( ) Rotational Random Shuffling ,2 1
|
|
- きみのしん あんさい
- 5 years ago
- Views:
Transcription
1 27 F13J011D
2 ( ) Rotational Random Shuffling ,2 1
3 (CHPCA) (RMT) Rotational Random Shuffling(RRS) A (PCA) 38 A B 42 B B B C 49 D 50 D D D E PCA CHPCA 54 2
4 1 1.1 (PCA) (RMT) [1 6] PCA RMT RMT Rotational Random Shuffling(RRS) [7] PCA (CHPCA) [8, 9] CHPCA ( ) 3
5 2000 [10] 350 [11] CHPCA [12] CHPCA RMT RRS [12] [19] ( ) 4
6 2 2.1 (CHPCA) (CHPCA) [8,9] x µ (t) (µ = 1,, N) y µ (t) ξ µ (t) = x µ (t) + iy µ (t), (2.1.1) y µ (t) x µ (t) y µ (t) = 1 π x µ (u) du. (2.1.2) t u u. y µ (t) x µ (t) π/2 {ξ µ (t)} C = 1 T ΞΞ, (2.1.3) Ξ = ξ 1 (t) ξ 2 (t). ξ n (t). (2.1.4) Ξ ξ µ (t) N T Ξ Ξ C Cα l = λ l α l, (2.1.5) α l α m = δ lm. (2.1.6) (CHPCA) λ l α l C 5
7 x µ µ = 1 µ = t 2.1: sin cos C C = N λ l = N, (2.1.7) l=1 N λ l αα, (2.1.8) l=1 λ 1 > λ 2 > > λ N CHPCA sin cos x 1 (t) = sin ( ) ( ) πt πt, x 2 (t) = cos (2.1.9) t = 0, 1, 2,, 99, 2.1 PCA ( ) C =, (2.1.10) C 12 = CHPCA ( ) e C π =. (2.1.11) 0.985e 0.515π 1 6
8 ( λ 1 = 1.985, α 1 = 1 2 ( λ = 0.015, α 2 = e 0.515π 1 e π ) ), (2.1.12), (2.1.13) 0.515π 0.5π C 12 = sin cos CHPCA 2.2 (RMT) ρ(λ) RMT ρ(λ) RMT λ λ 2.2: RMT N % 3. 7
9 C m = λ m N i=1 λ. (2.2.1) i m 70% 70% 70% RMT λ Q = T/(2N) > 1 N, T 1 ρ(λ) = Q (λ+ λ)(λ λ ), (2.2.2) 2π λ λ ± = ( 1 ± 1 Q ) 2. (2.2.3) [λ, λ + ] 2.2 λ Rotational Random Shuffling(RRS) RMT RMT 2.2 RMT Rotational Random Shuffling(RRS) RMT RRS 1 PCA Q = T/N CHPCA RMT Q = T/(2N) [13] 8
10 2.3: Rotational Random Shuffling RRS RMT 9
11 3 8 N = T = 360 (CPI) 2 45 (PPI) 3 23 (IPI) 3 10 (USD/JPY) 4 (Leading Index) 5 (Coincident Index) 5 (Lagging Index) 5 (Money Stock) 6 (CPI) (PPI) (PPI) ( ) ( / NEEDS 5 Composite Index M2 M2+CD, M2+CD 10
12 ) (IPI) (IPI) ( CIF ) (CI) (DI) CI ( ) DI ( ) CI DI 3 ( ) 1 r µ (t) = logx µ (t) logx µ (t 1). (3.0.1) r µ (t) = r µ(t) r µ σ µ (3.0.2) 11
13 CPI CGPI IPI USD/JPY Leading Index Coincident Index Lagging Index year 3.1: ( ) 0 1 r µ r µ σ µ r µ Phillips-Perron 4CPI 5CPI 6CPI 24CPI 25CPI 12
14 1: 1 CPI 46 PPI 2 CPI 47 PPI 3 CPI 48 PPI 4 CPI 49 PPI 5 CPI 50 PPI 6 CPI 51 PPI 7 CPI 52 PPI 8 CPI 53 PPI 9 CPI 54 PPI 10 CPI 55 PPI 11 CPI 56 PPI 12 CPI 57 PPI 13 CPI 58 PPI 14 CPI 59 PPI 15 CPI 60 PPI 16 CPI 61 PPI 17 CPI 62 PPI 18 CPI 63 PPI 19 CPI 64 PPI 20 CPI 65 PPI 21 CPI 66 PPI 22 CPI 67 PPI 23 CPI 68 PPI 24 CPI 69 IPI 25 CPI 70 IPI 26 CPI 71 IPI 27 CPI 72 IPI 28 CPI 73 IPI 29 CPI 74 IPI 30 CPI 75 IPI 31 CPI 76 IPI 32 CPI 77 IPI 33 CPI 78 IPI 34 CPI 79 USD/JPY 35 CPI 80 Leading Index 36 CPI 81 Coincident Index 37 CPI 82 Lagging Index 38 CPI 83 Money Stock 39 CPI 40 CPI 41 CPI 42 CPI 43 CPI 44 CPI 45 CPI 13
15 4 4.1 ρ(λ) λ RMT 4.1: 4.1 RMT λ + (= 2.82) 6 (λ 1 = 10.43, λ 2 = 5.97, λ 3 = 5.68, λ 4 = 3.98, λ 5 = 3.04, λ 6 = 2.86) 6 (31.97/83 = 0.385) 38.5% 6 RMT RRS 4.2 RRS RMT RRS 1000 λ l + 3σ l λ l RRS l σ l l l RRS λ l + 3σ l 6 RMT 14
16 ρ(λ) RMT λ l actual RRS RRS+3σ λ l 4.2: RRS (1000 ) α l 2 = N α li 2 = N. (4.2.1) i=1 4.3 Mode (84 ) CHPCA ±2σ 15
17 Eigenvector No.1 Eigenvector No.2 Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re Re Eigenvector No.3 Eigenvector No.4 Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re Re Eigenvector No.5 Eigenvector No.6 Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Im CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re Re 4.3: 16
