.. F x) = x ft)dt ), fx) : PDF : probbility density function) F x) : CDF : cumultive distribution function F x) x.2 ) T = µ p), T : ) p : x p p = F x
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- きょうすけ あさま
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1 L N ) LN3 ) III P3 ) III LP3 ) GEV ) SQRT-ET ) ) GPD) SLSC
2 .. F x) = x ft)dt ), fx) : PDF : probbility density function) F x) : CDF : cumultive distribution function F x) x.2 ) T = µ p), T : ) p : x p p = F x p ) x p = F p) x p T - µ : X x p 2) POT, Peks Over Threshold dt) AMS, Annul Mximum Series dt) µ = T T = p p = T ) N 2) T x T N AMS ) P X < x T ) N = T 4) P X x T ) N = ) N 5) T 3) 3) T N P AMS ) T N P x = x j N 6) S 2 = N C s = N x j x) 2 7) ) 3 xj x 8) S ˆσ 2 = N N S2 9) NN ) ˆγ = C s 0) N 2
3 .4 L PWM : Probbility Weighted Moments) L L Moments) L PWM) β r = 0 xf r df r = 0,, 2,... ) ) PWM L λ = β 0 2) λ 2 = 2β β 0 3) λ 3 = 6β 2 6β + β 0 4) PWM L b 0 = N b = b 2 = x j) 5) NN ) j )x j) 6) NN )N 2) j )j 2)x j) 7) x j) N j.5 F [x i) ] = i α N + 2α 8) N i x i) F [x i) ] α i ) Weibull Blom Cunnne Gringorten Hzen α
4 .6 3 ) Generlized Extreme Vlue distribution) 2 Gumbel distribution) 3 Weibull distribution with 3 prmeters) 2 SQRT exponentil-type distribution of mximum) 3 Generlized Preto distribution) 2 Exponentil distribution) 2 Norml distribution) 3 Log-Norml Distribution with 3 prmeters) III 3 Person type III distribution) III 3 Log-Person type III distribution) 3
5 2. 2. N ) x ) fx) = [ exp σ x 2π 2 ) ] 2 x µx σ x 9) 2) ) x µx F x) = Φ σ x Φz) = 2π z exp 2 ) t2 dt 20) 3) p x p z = x µ x σ x x = µ x + σ x z 2) x p = µ x + σ x z p z p p = Φz) z 22) 4) L ) b 0 = N x j) b = NN ) j )x j) 23) x j) N j b i = β i λ i λ = β 0 λ 2 = 2β β 0 24) L { µ x = λ σ x = πλ 2 25) 4
6 2.2 LN3 ) x 3 ) fx) = { exp x )σ y 2π 2 [ ] } 2 lnx ) µy σ y y = lnx ) 26) 2) ) lnx ) µy F x) = Φ σ y Φz) = 2π z exp 2 ) t2 dt 27) 3) p x p z = lnx ) µ y σ y x = + expµ y + σ y z) 28) x p = + expµ y + σ y z p ) z p p = Φz) z 29) 4) : ) = x 2 ) x N) x m x ) + x N) 2x m > 0 x ) + x N) 2x m µ y = N N lnx j ) 30) σ 2 y = N N [lnx j ) µ y ] 2 x ) x N) x m ) x j x j) 5) ) ) 3 xj x 3) x = N x j S 2 x = N x j x) 2 C sx = N S x NN ) µ x = x σ x = [N/N )] /2 S x γ x = C sx 32) N 2 x i ln x i ) ) γ x Bobee Robitille ) B C sx 3 γ x = C sxa + B C 3 sx ) 33) A = /N /N 2 B =.69/N /N 2 34) 5
7 µ x = + expµ y ) expσy/2) 2 35) σ x = expµ y ) expσy){expσ 2 y) 2 } 36) γ x = {expσy) 2 + 2} expσy) 2 37) σ y, µ y, 3 γ x = {expσy) 2 + 2} expσy) 2 x 3 + 3x 2 4 γx 2 = 0 where, x = expσy) 2 38) 3 x x = β + ) /3 β 2 + β ) /3 β 2 γ 2 where, β = + x 2 σ y = lnx) ) σ x µ y = ln xx ) = µ x expµ y ) expσy/2) 2 : x x 3 + 3x 2 4 γ 2 x = 0 39) 6) ) z = lnx ) µ y σ y z = A X + B 40) X = lnx ) A = σ y B = µ y σ y 4) z % µ y = B A σ y = A : ) = x ) δ δ: ) 6
