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Leslie Morgan
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5 ,

6 t = 0, 1,... δ [0, 1) c 0

7 G 0 [c 0, c 0 ] g 0 (c 0 ) > 0 c 0 (c 0, c 0 ) c 0 δ [0, 1) B π π B ( ) B B π B = B π B B (1 π) B B v 0 F 0 [v 0, v 0 ] f 0 (v 0 ) > 0 v 0 (v 0, v 0 ) v 0 v := K B + ˆK B v 0 K B ˆK B K B ˆK B v F [v, v] c := K S + ˆK S c 0 G [c, c] G(c) := G 0 ((c K S )/ ˆK S ) [c, c] F (v) := F 0 ((v K B )/ ˆK B ) [v, v] c := ˆK S c 0 +K S c := ˆK S c 0 +K S v := ˆK B v 0 + K B v := ˆK B v 0 + K B

8 f g F G Φ(v) := v 1 F (v) f(v) Γ(c) := c + G(c) g(c) Φ(v) Γ(c) c = v c = v ω = (ω t ) t=0 ω t( p) t p p t t p t G(p) 1 F (p) Φ(p) Γ(p) c 0 = v 0 c 0 = v 0 c v c v c > v v < c [c, v] G F 1 δ F

9 δ = 0 B > 1 B = 1 δ = 0 δ > 0 B πb δ = 0 ω t R ωt (p t ) t p t R ωt (p t ) = (p t ω t (p t ))(F (2) (p t ) F (1) (p t )) + v p t ( p ω t ( p))df (2) ( p), 1 F (1) (p t ) F (1) (v) := B=0 π BF (v) B F (2) (v) := F (1) (v) + (1 F (v)) B=1 π BBF (v) B 1

10 c ω p = (p t ) t=0 W S (c, p, ω) := t 1 (R ωt (p t ) c)(1 F (1) (p t )) δf (1) (p τ ), t=0 τ=0 x 1 τ=x y τ = 1 (y t ) T t=1 P (c) = (P t (c)) t=0 W S(c, p, ω) ω t t c p t = P t (c) ω t (p t )(F (2) (p t ) F (1) (p t )) + v p t ω t ( p)df (2) ( p) c p ω W I (c, p, ω) := ( v ) t 1 ω t (P t (c))(f (2) (P t (c)) F (1) (P t (c))) + ω t ( p)df (2) ( p) δf (1) (P τ (c)). P t(c) t=0 ω W I W I + W S τ=0 W (α, ω) := E c G [αw I (c, P (c), ω) + (1 α)(w I (c, P (c), ω) + W S (c, P (c), ω))], α [0, 1] α = 0 α = 1 α = 0 ω P (c) ω 1 F (1) (p t ) F (2) (p t ) F (1) (p t ) p F (2) W (α, ω) E c [α 0 W I (c, P (c), ω) + (1 α 0 )W S (c, P (c), ω)] α 0 = 1/(2 α)

11 F G α > 0 ω = (ω t ) t=0 p = (p t ) t=0

12 ω t = ω p t = p t ω p W S (c, p, ω) = (R ω (p) c)(1 F (p)), ( ) 1 F (p) := (1 F (1) (p)) δ t F (1) (p) t t=0 = 1 F (1)(p) 1 δf (1) (p) Φ ω (p) := p ω(p) (1 ω (p)) 1 F (p) f(p) ω Φ ω (p) := v ω(v) v p 1 δf (1) (v) Φ 1 δ ω(v)dv. Φ ω (p) Φ ω (p) p W S ω Φ ω (p) c P (c) = 1 Φ ω (c) 1 F 1 δ

13 Φ ω Φ ω Φ ω Φ ω Φ(v) := Φ 0 (v) R(p) := R 0 (p) Φ ω Φ ω (v) = v δ = 0 Φ ω Φ ω ω(p) = 0 p Φ ω(p) = bp b [0, 1] Φ b (v) := Φ ω (v) ω(p)=bp Φ b (v) := Φ ω (v) ω(p)=bp Φ b (v) = (1 b)φ(v) Φ b (p) = (1 b) Φ(p) b c P (c) = Φ 1 (c/(1 b)). P (c) ω ω P (c) ω P (c) W (α, ω) ω