18 +2σ 4.4 ( )e µ α l N N ξ(t) = ξ µ (t)e µ = a l (t)α l, (4.4.1) µ=1 l=1 a l (t) = α l ξ(t). (4.4.2) a l (t) l a la m = δ lm λ l. (4.4.3) a l (t) 3 6 ( ) 5 cosine similarity cos θ lα = α l α m α l α m. (4.4.4) θ lα 2 α l α m 4.4 α l α m cosine similarity N(=83) p
19 Eigenvector No.1 Eigenvector No Eigenvector No.3 Eigenvector No Eigenvector No.5 Eigenvector No : x y 84 ( ) (1000 ±2σ) 18
20 Mode signal No. 1 Mode signal Re Im time Mode signal No. 2 Mode signal time Re Im Mode signal No. 3 Mode signal Re Im time Mode signal No. 4 Mode signal time Re Im Mode signal No. 5 Mode signal time Re Im Mode signal No. 6 Mode signal time Re Im 4.5: 6 19
21 Mode signal No.1 Mode signal Year Mode signal No.2 Mode signal Year Mode signal No.3 Mode signal Year Mode signal No.4 Mode signal Year Mode signal No.5 Mode signal Year Mode signal No.6 Mode signal Year 4.6: 20
22 Mode signal No.1 Mode signal No Period Period Mode signal No.3 Mode signal No Period Period Mode signal No.5 Mode signal No Period Period : 21
23 2: α l (l = 1,, 6) α m (m = 1,, 5) cosine similarity N(=83) p l\m p cos θ % 2 22
24 π π x y ( ) ( ) ( ) (CPI,PPI) 68 CHPCA ( ) 23
25 3: 1 2 Abs 1 2 Mode 1 Mode 2 Abs θ[rad] Category Abs θ[rad] Category Leading Index Leading Index USD/JPY PPI PPI IPI IPI IPI IPI Coincident Index IPI IPI IPI PPI IPI IPI IPI IPI PPI USD/JPY IPI CPI IPI IPI IPI IPI IPI IPI Coincident Index IPI PPI CPI CPI Lagging Index PPI PPI PPI CPI CPI PPI Lagging Index PPI PPI CPI PPI PPI CPI PPI CPI PPI PPI PPI PPI CPI PPI PPI CPI CPI PPI PPI PPI PPI PPI CPI CPI CPI CPI CPI PPI PPI CPI CPI CPI CPI PPI PPI PPI CPI CPI PPI PPI CPI CPI PPI CPI CPI CPI PPI CPI CPI CPI CPI CPI PPI CPI CPI CPI CPI 24
26 Eigenvector No π 2 π 3π 2 2π θ Eigenvector No π 2 π 3π 2 2π θ 4.8: 1 2 x y π 25
27 θ θ 1 4.9: x 1 y 2 4: α l α m cosine similarity l\m
28 4.7 I l (t) S l (t) I l (t) = a l (t) 2 N l=1 a l(t) 2. (4.7.1) S l (t) = t t =1 I l(t ) T t =1 I l(t ). (4.7.2) 4.11 I l (t) S l (t) l ξ lµ (t) = a l (t)α lµ. (4.7.3) ξ µ (t) (a) (1985.3) 1 2 (b) (1991.3) (c) (1995.9) (d) ( ) (e)
29 Mode signal No.1 Relative Intensity Year Mode signal No.2 Relative Intensity Year Mode signal No.3 Relative Intensity Year Mode signal No.4 Relative Intensity Year Mode signal No.5 Relative Intensity Year Mode signal No.6 Relative Intensity Year 4.10: 28
30 Mode signal No.1 Mode signal No.2 Cumulative Intensity (a) (c) (d) (e) Cumulative Intensity (b) (d) Year Year 4.11: 1 (a) (b) (c) (d) (e) , ( 1 2 )
31 Mode 1 Mode 2 (a) Mode 1 USD/JPY Mode 2 USD/JPY Month Month (b) Mode 1 USD/JPY Mode 2 USD/JPY Month Month (c) Mode 1 USD/JPY Mode 2 USD/JPY Month Month (d) Mode 1 USD/JPY Mode 2 USD/JPY Month Month (e) Mode 1 USD/JPY Month Mode 2 USD/JPY Month 4.12: ( ) 1 2 (a): (1985.9) (b): (1991.3) (c): (d): ( ) (e): 30
32 Mode 1 Mode 2 (a) Mode 1 Coincident Index Mode 2 Coincident Index Month Month (b) Mode 1 Coincident Index Mode 2 Coincident Index Month Month (c) Mode 1 Coincident Index Mode 2 Coincident Index Month Month (d) Mode 1 Coincident Index Mode 2 Coincident Index Month Month (e) Mode 1 Coincident Index Month Mode 2 Coincident Index Month 4.13: ( ) 1 2 (a): (1985.9) (b): (1991.3) (c): (d): ( ) (e): 31
33 Mode 1 + Mode 2 Mode 1 Mode 2 USD/JPY Month USD/JPY 4.14: , ,2 19.8% IT ,2 7 32
