv (i) (t) = V (i) (t) dν a(ν)e i(ϕ(ν) 2πνt) (95) dν a(ν) sin (ϕ(ν) 2πνt) (96) ν ν δν/ν 1 ν ν = ν V (t) = A(t)e i(φ(t) 2πνt) = (A(t) e iφ(t) ) e 2πiνt (97) A Φ (95) (97) A(t) e iφ(t) = { V (t) = (A(t) e iφ(t) ) e 2πiνt = µ [ dµ a(µ) e iϕ(µ)] e 2πµt (98) µ ν ν (99) µ dµ [a(µ) e iϕ(µ)] } e 2πµt e 2πiνt (1) a(µ) µ = ν ν = {} µ = ν ν ν e 2πiνt ν µ (97) A(t) e iφ(t) A(t) Φ(t) ν A(t) Φ(t) 5 12

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

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

gr09.dvi

gr09.dvi .1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {

More information

1 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

= 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

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

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

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

1

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

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

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

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

『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

研修コーナー

研修コーナー l l l l l l l l l l l α α β l µ l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

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

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

<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

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

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

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

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

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

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

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

[ ] = 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

(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

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

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

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

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

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

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

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

O1-1 O1-2 O1-3 O1-4 O1-5 O1-6

O1-1 O1-2 O1-3 O1-4 O1-5 O1-6 O1-1 O1-2 O1-3 O1-4 O1-5 O1-6 O1-7 O1-8 O1-9 O1-10 O1-11 O1-12 O1-13 O1-14 O1-15 O1-16 O1-17 O1-18 O1-19 O1-20 O1-21 O1-22 O1-23 O1-24 O1-25 O1-26 O1-27 O1-28 O1-29 O1-30 O1-31 O1-32 O1-33 O1-34 O1-35

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

現代物理化学 1-1(4)16.ppt

現代物理化学 1-1(4)16.ppt (pdf) pdf pdf http://www1.doshisha.ac.jp/~bukka/lecture/index.html http://www.doshisha.ac.jp/ Duet -1-1-1 2-a. 1-1-2 EU E = K E + P E + U ΔE K E = 0P E ΔE = ΔU U U = εn ΔU ΔU = Q + W, du = d 'Q + d 'W

More information

lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d

lim lim lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d lim 5. 0 A B 5-5- A B lim 0 A B A 5. 5- 0 5-5- 0 0 lim lim 0 0 0 lim lim 0 0 d lim 5. d 0 d d d d d d 0 0 lim lim 0 d 0 0 5- 5-3 0 5-3 5-3b 5-3c lim lim d 0 0 5-3b 5-3c lim lim lim d 0 0 0 3 3 3 3 3 3

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

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

3/4/8:9 { } { } β β β α β α β β

3/4/8:9 { } { } β β β α β α β β α β : α β β α β α, [ ] [ ] V, [ ] α α β [ ] β 3/4/8:9 3/4/8:9 { } { } β β β α β α β β [] β [] β β β β α ( ( ( ( ( ( [ ] [ ] [ β ] [ α β β ] [ α ( β β ] [ α] [ ( β β ] [] α [ β β ] ( / α α [ β β ] [ ] 3

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

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

202

202 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 DS =+α log (Spread )+ β DSRate +γlend +δ DEx DS t Spread t 1 DSRate t Lend t DEx DS DEx Spread DS

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

,, 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

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

3 ( 9 ) ( 13 ) ( ) 4 ( ) (3379 ) ( ) 2 ( ) 5 33 ( 3 ) ( ) 6 10 () 7 ( 4 ) ( ) ( ) 8 3() 2 ( ) 9 81

3 ( 9 ) ( 13 ) ( ) 4 ( ) (3379 ) ( ) 2 ( ) 5 33 ( 3 ) ( ) 6 10 () 7 ( 4 ) ( ) ( ) 8 3() 2 ( ) 9 81 1 ( 1 8 ) 2 ( 9 23 ) 3 ( 24 32 ) 4 ( 33 35 ) 1 9 3 28 3 () 1 (25201 ) 421 5 ()45 (25338 )(2540 )(1230 ) (89 ) () 2 () 3 ( ) 2 ( 1 ) 3 ( 2 ) 4 3 ( 9 ) ( 13 ) ( ) 4 ( 43100 ) (3379 ) ( ) 2 ( ) 5 33 ( 3 )

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

<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

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

< 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

2

2 16 1050026 1050042 1 2 1 1.1 3 1.2 3 1.3 3 2 2.1 4 2.2 4 2.2.1 5 2.2.2 5 2.3 7 2.3.1 1Basic 7 2.3.2 2 8 2.3.3 3 9 2.3.4 4window size 10 2.3.5 5 11 3 3.1 12 3.2 CCF 1 13 3.3 14 3.4 2 15 3.5 3 17 20 20 20

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

22 / ( ) OD (Origin-Destination)

22 / ( ) OD (Origin-Destination) 23 2 15 22 / ( ) OD (Origin-Destination) 1 1 2 3 2.1....................................... 3 2.2......................................... 3 2.3.......................................... 5 2.4............................