14 Γ α (c) := αγ(c) + (1 α)c Γ(c) c t max b t Φ(vbt ) Γ α (c), v c

15 Γ α (c) Φ(v) Γ(c) Φ(v) Γ α (c) c P (c) := Φ 1 (Γ α (c)) P (c) Φ Γ α (c) c P (c) 1 F (P (c)) Γ 1 α (v) v V (c) c [c, Γ 1 α (v)] P (c) V (c) = Γ 1 α (v) c (1 F (P (y)))dy. ω t ( p) t = 0, 1,.. ω t (p) = ω(p) := p v p [ ] Γ 1 α ( Φ(v)) + δv (Γ 1 α ( Φ(v))) 1 F (p) f(v)dv.

16 c P (c) p P (c) δv (c) ω(p) (R ωt (p) c)(1 F 1 (p)) F 1 (p)δv (c) (p ω(p))[f (2) (p) F (1) (p)] + v p [y ω(y)f (2) (y)dy + F (1) (p)[c + δv (c)]. [ ] 0 = f (1) (p) Γ 1 α ( Φ(p)) δv (Γ 1 α ( Φ(p))) + c + δv (c), F (2) (p) F (1) (p) = f (1) (p)(1 F (p))/f(p) p = P (c) p δ = 0 π 1 = 1 α = 1 F G [0, 1] 1/24 B π B = 1 F P (c) = Φ 1 (c) α = 0 G(c) = c σ c [0, 1] σ > 0 ω(p) = p/(1 + σ) F π

17 F 0 G 0 v 0 c 0 [0, 1] K j := (K S j, ˆK S j, K B j, ˆK B j ) j 0 K 0 = (0, 1, 0, 1) c = Kj S + ˆK j S c 0 G j (c) = G 0 ((c Kj S )/ ˆK j S ) v = Kj B + ˆK j B v 0 F j ((v Kj B )/ ˆK j B )

18 F F j G G j K B j > 0 ˆKB j = 0 x v = λ j x + (1 λ j )v 0 λ j K B j = λ j x ˆK B j = 1 λ j K S j ˆK S j K B j ˆKB j K S j ˆK S j u B j u S j K j u B j u S j [c j, c j ] [v j, v j ] [c j, v j ] u S j := (v j c j )/(c j c j ) u B j := (v j c j )/(v j v j ) (c j, u S j, v j, u B j ) K j (c j, u S j, v j, u B j ) c > v j c v j c 0 u S j G 0 (u S j ) 1 F 0 (1 u B j ) c := (c 0 c j )/(v j c j ) ṽ := (v 0 c j )/(v j c j ) p := (p c j )/(v j c j ) c ṽ [c j, v j ] Gj F j G j ( c) := G 0(u S j c) G 0 (u S j ) Fj (ṽ) := 1 1 F 0(1 u B j (1 ṽ)) 1 F 0 (1 u B j ),

19 ω j ( p) = ω(p)/(v j c j ) c j v j c j < v j j u S j u B j j j lim j Fj (ṽ) = F (ṽ) := 1 (1 ṽ) β lim j Gj ( c) = G ( c) := c σ ω j ( p) α p/(α + σ) lim ω j( p) = j α α + σ p. G 0 F 0 [0, 1] G 0 (c 0 ) = c σ 0 σ > 0 F 0 (v 0 ) = 1 (1 v 0 ) β β > 0 G j ( c) = G 0 (u S j c)/g 0 (u S j ) = c σ 1 F j (ṽ) = (1 F 0 (1 u B j (1 ṽ))/(1 F 0 (1 u B j )) = (1 ṽ) β j G j F j j F j Φ F j