34 Mode Year Mode Year Mode 2 Mode Year 4.15:
35 Year Mode 1 + Mode 2 Mode 1 Mode 2 Coincident Index Coincident Index 4.16: 5: IT
36 RMT RRS ,2 CHPCA
37 2 1 2 IT
38 4 OB 37
39 A (PCA) ( ) P {x p }(p = 1, 2,, P ) {x p } M(M P ) {z m } z m = P w pm x p (m = 1, 2,, M) (A.0.1) p=1 z m m {w pm } 1 z 1 {x p } 1 m {z m } {z m }(m = 1, 2,, m 1) 1 P wpm 2 = 1 (A.0.2) p=1 A.1 P N {x np}(n = 1, 2,, N; p = 1, 2,, P ) { x p } {x np } x np = x np x p X x 11 x 12 x 1P x 21 x 22 x 2P X = (A.1.1) (A.1.2) x N1 x N2 x NP 1 z 1 (A.0.1) w 11 w 21 w 1 =. (A.1.3) w P 1 38
40 n 1 z 1 t n1 x n = ( x n1 x n2 x np ) (A.1.4) t n1 = P w p1 x np p=1 = x n w 1 (A.1.5) 1 z 1 t n1 1 N 1 1 t 11 t 21 t 1 = (A.1.6). t N1 t 1 = Xw 1 (A.1.7) 1 t 1 t 1 = 1 N = 1 N = 1 N = 1 N N n=1 t n1 N x n w 1 n=1 ( N P ) w p1 x np n=1 p=1 ( P N ) w p1 x np p=1 1 z 1 σ 2 z 1 n=1 = 0 (A.1.8) σz 2 1 = 1 N 1 t 1 t 1 = 1 N 1 (Xw 1) (Xw 1 ) = w1 Vw 1 0 (A.1.9) 39
41 V V = 1 N 1 X X (A.1.10) (i, j) v ij v ij = = 1 N 1 1 N 1 N x ni x nj n=1 N (x ni x i )(x nj x j ) (A.1.11) n=1 v ij = v ji V = V 1 z 1 (A.0.2) σ 2 z 1 Lagrange Lagrange λ J 1 = w 1 Vw λ(w 1 w 1 1) (A.1.12) J 1 w 1 J 1 w 1 0 J 1 w 1 = = J 1 w 11 J 1 w 21. J 1 w P 1 2 P p=1 v 1pw p1 2 P p=1 v 2pw p1. 2 2λ P p=1 v P pw p1 = 2Vw 1 2λw 1 w 11 w 21. w P 1 = 0 (A.1.13) (V λi)w 1 = 0 (A.1.14) Lagrange λ det V λi = 0 (A.1.15) 40
42 Lagrange λ 1 z 1 w 1 V V P λ P 1 z 1 σ 2 z 1 ( )w 1 1 z 1 σ 2 z 1 (A.1.9) w 1 : (A.1.15) (A.1.9) (A.1.15) w 1 w 1 = 1 σ 2 z 1 = w 1 Vw 1 = w 1 λw 1 = λ (A.1.16) 1 z 1 σ 2 z 1 V λ 1 z 1 σ 2 z 1 V w 1 2 {w m }(m = 2, 3,, M) 1 m z m (m = 2, 3,, M) σ 2 z m {z m }(m = 1, 2,, m 1) 1 m (A.0.2) m w m m 1 {w i }(i = 1, 2,, m 1) (V λ i I)w i = 0 (A.1.17) { wi T 1 (i = j) w j = (A.1.18) 0 (i j) m w m Lagrange J m = w mvw m λ m (w mw m 1) m 1 i=1 µ i w mw i (A.1.19) J m w m m z m (m = 2,, M) z m (m = 1, 2,, m 1) 41
43 J m w m 0 m 1 J m = 2Vw m 2λ m w m µw i w m i=1 = 0 (A.1.20) w j (j = 1, 2,, m 1) (A.1.18) w j Vw m µ j = 0 (j = 1, 2,, m 1) (A.1.21) 1 V = V (A.1.17) (A.1.21) w j Vw m = w mvw j = w mλ j w j = 0 (j = 1, 2,, m 1) (A.1.22) µ j = 0 (j = 1, 2,, m 1) (A.1.23) (A.1.20) (V λ m I)w m = 0 (A.1.24) (A.1.17) m z m σ 2 z m V m 1 m 1 m z m σ 2 z m V m w m B B.1 T N 1 N T H H C = HH C N N 42
44 ρ(λ) 0.4 P(u) B.1: λ Gauss u Porter-Thomas N(0, 1) H = [h ij ] h ij N(0, 1) (B.1.1) C = HH Gauss Wishart λ B.1(a) λ B.1(b) = 10 6 N j=1 u2 ij = N, N = 1000 B.1(b) 1956 C.E.Porter R.G.Thomas [14] Porter-Thomas Gauss P (u i,j ) = 1 ( ) exp u2 i,j (B.1.2) 2π 2 Gauss U( 0.5, 0.5) N = H = [h ij ], h ij U( 0.5, 0.5) (B.1.3) C = HH λ B.2(a) B.1(a) λ Gauss 43
45 ρ(λ) 6 4 P(u) B.2: λ u Porter-Thomas B.2(b) = 10 6 N j=1 u2 ij = N, N = 1000 B.1(b) u ij Porter-Thomas B.1(a) B.2(a) 1988 A.Edelman [15] A.M.Sengupta P.P.Mitra [16] ρ(λ) Q := T/N N T ρ(λ) = Q (λ+ λ)(λ λ ) 2πσ 2 λ (B.1.4) σ 2 H N(0, 1) σ 2 = 1 U( 0.5, 0.5) σ 2 = 1/12 ( λ ± = σ ) 1 Q ± 2 Q (B.1.5) λ [λ, λ + ] [17] (B.1.4) B.1.1 (B.1.4) Dirac Dirac 44
46 Dirac 1 ϵ δ(x) = lim (B.1.6) ϵ 0 π x 2 + ϵ 2 δ(x) = lim I 1 1 (B.1.7) ϵ 0 π x iϵ I z i z = Rz + iiz Rz z I z δ(x) = 1 e ikx dk (B.1.8) 2π N A = [a ij ] (i, j) ã ij A det(a) N det(a) = a ij ã ij (B.1.9) i=1 j Gauss Gauss a π dxe ax2 = (B.1.10) a A N ( ) N N dx i exp x i a ij x j = πn/2 = π N det(a 1 ) (B.1.11) det (A) i=1 i,j=1 A 1 A A 1 = det(a) ij = ( 1) i+j minor(a) ij (B.1.12) (B.1.13) minor(a) ij A i j A 1 ij = ( 1)i+j minor(a) ij det(a) (B.1.14) 45
