.. ( )T p T = p p = T () T x T N P (X < x T ) N = ( T ) N (2) ) N ( P (X x T ) N = T (3) T N P T N P 0
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1 L (N ) (LN3 ) III (P3 ) III (LP3 ) (GEV ) (SQRT-ET ) (3 ) No () No (2) No L-moments B2( ) Vol.B2-65 No pp6-65 Derek A. Roff( ) ( )
2 .. ( )T p T = p p = T () T x T N P (X < x T ) N = ( T ) N (2) ) N ( P (X x T ) N = T (3) T N P T N P x = x j (4) N S 2 = N C s = N (x j x) 2 (5) ( ) 3 xj x (6) S ˆσ 2 = N N S2 (7) N(N ) ˆγ = C s (8) N 2.3 L (PWM : Probbility Weighted Moments) L (L Moments) L (PWM) β r = 0 xf r df (r = 0,, 2,... ) (9) PWM L λ = β 0 (0) λ 2 = 2β β 0 () λ 3 = 6β 2 6β + β 0 (2) PWM L
3 b 0 = N b = b 2 = x (j) (3) N(N ) (j )x (j) (4) N(N )(N 2) (j )(j 2)x (j) (5) x (j) N j.4 F [x (i) ] = i α N + 2α (6) N i x (i) F [x (i) ] α i ( ) Weibull Blom Cunnne Gringorten Hzen α
4 2. 2. (N ) x () f(x) = [ exp 2πσx 2 ( ) ] 2 x µx σ x (7) (2) ( ) x µx F (x) = Φ σ x Φ(z) = 2π z exp ( 2 ) t2 dt (8) (3) p x p z = x µ x σ x x = µ x + σ x z (9) x p = µ x + σ x z p z p p = Φ(z) z (20) (4) (L ) b 0 = N x (j) b = N(N ) (j )x (j) b 2 = N(N )(N 2) (j )(j 2)x (j) (2) 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 (22) L { µ x = λ σ x = πλ 2 (23) 3
5 2.2 (LN3 ) x 3 () f(x) = { (x ) exp 2πσ y 2 [ ] } 2 ln(x ) µy σ y y = ln(x ) (24) (2) ( ) ln(x ) µy F (x) = Φ σ y Φ(z) = 2π z exp ( 2 ) t2 dt (25) (3) p x p z = ln(x ) µ y σ y x = + exp(µ y + σ y z) (26) x p = + exp(µ y + σ y z p ) z p p = Φ(z) z (27) (4) ( ) = x 2 () x (N) x m x () + x (N) 2x m > 0 x () + x (N) 2x m µ y = N N ln(x j ) (28) σ 2 y = N N [ln(x j ) µ y ] 2 x () x (N) x m ( ) x j x (j) (5) ( ) ( ) 3 xj x (29) x = N x j S 2 x = N (x j x) 2 C sx = N S x N(N ) µ x = x σ x = [N/(N )] /2 S x γ x = C sx (30) N 2 γ x Bobee Robitille ( ) ( ) B C sx 3 γ x = C sx (A + B C 3 sx ) (3) A = /N /N 2 B =.69/N /N 2 (32) µ x = + ɛ φ σ x = ɛ φ(φ ) γ x = (φ + 2) φ (33) ɛ = exp(µ y ) φ = exp ( 2 σ ) y (34) 4
6 [ φ = β + ] /3 [ β 2 + β ] /3 β 2 (β = + γ x 2 ) 2 (35) σ x ɛ = φ(φ ) (36) σ y = ln φ µ y = ln ɛ = µ x ɛ φ (37) 5
7 2.3 III (P3 ) x III Person type 3 distribution ( ) () f(x) = Γ(b) ( x c ) b ( exp x c ) > 0 : c x < (38) (2) ( ) x c F (x) = G G(w) = Γ(b) w 0 t b exp( t)dt ( > 0) (39) (3) p x p w = x c x = c + w (40) x p = c + w p w p p = G(w) w (4) (4) ( ) ( ) 3 xj x (42) x = N x j S 2 x = N (x j x) 2 C sx = N S x N(N ) µ x = x σ x = [N/(N )] /2 S x γ x = C sx (43) N 2 γ x Bobee Robitille ( III ) ( ) B C sx 2 γ x = C sx (A + B C 2 sx ) (44) A = + 6.5/N /N 2 B =.48/N /N 2 (45) 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 (46) (47) γ x < 0 < 0 w p p γ x III 6
8 2.4 III (LP3 ) x y = ln x III () f(x) = Γ(b) x ( ln x c ) b ( exp ln x c ) > 0 : exp(c) < x < (48) (2) ( ) ln x c F (x) = G G(w) = Γ(b) w 0 t b exp( t)dt ( > 0) (49) (3) p x p w = ln x c x = exp(c + w) (50) x p = exp(c + w p ) w p p = G(w) w (5) (4) ( ) ( ) 3 yj ȳ (52) y j = ln x j ȳ = N y j S 2 y = N (y j ȳ) 2 C sy = N S y N(N ) µ y = ȳ σ y = [N/(N )] /2 S y γ y = C sy (53) N 2 γ x Bobee Robitille ( III ) ( ) B C sy 2 γ y = C sy (A + B C 2 sy ) (54) A = + 6.5/N /N 2 B =.48/N /N 2 (55) 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 (56) (57) γ y < 0 < 0 w p p γ y b b Wilson-Hilferty p x p x p = exp(µ y + σ y K p ) K p = 2 ( + γ yz p γ y 2 ) 2 (58) γ y 6 36 γ y z p N(0, ) Wilson-Hilferty b < 0, 000 γ y < 0 z p p K p γ y γ y p γ y 7