8 2.3 III P3 ) x III Person type 3 distribution ) ) fx) = Γb) x c ) b exp x c ) > 0 : c x < 42) c b 2) ) x c F x) = G Gw) = Γb) w 0 t b exp t)dt > 0) 43) 3) p x p w = x c x = c + w 44) x p = c + w p w p p = Gw) w 45) 4) ) ) 3 xj x 46) x = N x j S 2 x = N x j x) 2 C sx = N S x µ x = x σ x = [N/N )] /2 S x γ x = ) NN ) C sx 47) N 2 γ x Bobee Robitille III ) ) B C sx 2 γ x = C sx A + B C 2 sx ) 48) A = + 6.5/N /N 2 B =.48/N /N 2 49) µ x = c + b σ x 2 = 2 b γ x = 2 b 2 b = 4/γ x b > 0) = σ x / b γ x < 0 = σ x / b < 0) c = µ x b 50) 5) γ x < 0 < 0 w p p γ x III 7
9 2.4 III LP3 ) x y = ln x III ) fx) = Γb) x ln x c ) b exp ln x c ) > 0 : expc) < x < 52) c b 2) ) ln x c F x) = G Gw) = Γb) w 0 t b exp t)dt > 0) 53) 3) p x p w = ln x c x = expc + w) 54) x p = expc + w p ) w p p = Gw) w 55) 4) ) ) 3 yj ȳ 56) y j = ln x j ȳ = N y j S 2 y = N y j ȳ) 2 C sy = N S y µ y = ȳ σ y = [N/N )] /2 S y γ y = ) NN ) C sy 57) N 2 γ x Bobee Robitille III ) ) B C sy 2 γ y = C sy A + B C 2 sy ) 58) A = + 6.5/N /N 2 B =.48/N /N 2 59) µ y = c + b σ y 2 = 2 b γ y = 2 b 2 b = 4/γ y b > 0) = σ y / b γ y < 0 = σ y / b < 0) c = µ y b 60) 6) γ y < 0 < 0 w p p γ y b b Wilson-Hilferty p x p x p = expµ y + σ y K p ) K p = 2 + γ yz p γ y 2 ) 2 62) γ y 6 36 γ y 8
10 z p N0, ) Wilson-Hilferty b < 0, 000 γ y < 0 z p p K p γ y γ y p γ y 9
11 2.5 x Gumbel distribution) ) fx) = [ exp x c exp x c )] < x < 63) c 2) [ F x) = exp exp x c )] 64) 3) p x p [ p = exp exp x c )] x = c ln[ lnp)] 65) x p = c ln[ lnp)] 66) 4) L ) b 0 = N x j) b = NN ) j )x j) 67) x j) N j b i = β i λ i λ = β 0 λ 2 = 2β β 0 68) L { = λ 2 / ln 2 c = λ ) 0
12 2.6 GEV ) x Generlized Extreme Vlue distribution) k = 0 ) fx) = k x c ) [ /k exp k x c ) ] /k k 0) 70) c k 2) F x) = exp [ k x c ) ] /k k 0) 7) 3) p x p p = exp [ k x c ) ] /k x = c + k { [ lnp)] k} 72) x p = c + k { [ lnp)] k} 73) 4) L ) b 0 = N x j) b = NN ) j )x j) b 2 = NN )N 2) j )j 2)x j) 74) x j) N j b i = β i λ i λ = β 0 λ 2 = 2β β 0 λ 3 = 6β 2 6β + β 0 75) L k = d d 2 d = 2λ 2 ln2) λ 3 + 3λ 2 ln3) kλ 2 = 2 k ) Γ + k) c = λ [ Γ + k)] k 76)
13 2.7 SQRT-ET ) x SQRT exponentil-type distribution of mximum) ) fx) = b [ 2 exp bx + ) bx exp )] bx x 0) 77) 2) [ F x) = exp + ) bx exp )] bx x 0) 78) 3) p x p [ p = exp + ) bx exp )] bx = exp [ + t p ) exp t p )] t p = bx) 79) x = t p 2 ln + t p ) t p = ln [ ] b lnp) 80) x p = t p 2 b ln + t p ) t p = ln [ ] lnp) 8) ) t p gt p ) = ln + t p ) t p ln [ ] lnp) 82) g t p ) = + t p 83) gt p ) g0) > 0 Newton- Rphson gt p ) = 0 t p t pn+) = t pn) gt pn)) g t pn) ) n) 84) t p p x p x t p = b x mx t p 4) ) b L L, b) = ln fx j ) = N ln + N ln b N ln 2 bxj exp ) bx j + bxj exp ) bx j 85) 2