47 2 Banach Banach Hilbert N T T N N N H Hu = λu (B.1.15) N u 0 λ H u λ λ H N v (λ1 H)u = v (B.1.16) u 1 u = (λ1 H) 1 v (B.1.17) H Banach X D(H) z Hu = zu u D(H) u = 0 (z1 H) 1 X z z R(z) := (z1 H) 1 (B.1.18) H Hu = zu u D(H) u 0 Hu = zu u D(H) u = 0 (z1 H) 1 X z (Spectrum) z H B.1.2 H = [h ij ] N N λ i (i = 1, 2,, N) ρ(λ) ρ(λ) = 1 N N δ(λ λ i ) i=1 46 (B.1.19)
48 H G(λ) G ij (λ) = (λ1 H) 1 ij (B.1.20) G(λ) (Trace ) TrG(λ) = N G ii (λ) = i=1 N 1 λ λ i i=1 (B.1.21) (B.1.7) (B.1.19) ρ(λ) = 1 N N i=1 1 lim ϵ 0 π I 1 λ λ i iϵ 1 = lim ITrG(λ iϵ) ϵ 0 Nπ (B.1.22) G(λ) (B.1.22) ρ(λ) (B.1.21) TrG(λ) = N 1 = λ λ i=1 i λ log (λ λ i ) i = log det(λ1 C) := λ λ Z(λ) (B.1.23) Z(λ) := log det(λ1 C) (B.1.22) (B.1.23) ρ(λ) Gauss (B.1.11) Z(λ) = log det(λ1 C) = 2 log I(λ), (B.1.24) ( ) I(λ) := exp λ N x 2 i 1 N T N ( ) dxi x i x j h ik h jk 2 2 2π i=1 i,j=1 k=1 i=1 (B.1.25) (B.1.23) TrG(λ) TrG(λ) N H (B.1.24) H (B.1.24) N (B.1.24) Z(λ) = 2 log I(λ) = 2 log I(λ) (B.1.26) 47
49 (B.1.26) N T h ik 0 σ 2 /T Gauss (B.1.25) (B.1.26) exp ( 1 2 N i,j=1 k=1 ) ( T x i x j h ik h jk = 1 σ2 T ) T/2 N x 2 i (B.1.27) i=1 q := σ2 T N i=1 x 2 i (B.1.28) (B.1.8) Dirac ( ) N [ ( δ q σ2 1 x 2 i := T 2π exp iζ q σ2 T i=1 N i=1 x 2 i )] dζ (B.1.29) (B.1.25) x i [ T i { Z(λ) = 2 log exp N 4π i 2 ( log(λ σ 2 z) + Q log(1 q) + Qqz )} ] dqdz z := 2iζ/T Q := T/N Q T N (B.1.30) (B.1.30) Qq = z = σ 2 λ σ 2 z, 1 1 q (B.1.31) (B.1.32) (B.1.31) (B.1.32) q(λ) q(λ) = σ2 (1 Q) + Qλ ± [σ 2 (1 Q) + Qλ] 2 4σ 2 Qλ 2Qλ (B.1.33) (B.1.30) (B.1.23) G(λ) = N λ σ 2 z(λ) = NQq(λ) σ 2 (B.1.34) 48
50 (B.1.22) 4σ 2 Qλ [σ 2 (1 Q) + Qλ] 2 ρ(λ) = 2πλσ 2 Q (λ+ λ)(λ λ ) = 2πσ 2 λ ( λ ± = σ ) 1 Q ± 2 Q (B.1.35) (B.1.36) (B.1.37) (B.1.4) λ [λ, λ + ] λ + C x(t) Hilbert y(t) y(t) = 1 x(u) π t u du, (C.0.38) du Hilbert x(t) : x(ω) = x(t) = 1 2π x(t)e iωt dt, x(ω)e iωt dω. y(t) y(ω) y(ω) = = = 1 π = 1 π y(t)e iωt dt ) dt e iωt ( 1 π du x(u) t u dωx(u)e iωu dt e iω(t u) 1 t u du x(u)e iωu dt 1 t e iωt = 1 π x(ω) dt 1 t e iωt 49 (C.0.39) (C.0.40) (C.0.41) (C.0.42) (C.0.43) (C.0.44) (C.0.45)
51 dt 1 t e iωt = { iπ (ω > 0) iπ (ω < 0) 1 (ω > 0) y(ω) = ix(ω) sgn(ω), sgn = 0 (ω = 0) 1 (ω < 0). y(t) = 1 2π (C.0.46) (C.0.47) x(ω) e i π 2 e iωt sgn(ω) dω, (C.0.48) y(t) x(t) Hilbert Hilbert x t T 1 x k = x t = 1 T t=0 T 1 k=0 2πkt i x t e T x k e i 2πkt T Hilbert T 1 2πkt i x(t) = x k e T x k e i π T 2 sgn(k 2 t=0 sgn(k T 2 ) = (C.0.49) (C.0.50) ), (C.0.51) 1 (k > T ) 2 0 (k = T ) 2 (C.0.52) 1 (k < T ) 2 x k π/2 π/2 y t x t π/2 D 50
52 D.1 x µ (t) = x µ(t) x µ (t 12). (D.1.1) N(0, 1) 12 D.2 ρ(λ) RMT λ D.1: D.1 RMT λ + (= 2.85) 5 (λ 1 = 10.5, λ 2 = 6.26, λ 3 = 3.56, λ 4 = 3.07, λ 5 = 2.85) D.2 RRS 1000 λ l + 3σ l λ l RRS l σ l 5 51
53 ρ(λ) RMT λ l actual RRS RRS+3σ λ l D.2: RRS (1000 ) D.3 D
54 Eigenvector No.1 Eigenvector No CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re Eigenvector No.3 Re Eigenvector No CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re Re Eigenvector No CPI PPI IPI USD/JPY Leading Index Coincident Index Lagging Index Money Stock Re D.3: 5 53