20 G 0 g 0 (c 0 ) c 4 0(1 c 0 ) 4 [0, 1] c 0 [0, u] u {1, 0.7, 0.5, 0.3, 0.2} u

21 u = 1 u = 0.7 u = 0.5 u = 0.3 u = 0.2 G u (c) = G 0 (c+u(c c))/g 0 (c+u(c c)) u {1, 0.7, 0.5, 0.3, 0.2} [0, 1] g 0 (c) c 4 (1 c) 4 [0, 1] G(1) = 1 G 0 (0.7) = 0.9 G 0 (0.5) = 0.5 G 0 (0.3) = 0.1 G 0 (0.2) = 0.02 δ = 0 G 0 G 0 (c 0 ) = c σ 0 c 0 [0, 1] Γ α,0 (c 0 ) := c 0 + αg 0 (c 0 )/g 0 (c 0 ) = c 0 (1 + α/σ) ω(p) = p E v0 F 0 [Γ 1 α,0(φ 0 (v 0 )) v 0 p] = p Γ 1 α,0(e v0 F 0 [Φ 0 (v 0 ) v 0 p]). Γ 1 α,0 E v0 F 0 [Φ 0 (v 0 ) v 0 p] = p ω(p) = p Γ 1 α,0(p) = pα/(α + σ) j G 0

22 u = 1 u = 0.7 u = 0.5 u = 0.3 u = 0.2 ω( ) F u (v) = 1 [1 F 0 (v u(v v))]/[1 F 0 (v u(v v))], G u (c) = G 0 (c + u(c c))/g 0 (c + u(c c)) u {1, 0.7, 0.5, 0.3, 0.2} [0, 1] f(x) = g(x) x 4 (1 x) 4 V ( ) G 0 (u) 1 F 0 (1 u) 0 u ω(p) j u S j u B j G (c) := ( ) σ c c 1 F (v) := c c ( ) β v v, v v c = v = c j v = c = v j G (c)

23 Γ α(c) = c(1 + α/σ) αc/σ p ω(p) = p Γ 1 α (p), v G (ˆp) Γ 1 α (v) G( c) = c σ σ σ ω( p) = α p/(σ + α) [c, v] G j ( c) Φ 1 (Γ α (c)) δ > 0 Γ α (c) v p E[c c p] (v (αp + (1 α)e[c c p]))g (p) (v Γ α(p))g (p) = 0

24 Kj B Kj S Kj B +Kj S v 0 c 0 p p µ p µ µ µ 0 β = e βµ γ = e γµ

25 δ = e δµ v 0 t=0 (v 0 p) β(1 β) t δt = (v 0 p)β β = β/[1 (1 β δ)] v = v 0 β(v 0 p) = βp+(1 β)v 0 K B = βp ˆK B = 1 β v = βp + (1 β)v 0 p µ c = c 0 + γ(p c 0 ) = γp + (1 γ)c 0 γ = γ/[1 (1 γ) δ] c = γp + (1 γ)c 0 p µ 0 µ 0 [c, v] p v F v F v = v 0 δv B (v 0 ) V B (v 0 )

26 δ δ v = v 0 δv B (v 0 ) [c, v] π 1 = 1 δ = 0 F G v c Φ Γ 1 α δ = 0 ω(p) = p E[Γ 1 α (Φ(v)) v p]. G(p) 1 F (p) p v η d (v) := vf(v)/(1 F (v)) c η s (c) := cg(c)/g(c)

27 η s α η s (c) α Γ α (c) = c(1 + α/η s (c)) α η s (c) η d Φ(v) = v(1 1/η d (v)) G Γ α ω η d Γ η d Γ 1 α (x) v p ω(p) = + {}}{{}}{ Γ 1 [Γ 1 α (v)] α (p) + v F [Φ(v) v v p] 2 [Γ α (v)] (n) {E v F [(Φ(v) v) n v p] E v F [Φ(v) v v p] n }, n! n=3 }{{} [Γ 1 α (v)] (n) n Γ 1 α (v) x Γ 1 α (x) p Γ 1 α (p) Γ α Γ α p η d G Γ α ω(p) F α = 1 Γ 1 1 (x) [Γ 1 1 (x)] = γ 2 G Γ 1 1 Γ 1