More information

2 FIG. 1: : n FIG. 2: : n (Ch h ) N T B Ch h n(z) = (sin ϵ cos ω(z), sin ϵ sin ω(z), cos ϵ), (1) 1968 Meyer [5] 50 N T B Ch h [4] N T B 10 nm Ch h 1 µ

2 FIG. 1: : n FIG. 2: : n (Ch h ) N T B Ch h n(z) = (sin ϵ cos ω(z), sin ϵ sin ω(z), cos ϵ), (1) 1968 Meyer [5] 50 N T B Ch h [4] N T B 10 nm Ch h 1 µ : (Dated: February 5, 2016), (Ch), (Oblique Helicoidal) (Ch H ), Twist-bend (N T B ) I. (chiral: ) (achiral) (n) (Ch) (N ) 1996 [1] [2] 2013 (N T B ) [3] 2014 [4] (oblique helicoid) 2016 1 29 Electronic

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

PRML pdf PRML (http://critter.sakura.ne.jp) N x t y(x, w) = w 0 + w 1 x + w 2 x w M x m = M w j x j (1.1) j=0 E(w) = 1 {y(x n, w) t n } 2

PRML pdf PRML (http://critter.sakura.ne.jp) N x t y(x, w) = w 0 + w 1 x + w 2 x w M x m = M w j x j (1.1) j=0 E(w) = 1 {y(x n, w) t n } 2 critter twitter ( PRML) PRML PRML PRML PRML 1. 2. 3. PRML PRML 110 PRML 700 1 PRML pdf PRML (http://critter.sakura.ne.jp) 1 1.1 N x t y(x, w) = w 0 + w 1 x + w 2 x 2 + + w M x m = M w j x j (1.1) j=0 E(w)

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

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

untitled

untitled 1 (1) (2) (3) (4) (1) (2) (3) (1) (2) (3) (1) (2) (3) (4) (5) (1) (2) (3) (1) (2) 10 11 12 2 2520159 3 (1) (2) (3) (4) (5) (6) 103 59529 600 12 42 4 42 68 53 53 C 30 30 5 56 6 (3) (1) 7 () () (()) () ()

More information

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

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 III 1 2005 Jan 30th, 2006 I : II : I : [ I ] 12 13 9 (Landau and Lifshitz, Quantum Mechanics chapter 12, 13, 9: Pergamon Pr.) [ ] ( ) (H. Georgi, Lie algebra in particle physics, Perseus Books) [ ] II

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

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

176 B B.1: ( ) ( ) ( ) (2 2 ) ( ) ( ) ( ) (quantitative nondestructive evaluation:qnde) (1) X X X X CT(computed tomography)

176 B B.1: ( ) ( ) ( ) (2 2 ) ( ) ( ) ( ) (quantitative nondestructive evaluation:qnde) (1) X X X X CT(computed tomography) B 1) B.1 B.1.1 ( ) B.1 1 50 100 m B.1.2 (nondestructive testing:ndt) (nondestructive inspection:ndi) (nondestructive evaluation:nde) 175 176 B B.1: ( ) ( ) ( ) (2 2 ) ( ) ( ) ( ) (quantitative nondestructive

More information

Q E Q T a k Q Q Q T Q =

Q E Q T a k Q Q Q T Q = i 415 q q q q Q E Q T a k Q Q Q T Q = 10 30 j 19 25 22 E 23 R 9 i i V 25 60 1 20 1 18 59R1416R30 3018 1211931 30025R 10T1T 425R 11 50 101233 162 633315 22E1011 10T q 26T10T 12 3030 12 12 24 100 1E20 62

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

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

KamLAND (µ) ν e RSFP + ν e RSFP(Resonant Spin Flavor Precession) ν e RSFP 1. ν e ν µ ν e RSFP.ν e νµ ν e νe µ KamLAND νe KamLAND (ʼ4). kton-day 8.3 < E ν < 14.8 MeV candidates Φ(νe) < 37 cm - s -1 P(νe

More information

layout_04.indd

layout_04.indd 第 三 種 電 気 主 任 技 術 者 理 論 編 1. 2. 3. 4. 5. 6. 7. 8. 9. Y 10. 11. 12. RL RC 13. 14. RLC 15. RLC 16. 17. 18. 19. 20. 21. Y 22. Y 23. 5 6 7 8 9 10 12 15 17 19 21 23 25 32 35 37 39 41 43 45 47 49 50 52 24. 25.

More information

建築設備学_07(熱負荷計算).ppt

建築設備学_07(熱負荷計算).ppt p. p. p.7 p. q w q w q GT q IT =q IS +q IL () () q HT = q HS + q HL q ET =q ES +q EL 1 () q s [W]C p ρ m /h Δt 1000/00 [W]0.4 m /h Δt q L [W]γ γ [m /h] Δx[g/kg(DA)] 1000/00 [W]4 [m /h] Δx[g/kg(DA)] C p

More information