55 E PCA CHPCA Eigenvector No.1 PCA Eigenvector No.1 Im Im φ = 0.02π Re PCA Eigenvector No Re Eigenvector No Im Re Im Re φ = 0.4π PCA Eigenvector No E.1: CHPCA PCA CHPCA PCA 2 PCA RMT RRS 8 E.1 CHPCA PCA CHPCA PCA CHPCA ϕ Re Im Re[ α l ] Im[ α l ] PCA α k cosine similarity 54
56 Re[ α 1 ] α 1 Re[ α 1 ] α 1 Im[ α 1 ] α 2 Im[ α 1 ] α 2 Re[ α 2 ] α 2 Re[ α 2 ] α 2 Im[ α 2 ] α 4 Im[ α 2 ] α 4 = = = = CHPCA PCA CHPCA 1 Re PCA 1 Im PCA 2 PCA CHPCA π/2 1 PCA 1 CHPCA ϕ Re Im PCA cosine similarity Re[ α i ] α j Re[ α i ] α j + Im[ α i] α k Im[ α i ] α k (E.0.1) j, k 55
57 [1] L. Laloux, P. Cizeau, J. P. Bouchaud, and M. Potters, Phys. Rev. Lett. 83, 1467(1999). [2] V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, T. Guhr, and H. E. Stanley, Phys. Rev. E 65, (2002) [3] A. Utsugi, K. Ino, and M. Oshikawa, Phys. Rev. E 70, (2004) [4] D. H. Kim and H. Jeong, Phys. Rev. E 72, (2005) [5] V. Kulkarni and N. Deo, Eur. Phys. J. B. 60, 101(2007) [6] R. K. Pan and S. Sinba, Phys. Rev. E 76, (2007) [7] H. Iyetomi et al., Phys. Rev. E83, (2011) [8] T. Barnett, Mon. Wea. Rev. 111, 756(1983). [9] A. Hannachi, I. T. Jolliffe, and D. B. Stephenson, Int. J. Climatol. 27, 1119(2007) [10] Klenow, Peter J and Benjamin A Malin. Handbook of Monetary Economics 3A, North-Holland, 2011, chapter 6. [11] Journal of Political Economy, Octoberr 2004, 112, [12] Yoshikawa, Hiroshi and Aoyama, Hideaki and Fujiwara, Yoshi and Iyetomi, Hiroshi, Deflation/Inflation Dynamics: Analysis Based on Micro Prices (February 15, 2015). Available at SSRN: or [13] Y. Arai, T. Yoshikawa and H. Iyetomi, Frontiers in Arti cial Intelligence and Applications, vol. 255, pp , [14] C.E.Porter and R.G.Thomas. Fluctuations of nucrear reaction widths. Physical Review, Vol. 104, No. 2, pp , Oct [15] A.Edelman. Eigenvalues and Condition Numbers of Random Matrices. SIAM Journal on Matrix Analysis and Applictions, Vol. 9, No. 4, pp , [16] A.M.Sengupta and P.P.Mitra. Distributions of singular values for some random matrices. Physical Review E, Vol. 60, No. 3, pp ,
58 [17] J.P.Bouchud and M.Potters. Theory of Financial Risk and Derivate Pricing: From Statistical Physics to Rsik Management. Cambridge University Press, 2nd edition, ( ) [18] 2008 [19] :
meiji_resume_1.PDF
β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E
More information() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
More informationII No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
More information20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
More information構造と連続体の力学基礎
II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton
More informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
More information1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0
1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0 0 < t < τ I II 0 No.2 2 C x y x y > 0 x 0 x > b a dx
More informationx A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ
More informationii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx
i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x
More information振動と波動
Report JS0.5 J Simplicity February 4, 2012 1 J Simplicity HOME http://www.jsimplicity.com/ Preface 2 Report 2 Contents I 5 1 6 1.1..................................... 6 1.2 1 1:................ 7 1.3
More information4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
More informationn ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................