14 L b 0 b L b = 0 = N bxj 2N N bx j) exp ) = 86) bx j L 0 b 2 L = 0 = N N exp ) N bx j + bxj exp ) = 2 87) bx j L = 2 hb) = b) 2 b) = 0 b 2 > 0 > 0 b > 0 b > N ) 2 2N 88) xj b b C ) /* Bisection method */ b=bb; /* >0 b ) */ b2=b+0.5; /* b+0.5 b2 */ bb=0.5*b+b2); /* bb */ f=fsqrnd,dtx,b,&,&2); /* hb) */ f2=fsqrnd,dtx,b2,&,&2); /* hb2) */ ff=fsqrnd,dtx,bb,&,&2); /* hbb) */ do{ /* */ iff*ff<0.0)b2=bb; ifff*f2<0.0)b=bb; ifff==0.0)brek; if0.0<f*ff&&0.0<ff*f2){b=b2;b2=b+0.5;} /* */ bb=0.5*b+b2); /* 0.5 */ f=fsqrnd,dtx,b,&,&2); f2=fsqrnd,dtx,b2,&,&2); ff=fsqrnd,dtx,bb,&,&2); }while0.00<fbs-2)); /* hb) <0.00 */ 3
15 2.8 3 ) x 3 Weibull distribution) ) fx) = k ) [ k ) ] k x c x c exp k = 0) 89) c k 2) [ ) ] k x c F x) = exp k 0) 90) 3) p x p [ ) ] k x c p = exp x = c + [ ln p)] /k 9) x p = c + [ ln p)] /k 92) 4) L *) ) b 0 = N x j) b = NN ) j )x j) b 2 = NN )N 2) j )j 2)x j) 93) x j) N j b i = β i λ i λ = β 0 λ 2 = 2β β 0 λ 3 = 6β 2 6β + β 0 94) L k = 285.3τ τ τ τ τ τ τ = λ 3 /λ 2 λ 2 = 2 /k ) Γ + /k) c = λ Γ + /k) 95) *) L-moments B2 ) Vol.B2-65 No pp k λ 3 τ 3 = λ3/λ 2 A k 5) ) 3 c k I) c 4
16 x F x) x [ ) ] k x c F x) = exp 96) 2 ln{ ln[ F x)]} = k lnx c) k ln Y = A X + B 97) Y i = ln{ ln[ F x i )]} X i = lnx i c) k = A = exp B/A) c c c 3 k c c k k Newton-Rphson II) k c t = x c ft) = k ) [ k ) ] k t t exp 98) L = N i= ln ft i) k Newton-Rphson k c L k = 0 k + N i= ln t i N N i= [ln t i) t i k ] N i= t i k = 0 99) L = 0 = N i= t ) /k i k 00) N gk) = k + T 0 N T 2k) T k) g k) = k 2 T 3k) T k) [T 2 k)] 2 [T k)] 2 0) T 0 = ln t i T k) = i= i= t i k T 2 k) = [ln t i t k i ] T 3 k) = [ln t i ln t i t k i ] 02) i= i= k n k n+ = k n gk n) g k n ) 03) k N i= = t ) /k i k 04) N 5
17 2.9 x Exponentil distribution) ) fx) = exp x c ) 05) c 2) F x) = exp x c ) 06) 3) p x p p = exp x c ) x = c ln p) 07) x p = c ln p) 08) 4) L ) b 0 = N x j) b = NN ) j )x j) 09) x j) N j b i = β i λ i λ = β 0 λ 2 = 2β β 0 0) L { = 2λ 2 c = λ ) 6
18 2.0 GPD) x Generlized Preto distribution) k = 0 ) fx) = c k 2) k x c ) /k k 0) 2) F x) = k x c ) /k k 0) 3) 3) p x p p = k x c ) /k x = c + { p) k } 4) k x p = c + k { p) k} 5) 4) L ) b 0 = N x j) b = NN ) j )x j) b 2 = NN )N 2) j )j 2)x j) 6) x j) N j b i = β i λ i λ = β 0 λ 2 = 2β β 0 λ 3 = 6β 2 6β + β 0 7) L k = λ 2 3λ 3 λ 2 + λ 3 8) = + k)2 + k)λ 2 c = λ 2 + k)λ 2 7