9 2.5 x (Gumbel distribution) () f(x) = [ exp x c ( exp x c )] < x < (59) (2) [ ( F (x) = exp exp x c )] (60) (3) p x p [ ( p = exp exp x c )] x = c ln[ ln(p)] (6) x p = c ln[ ln(p)] (62) (4) (L ) b 0 = N x (j) b = N(N ) (j )x (j) b 2 = N(N )(N 2) (j )(j 2)x (j) (63) 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 (64) L { = λ 2 / ln 2 c = λ (65) 8
10 2.6 (GEV ) x (Generlized Extreme Vlue distribution) () f(x) = ( k x c ) [ /k ( exp k x c ) ] /k (k 0) (66) (2) F (x) = exp [ ( k x c ) ] /k (k 0) (67) (3) p x p p = exp [ ( k x c ) ] /k x = c + k { [ ln(p)] k} (68) x p = c + k { [ ln(p)] k} (69) (4) (L ) b 0 = N x (j) b = N(N ) (j )x (j) b 2 = N(N )(N 2) (j )(j 2)x (j) (70) 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 = d d 2 d = 2λ 2 ln(2) λ 3 + 3λ 2 ln(3) kλ 2 = ( 2 k ) Γ( + k) c = λ [ Γ( + k)] k (72) 9
11 2.7 (SQRT-ET ) x (SQRT exponentil-type distribution of mximum) () f(x) = b [ 2 exp ( bx + ) ( bx exp )] bx (x 0) (73) (2) [ ( F (x) = exp + ) ( bx exp )] bx (x 0) (74) (3) p x p [ ( p = exp + ) ( bx exp )] bx = exp [ ( + t p ) exp( t p )] (t p = bx) (75) x = t p 2 ln( + t p ) t p = ln [ ] b ln(p) (76) x p = t p 2 b ln( + t p ) t p = ln [ ] ln(p) (77) ( ) t p g(t p ) = ln( + t p ) t p ln [ ] ln(p) (78) g (t p ) = + t p (79) g(t p ) g(0) > 0 Newton- Rphson g(t p ) = 0 t p t p(n+) = t p(n) g(t p(n)) g (t p(n) ) (n) (80) t p p x p x t p = b x mx t p (4) ( ) b L L(, b) = ln f(x j ) = N ln + N ln b N ln 2 bxj exp ( ) bx j + bxj exp ( ) bx j (8) 0
12 L b 0 b L b = 0 = N bxj 2N N (bx j) exp ( ) = (82) bx j L 0 b 2 L = 0 = N N exp ( ) N bx j + bxj exp ( ) = 2 (83) bx j L = 2 h(b) = (b) 2 (b) = 0 b 2 > 0 > 0 b > 0 b > ( N ) 2 2N (84) 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=fsqr(nd,dtx,b,&,&2); /* h(b) */ f2=fsqr(nd,dtx,b2,&,&2); /* h(b2) */ ff=fsqr(nd,dtx,bb,&,&2); /* h(bb) */ do{ /* */ if(f*ff<0.0)b2=bb; if(ff*f2<0.0)b=bb; if(ff==0.0)brek; if(0.0<f*ff&&0.0<ff*f2){b=b2;b2=b+0.5;} /* */ bb=0.5*(b+b2); /* 0.5 */ f=fsqr(nd,dtx,b,&,&2); f2=fsqr(nd,dtx,b2,&,&2); ff=fsqr(nd,dtx,bb,&,&2); }while(0.00<fbs(-2)); /* h(b) <0.00 */
13 2.8 (3 ) x 3 (Weibull distribution) () f(x) = k ( ) [ k ( ) ] k x c x c exp (k = 0) (85) (2) [ ( ) ] k x c F (x) = exp (k 0) (86) (3) p x p [ ( ) ] k x c p = exp x = c + [ ln ( p)] /k (87) x p = x = c + [ ln ( p)] /k (88) (4) (L *) ) b 0 = N x (j) b = N(N ) (j )x (j) b 2 = N(N )(N 2) (j )(j 2)x (j) (89) 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 (90) L k = 285.3τ τ τ τ τ τ τ = λ 3 /λ 2 λ 2 = ( 2 /k ) Γ( + /k) c = λ Γ( + /k) (9) *) L-moments B2( ) Vol.B2-65 No pp k λ 3 τ 3 = λ3/λ 2 A k (5) ( ) 3 c k (I) c 2
14 x F (x) x [ ( ) ] k x c F (x) = exp (92) 2 ln{ ln[ F (x)]} = k ln(x c) k ln Y = A X + B (93) Y i = ln{ ln[ F (x i )]} X i = ln(x i c) k = A = exp( B/A) c c c 3 k c c k k Newton-Rphson (II) k c t = x c f(t) = k ( ) [ k ( ) ] k t t exp (94) L = N i= ln f(t 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 (95) L = 0 = ( N i= t ) /k i k (96) N g(k) = k + T 0 N T 2(k) T (k) g (k) = k 2 T 3(k) T (k) [T 2 (k)] 2 [T (k)] 2 (97) 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 ] (98) i= i= k n k n+ = k n g(k n) g (k n ) (99) k ( N i= = t ) /k i k (00) N 3