More information1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2
filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin
More informationNo δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
More information3. ( 1 ) Linear Congruential Generator:LCG 6) (Mersenne Twister:MT ), L 1 ( 2 ) 4 4 G (i,j) < G > < G 2 > < G > 2 g (ij) i= L j= N
RMT 1 1 1 N L Q=L/N (RMT), RMT,,,., Box-Muller, 3.,. Testing Randomness by Means of RMT Formula Xin Yang, 1 Ryota Itoi 1 and Mieko Tanaka-Yamawaki 1 Random matrix theory derives, at the limit of both dimension
More informationf(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f
22 A 3,4 No.3 () (2) (3) (4), (5) (6) (7) (8) () n x = (x,, x n ), = (,, n ), x = ( (x i i ) 2 ) /2 f(x) R n f(x) = f() + i α i (x ) i + o( x ) α,, α n g(x) = o( x )) lim x g(x) x = y = f() + i α i(x )
More information2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =
1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,
More informationII 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K
II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F
More informationII 1 II 2012 II Gauss-Bonnet II
II 1 II 212 II Gauss-Bonnet II 1 1 1.1......................................... 1 1.2............................................ 2 1.3.................................. 3 1.4.............................................
More informationuntitled
18 1 2,000,000 2,000,000 2007 2 2 2008 3 31 (1) 6 JCOSSAR 2007pp.57-642007.6. LCC (1) (2) 2 10mm 1020 14 12 10 8 6 4 40,50,60 2 0 1998 27.5 1995 1960 40 1) 2) 3) LCC LCC LCC 1 1) Vol.42No.5pp.29-322004.5.
More information211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,
More 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 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 informationv v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [
More informationZ: Q: R: C: sin 6 5 ζ a, b
Z: Q: R: C: 3 3 7 4 sin 6 5 ζ 9 6 6............................... 6............................... 6.3......................... 4 7 6 8 8 9 3 33 a, b a bc c b a a b 5 3 5 3 5 5 3 a a a a p > p p p, 3,
More informationH 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [
3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e
More information1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
More informationUntitled
II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
More information量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
More informationhttp://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
More information( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1
2009 1 ( ) ( 40 )+( 60 ) 1 1. 2. Schrödinger 3. (a) (b) (c) http://goofy.phys.nara-wu.ac.jp/ yoshioka/education-09.html pdf 1 1. ( photon) ν λ = c ν (c = 3.0 108 /m : ) ɛ = hν (1) p = hν/c = h/λ (2) h
More informationIA
IA 31 4 11 1 1 4 1.1 Planck.............................. 4 1. Bohr.................................... 5 1.3..................................... 6 8.1................................... 8....................................
More informationtomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.
tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i
More information2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................
More information30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
More information(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)
1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y
More information微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
More informationhttp://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................
More information第5章 偏微分方程式の境界値問題
October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ
More informationA 2 3. m S m = {x R m+1 x = 1} U + k = {x S m x k > 0}, U k = {x S m x k < 0}, ϕ ± k (x) = (x 0,..., ˆx k,... x m ) 1. {(U ± k, ϕ± k ) 0 k m} S m 1.2.
A A 1 A 5 A 6 1 2 3 4 5 6 7 1 1.1 1.1 (). Hausdorff M R m M M {U α } U α R m E α ϕ α : U α E α U α U β = ϕ α (ϕ β ϕβ (U α U β )) 1 : ϕ β (U α U β ) ϕ α (U α U β ) C M a m dim M a U α ϕ α {x i, 1 i m} {U,
More informationQMII_10.dvi
65 1 1.1 1.1.1 1.1 H H () = E (), (1.1) H ν () = E ν () ν (). (1.) () () = δ, (1.3) μ () ν () = δ(μ ν). (1.4) E E ν () E () H 1.1: H α(t) = c (t) () + dνc ν (t) ν (), (1.5) H () () + dν ν () ν () = 1 (1.6)
More information( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )
( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)
More informationall.dvi
72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G
More informationHanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence
Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................
More 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 informationII A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )
II 214-1 : October 2, 214 Version : 1.1 Kawahira, Tomoki TA (Kondo, Hirotaka ) http://www.math.nagoya-u.ac.jp/~kawahira/courses/14w-biseki.html pdf 1 2 1 9 1 16 1 23 1 3 11 6 11 13 11 2 11 27 12 4 12 11
More information006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
More information2 7 V 7 {fx fx 3 } 8 P 3 {fx fx 3 } 9 V 9 {fx fx f x 2fx } V {fx fx f x 2fx + } V {{a n } {a n } a n+2 a n+ + a n n } 2 V 2 {{a n } {a n } a n+2 a n+
R 3 R n C n V??,?? k, l K x, y, z K n, i x + y + z x + y + z iv x V, x + x o x V v kx + y kx + ky vi k + lx kx + lx vii klx klx viii x x ii x + y y + x, V iii o K n, x K n, x + o x iv x K n, x + x o x
More information[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin Clifford Spin 10 A 12 B 17 1 Cliffo
[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin + 8 5 Clifford Spin 10 A 12 B 17 1 Clifford Spin D Euclid Clifford Γ µ, µ = 1,, D {Γ µ, Γ ν
More information20 4 20 i 1 1 1.1............................ 1 1.2............................ 4 2 11 2.1................... 11 2.2......................... 11 2.3....................... 19 3 25 3.1.............................