19 3. 3. Q-Q quntile-quntile) r = N X i Y i X i Y i [N X 2 i X i ) 2 ] [N Y i2 Y i ) 2 ] 3.2 SLSC r : N : X i Y i : : SLSC = N j=n s j r j ) r 0.99 r 0.0 9) s i r i Normlized vrible by prmeters Normlized vrible by Plotting position formul r 0.99 Normlized vlue corresponding to the non-exceednce probbility of 99% r 0.0 Normlized vlue corresponding to the non-exceednce probbility of % Distribution S i r i LN3 lnx i ) µ y σ y qnormp i) %-point of SND) LP3 Gumbel exp ln x i c xi c ) qgmmp i, shpe = b, rte = ) %-point of Gmm Distribution) lnp i ) GEV k x i c ) /k lnp i) SQRT-ET exp{ln + bx i ) bx i } lnp i ) Weibull ) k xi c ln p i) Exponentil GPD k ln x i c ) k xi c ln p i ) ln p i ) 8
20 4. Jckknife JckKnife ) 2 3 N x i) ˆθ i N ˆθ i) n ˆθ i) N ˆθ ) ˆθ ) = N ˆθ i) 20) i= 4 N x i) ˆθ N ˆθ ) jckknife θ θ = N ˆθ N ) ˆθ ) 2) 5 θ SE) SE) = N ˆθi) N ˆθ ) 2 ) 22) i= 5. bootstrp N x i) N N θ i) θ i) bootstrp ) B bootstrp ˆ θ = B B θ i) 23) bootstrp bootstrp percentile method) i= 9
21 6. N x ɛ β 0 ɛ 0 ɛ 0 = β 0 ) /N β 5%) 24) N F x) )q % u ɛ u ɛ Φ q = F x ɛ ) = Φu ɛ ) 25) 5 N F, M ) F M= N ) F = ) M 2 u ɛ M + 26) 6 F 2ɛ ) M FM 2ɛ) 2 = u ɛ M + ɛ ɛ 0 ɛ ɛ 0 ɛ > ɛ 0 ) ) 27) 28) ) F F % 20
22 No ) No ) No L-moments B2 ) Vol.B2-65 No pp6-65 Derek A. Roff ) ) KADOYA Mutsumi : Appliction of Extreme Vlue Distribution in Hydrologic Frequency Anlysis Prt II. Singulr Hydrologic Amount nd Rejection Test, Cittion Bulletins - Disster Prevention Reserch Institute, Kyoto University 964), 66: 33-44, URL 2
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IA September 25, 2017 ( ) I = [a, b], f (x) I = (a 0 = a < a 1 < < a m = b) I ( ) (partition) S (, f (x)) = w (I k ) I k a k a k 1 S (, f (x)) = I k 2
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A. A. A-- [ ] f(x) x = f 00 (x) f 0 () =0 f 00 () > 0= f(x) x = f 00 () < 0= f(x) x = A--2 [ ] f(x) D f 00 (x) > 0= y = f(x) f 00 (x) < 0= y = f(x) P (, f()) f 00 () =0 A--3 [ ] y = f(x) [, b] x = f (y)
1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1
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1. x { e 1,..., e n } x = x1 e1 + + x n en = (x 1,..., x n ) X, Y [X, Y ] Intrinsic ( ) Intrinsic M m P M C P P M P M v 3 v : C P R 1 f, g C P, λ R (1) v(f + g) = v(f) + v(g) (2) v(λf) = λv(f) (3) v(fg)
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II 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
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* 10 100 1 ( ) ( ) 20 2 X f( ) (pdf: probability density function) F( ) X (cdf: cumulative distribution function) (2.1) X ( ) p ( ) W( ) 1 F( ) p p 1(p) W( p) ( p) X 1/T T ( ) T p T T T p 1 1/T p F( )
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
() 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.
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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
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M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -
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(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x