15 3. Jckknife (JckKnife ) 2 3 N x (i) ˆθ i N ˆθ (i) n ˆθ (i) N ˆθ ( ) ˆθ ( ) = N ˆθ (i) (0) i= 4 N x (i) ˆθ N ˆθ ( ) jckknife θ θ = N ˆθ (N ) ˆθ ( ) (02) 5 θ (SE) (SE) = N (ˆθ(i) N ˆθ ) 2 ( ) (03) i= 4. bootstrp 2 3 N x (i) ˆθ N N θ (i) θ (i) (bootstrp ) B bootstrp ˆ θ = B B θ (i) (04) i= 4 bootstrp bootstrp (percentile method) 4
16 N x ɛ N F (x) ( )q u ɛ u ɛ Φ q = F (x ɛ ) = Φ(u ɛ ) (05) 4 N F (, M ) F M(= N ) F = ( ) M 2 u ɛ M + (06) F 2ɛ ( ) M FM (2ɛ) 2 = u ɛ M + (07) 5 β ɛ ɛ 0 = ( β 0 ) /N (β 5%) (08) 5 ɛ ɛ 0 ɛ ɛ 0 ( ) ɛ > ɛ 0 ( ) (09) 5
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1 1 ϕ ϕ ϕ S F F = ϕ (1) S 1: F 1 1 (1) () (3) I 0 3 I I d θ = L () dt θ L L θ I d θ = L = κθ (3) dt κ T I T = π κ (4) T I κ κ κ L l a θ L r δr δl L θ ϕ ϕ = rθ (5) l : l r δr θ πrδr δf (1) (5) δf = ϕ πrδr
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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
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3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
( ) 2002 1 1 1 1.1....................................... 1 1.1.1................................. 1 1.1.2................................. 1 1.1.3................... 3 1.1.4......................................
2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n
. X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n
DVIOUT
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)
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
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r d 2r d l d (a) (b) (c) 1: I(x,t) I(x+ x,t) I(0,t) I(l,t) V in V(x,t) V(x+ x,t) V(0,t) l V(l,t) 2: 0 x x+ x 3: V in 3 V in x V (x, t) I(x, t
1 1 2 2 2r d 2r d l d (a) (b) (c) 1: I(x,t) I(x+ x,t) I(0,t) I(l,t) V in V(x,t) V(x+ x,t) V(0,t) l V(l,t) 2: 0 x x+ x 3: V in 3 V in x V (x, t) I(x, t) V (x, t) I(x, t) V in x t 3 4 1 L R 2 C G L 0 R 0
II (No.2) 2 4,.. (1) (cm) (2) (cm) , (
II (No.1) 1 x 1, x 2,..., x µ = 1 V = 1 k=1 x k (x k µ) 2 k=1 σ = V. V = σ 2 = 1 x 2 k µ 2 k=1 1 µ, V σ. (1) 4, 7, 3, 1, 9, 6 (2) 14, 17, 13, 11, 19, 16 (3) 12, 21, 9, 3, 27, 18 (4) 27.2, 29.3, 29.1, 26.0,
1 1.1 [ 1] velocity [/s] 8 4 (1) MKS? (2) MKS? 1.2 [ 2] (1) (42.195k) k 2 (2) (3) k/hr [ 3] t = 0
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基礎数学I
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18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α
18 I ( ) (1) I-1,I-2,I-3 (2) (3) I-1 ( ) (100 ) θ ϕ θ ϕ m m l l θ ϕ θ ϕ 2 g (1) (2) 0 (3) θ ϕ (4) (3) θ(t) = A 1 cos(ω 1 t + α 1 ) + A 2 cos(ω 2 t + α 2 ), ϕ(t) = B 1 cos(ω 1 t + α 1 ) + B 2 cos(ω 2 t
70 : 20 : A B (20 ) (30 ) 50 1
70 : 0 : A B (0 ) (30 ) 50 1 1 4 1.1................................................ 5 1. A............................................... 6 1.3 B............................................... 7 8.1 A...............................................
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)
No.004 [1] J. ( ) ( ) (1968) [2] Morse (1997) [3] (1988) 1
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No.004 [1] J. ( ) ( ) (1968) [2] Morse (1997) [3] (1988) 1 1 (1) 1.1 X Y f, g : X Y { F (x, 0) = f(x) F (x, 1) = g(x) F : X I Y f g f g F f g 1.2 X Y X Y gf id X, fg id Y f : X Y, g : Y X X Y X Y (2) 1.3