More informationnote1.dvi
(1) 1996 11 7 1 (1) 1. 1 dx dy d x τ xx x x, stress x + dx x τ xx x+dx dyd x x τ xx x dyd y τ xx x τ xx x+dx d dx y x dy 1. dx dy d x τ xy x τ x ρdxdyd x dx dy d ρdxdyd u x t = τ xx x+dx dyd τ xx x dyd
More information20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
More informationI ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
More informationω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
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 informationRadiation from moving charges#1 Liénard-Wiechert potential Yuji Chinone 1 Maxwell Maxwell MKS E (x, t) + B (x, t) t = 0 (1) B (x, t) = 0 (2) B (x, t)
Radiation from moving harges# Liénard-Wiehert potential Yuji Chinone Maxwell Maxwell MKS E x, t + B x, t = B x, t = B x, t E x, t = µ j x, t 3 E x, t = ε ρ x, t 4 ε µ ε µ = E B ρ j A x, t φ x, t A x, t
More informationall.dvi
5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0
More informationp = mv p x > h/4π λ = h p m v Ψ 2 Ψ
II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π
More informationchap9.dvi
9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j =
More informationuntitled
0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.
More informationPart () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
More informationSFGÇÃÉXÉyÉNÉgÉãå`.pdf
SFG 1 SFG SFG I SFG (ω) χ SFG (ω). SFG χ χ SFG (ω) = χ NR e iϕ +. ω ω + iγ SFG φ = ±π/, χ φ = ±π 3 χ SFG χ SFG = χ NR + χ (ω ω ) + Γ + χ NR χ (ω ω ) (ω ω ) + Γ cosϕ χ NR χ Γ (ω ω ) + Γ sinϕ. 3 (θ) 180
More information( )
7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................
More informationA = 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 information6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2
1 6 6.1 (??) (P = ρ rad /3) ρ rad T 4 d(ρv ) + PdV = 0 (6.1) dρ rad ρ rad + 4 da a = 0 (6.2) dt T + da a = 0 T 1 a (6.3) ( ) n ρ m = n (m + 12 ) m v2 = n (m + 32 ) T, P = nt (6.4) (6.1) d [(nm + 32 ] )a
More informationI A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google
I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59
More 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 informationphs.dvi
483F 3 6.........3... 6.4... 7 7.... 7.... 9.5 N (... 3.6 N (... 5.7... 5 3 6 3.... 6 3.... 7 3.3... 9 3.4... 3 4 7 4.... 7 4.... 9 4.3... 3 4.4... 34 4.4.... 34 4.4.... 35 4.5... 38 4.6... 39 5 4 5....
More information4 2016 3 8 2.,. 2. Arakawa Jacobin., 2 Adams-Bashforth. Re = 80, 90, 100.. h l, h/l, Kármán, h/l 0.28,, h/l.., (2010), 46.2., t = 100 t = 2000 46.2 < Re 46.5. 1 1 4 2 6 2.1............................
More information第10章 アイソパラメトリック要素
June 5, 2019 1 / 26 10.1 ( ) 2 / 26 10.2 8 2 3 4 3 4 6 10.1 4 2 3 4 3 (a) 4 (b) 2 3 (c) 2 4 10.1: 3 / 26 8.3 3 5.1 4 10.4 Gauss 10.1 Ω i 2 3 4 Ξ 3 4 6 Ξ ( ) Ξ 5.1 Gauss ˆx : Ξ Ω i ˆx h u 4 / 26 10.2.1
More information21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
More informationS I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d
S I.. http://ayapin.film.s.dendai.ac.jp/~matuda /TeX/lecture.html PDF PS.................................... 3.3.................... 9.4................5.............. 3 5. Laplace................. 5....
More informationDirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m
Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 p µ γ µ + mp ν γ ν + m 5.1 γ p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 1 2 p µp ν {γ µ, γ ν } + m 2 5.2 p m p p µ γ µ {, } 10 γ {γ µ, γ ν } 2η µν 5.3 p µ γ µ + mp
More information1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
More informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................
More information医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
More informationm(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ
More information( ) sin 1 x, cos 1 x, tan 1 x sin x, cos x, tan x, arcsin x, arccos x, arctan x. π 2 sin 1 x π 2, 0 cos 1 x π, π 2 < tan 1 x < π 2 1 (1) (
6 20 ( ) sin, cos, tan sin, cos, tan, arcsin, arccos, arctan. π 2 sin π 2, 0 cos π, π 2 < tan < π 2 () ( 2 2 lim 2 ( 2 ) ) 2 = 3 sin (2) lim 5 0 = 2 2 0 0 2 2 3 3 4 5 5 2 5 6 3 5 7 4 5 8 4 9 3 4 a 3 b
More informationI-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 informationkawa (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 informationii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.