![(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x](/thumbs/104/162595168.jpg "(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x")
Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k
1 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
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II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R
![II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R](/thumbs/95/125540821.jpg "II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R")
II Karel Švadlenka 2018 5 26 * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* 5 23 1 u = au + bv v = cu + dv v u a, b, c, d R 1.3 14 14 60% 1.4 5 23 a, b R a 2 4b < 0 λ 2 + aλ + b = 0 λ =
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5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E
5 5.1 E 1, E 2 N 1, N 2 E tot N tot E tot = E 1 + E 2, N tot = N 1 + N 2 S 1 (E 1, N 1 ), S 2 (E 2, N 2 ) E 1, E 2 S tot = S 1 + S 2 2 S 1 E 1 = S 2 E 2, S 1 N 1 = S 2 N 2 2 (chemical potential) µ S N
4. 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
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13 13.1 O r F R = m d 2 r dt 2 m r m = F = m r M M d2 R dt 2 = m d 2 r dt 2 = F = F (13.1) F O L = r p = m r ṙ dl dt = m ṙ ṙ + m r r = r (m r ) = r F N. (13.2) N N = R F 13.2 O ˆn ω L O r u u = ω r 1 1:
n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
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ＪＭＰ V4 による生存時間分析
V4 1 SAS 2000.11.18 4 ( ) (Survival Time) 1 (Event) Start of Study Start of Observation Died Died Died Lost End Time Censor Died Died Censor Died Time Start of Study End Start of Observation Censor
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
S 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....
, 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x 2 + x + 1). a 2 b 2 = (a b)(a + b) a 3 b 3 = (a b)(a 2 + ab + b 2 ) 2 2, 2.. x a b b 2. b {( 2 a } b )2 1 =
x n 1 1.,,.,. 2..... 4 = 2 2 12 = 2 2 3 6 = 2 3 14 = 2 7 8 = 2 2 2 15 = 3 5 9 = 3 3 16 = 2 2 2 2 10 = 2 5 18 = 2 3 3 2, 3, 5, 7, 11, 13, 17, 19.,, 2,.,.,.,?.,,. 1 , 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x
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
.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,
[ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b
1 9 v.0.1 c (2016/10/07) Minoru Suzuki T µ 1 (7.108) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1)
1 9 v..1 c (216/1/7) Minoru Suzuki 1 1 9.1 9.1.1 T µ 1 (7.18) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1) E E µ = E f(e ) E µ (9.1) µ (9.2) µ 1 e β(e µ) 1 f(e )
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