24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)
More informationd ϕ i) t d )t0 d ϕi) ϕ i) t x j t d ) ϕ t0 t α dx j d ) ϕ i) t dx t0 j x j d ϕ i) ) t x j dx t0 j f i x j ξ j dx i + ξ i x j dx j f i ξ i x j dx j d )
23 M R M ϕ : R M M ϕt, x) ϕ t x) ϕ s ϕ t ϕ s+t, ϕ 0 id M M ϕ t M ξ ξ ϕ t d ϕ tx) ξϕ t x)) U, x 1,...,x n )) ϕ t x) ϕ 1) t x),...,ϕ n) t x)), ξx) ξ i x) d ϕi) t x) ξ i ϕ t x)) M f ϕ t f)x) f ϕ t )x) fϕ
More information. ev=,604k m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H ev Saha x x = 3/ πme kt g i g e n
003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4.........
More information1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji
8 4 2018 6 2018 6 7 1 (Contents) 1. 2 2. (1) 22 3. 31 1. Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji SETO 22 3. Editorial Comments Tadashi
More information() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.
() 6 f(x) [, b] 6. Riemnn [, b] f(x) S f(x) [, b] (Riemnn) = x 0 < x < x < < x n = b. I = [, b] = {x,, x n } mx(x i x i ) =. i [x i, x i ] ξ i n (f) = f(ξ i )(x i x i ) i=. (ξ i ) (f) 0( ), ξ i, S, ε >
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 informationW u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
More informationLLG-R8.Nisus.pdf
d M d t = γ M H + α M d M d t M γ [ 1/ ( Oe sec) ] α γ γ = gµ B h g g µ B h / π γ g = γ = 1.76 10 [ 7 1/ ( Oe sec) ] α α = λ γ λ λ λ α γ α α H α = γ H ω ω H α α H K K H K / M 1 1 > 0 α 1 M > 0 γ α γ =
More information(2004 ) 2 (A) (B) (C) 3 (1987) (1988) Shimono and Tachibanaki(1985) (2008) , % 2 (1999) (2005) 3 (2005) (2006) (2008)
,, 23 4 30 (i) (ii) (i) (ii) Negishi (1960) 2010 (2010) ( ) ( ) (2010) E-mail:fujii@econ.kobe-u.ac.jp E-mail:082e527e@stu.kobe-u.ac.jp E-mail:iritani@econ.kobe-u.ac.jp 1 1 16 (2004 ) 2 (A) (B) (C) 3 (1987)
More information) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
More informationv_-3_+2_1.eps
I 9-9 (3) 9 9, x, x (t)+a(t)x (t)+b(t)x(t) = f(t) (9), a(t), b(t), f(t),,, f(t),, a(t), b(t),,, x (t)+ax (t)+bx(t) = (9),, x (t)+ax (t)+bx(t) = f(t) (93), b(t),, b(t) 9 x (t), x (t), x (t)+a(t)x (t)+b(t)x(t)
More information2016 ǯ¥Î¡¼¥Ù¥ëʪÍý³Ø¾Þ²òÀ⥻¥ß¥Ê¡¼ Kosterlitz-Thouless ž°Ü¤È Haldane ͽÁÛ
2016 Kosterlitz-Thouless Haldane Dept. of Phys., Kyushu Univ. 2016 11 29 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER ( ) ( ) (Dirac,
More information( ) s n (n = 0, 1,...) n n = δ nn n n = I n=0 ψ = n C n n (1) C n = n ψ α = e 1 2 α 2 n=0 α, β α n n! n (2) β α = e 1 2 α 2 1
(3.5 3.8) 03032s 2006.7.0 n (n = 0,,...) n n = δ nn n n = I n=0 ψ = n C n n () C n = n ψ α = e 2 α 2 n=0 α, β α n n (2) β α = e 2 α 2 2 β 2 n=0 =0 = e 2 α 2 β n α 2 β 2 n=0 = e 2 α 2 2 β 2 +β α β n α!
More informationI A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
More information64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k
63 3 Section 3.1 g 3.1 3.1: : 64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () 3 9.8 m/s 2 3.2 3.2: : a) b) 5 15 4 1 1. 1 3 14. 1 3 kg/m 3 2 3.3 1 3 5.8 1 3 kg/m 3 3 2.65 1 3 kg/m 3 4 6 m 3.1. 65 5
More information‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í
Markov 2009 10 2 Markov 2009 10 2 1 / 25 1 (GA) 2 GA 3 4 Markov 2009 10 2 2 / 25 (GA) (GA) L ( 1) I := {0, 1} L f : I (0, ) M( 2) S := I M GA (GA) f (i) i I Markov 2009 10 2 3 / 25 (GA) ρ(i, j), i, j I
More informationB ver B
B ver. 2017.02.24 B Contents 1 11 1.1....................... 11 1.1.1............. 11 1.1.2.......................... 12 1.2............................. 14 1.2.1................ 14 1.2.2.......................
More informationI 1
I 1 1 1.1 1. 3 m = 3 1 7 µm. cm = 1 4 km 3. 1 m = 1 1 5 cm 4. 5 cm 3 = 5 1 15 km 3 5. 1 = 36 6. 1 = 8.64 1 4 7. 1 = 3.15 1 7 1 =3 1 7 1 3 π 1. 1. 1 m + 1 cm = 1.1 m. 1 hr + 64 sec = 1 4 sec 3. 3. 1 5 kg
More informationgr09.dvi
.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
More informationGmech08.dvi
145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2
More information