28 ω(p) = p Γ 1 (p) (γ 2 /2) [Φ(v) v v p] Γ 1 1 γ 2 < 0 Γ 1 1 Γ 1 1 F γ 2 < 0 γ 2 > 0 γ 2 = 0 Γ 1 α Γ Γ(c) = c + G(c)/g(c) G(c) = 1 G G ˆF F ˆη d (v) η d (v) ˆη d (v) > η d (v) v < v Φ(v) = v(1 1/η d (v)) ˆΦ(v) > Φ(v) (ˆΦ(v) v) 2 < (Φ(v) v) 2 v < v F ˆF ˆf(v)/(1 ˆF (v)) = ˆη d (v)/v > η d (v)/v = f(v)/(1 F (v)) E[ˆv ˆv p] E[v v p] p (ˆΦ(v) v) 2 < (Φ(v) v) 2 E[(ˆΦ(ˆv) v) 2 ˆv p] E[(Φ(v) v) 2 v p] ˆF F γ 2 < 0 [Φ(v) v v p] = E[(Φ(v) v) 2 v p] E[Φ(v) v v p] 2 = E[(Φ(v) v) 2 v p] (p v) 2

29 δ t F t πb t ω t t

30 ω p

31 σ

36 [(R ω (p) c)(1 F (p))] = [ Φ ω (p) c]f (p) Φ ω (p) := R ω (p) R ω(p) 1 F (p). f (p) Φ ω R ω (p) = (p ω(p))(f (2)(p) F (1) (p)) + v p (v ω(v))df (2)(v) 1 F (1) (p) R ω (p) = v p Φ ω(v)df (1) (v) 1 F (1) (p) Φ ω (p) := p ω(p) (1 ω (p)) 1 F (p) f(p) R ω R ω (v) = v [R ω (p)(1 F (1) (p))] R ω F (2) (p) F (1) (p) f (1) (p) = 1 F (p). f(p) f (1) (v) f (1) (v) = f(v) B=1 π BBF (v) B 1 F (2) (v) F (1) (v) = (1 F (v)) B=1 π BBF (v) B 1

37 R ω(p) = f (1)(p) 1 F (1) (p) (R ω(p) Φ ω (p)) f (1) (p) 1 F (p) 1 F (1) (p) f (p) = 1 δf (1)(p) 1 δ Φ ω (p) = R ω (p) (R ω (p) Φ ω (p)) 1 δf (1)(p) 1 δ Φ ω(p) = 1 δf (1)(p) Φ 1 δ ω(p) Φ ω Φ ω Φ ω (p) v p = v Φ ω ( Φ ω (p) c)f (p) = 0 1 Φ ω (c) c Φ β lim v v d dv [ ] 1 F (v) f(v) = lim v v d [v Φ(v)] = β. dv F F F j F (ṽ) = 1 (1 ṽ) β ṽ G j

38 β := lim v v 1 Φ (v) σ := lim c c Γ (c) 1 β := 1/β σ := 1/σ (v v)f(v) lim v v 1 F (v) = lim v v v v v Φ(v) = lim v v 1 1 Φ (v) = β. p k = R(p) k 1 F (k) := 1 F (R 1 (k)) ω(k) k ω(k) ω(k) = k v k Γ 1 α (Φ(v))f(v)dv. 1 F (k) ω (1 F (k))(k ω(k) c). ω(k) v max Γ 1 α (Φ(v))f(v)dv (1 F (k))c. k k 0 = f(k(c)) [ Γ 1 α (Φ(k)) c ], Φ(k) = Γ α (c) Φ(v)

39 F G ω α p ω α+σ α p α+σ F G [v, v] [c, c] F (v) = 1 [(v v)/(v v)] β G (c) = [(c c)/(c c)] σ Γ α(c) = c+(c c)α/σ ω (p) = p E v [Γ 1 α (Φ (v)) v p] = p Γ 1 (E v [Φ (v) v p]) = p Γ 1 (p), α α Γ α p F Φ E v [Φ(v) v p] = p Γ α [ ] α ω (p) = (p c). α + σ ω F G ω p F G [c j, (v j c j )/u S j + c j ] [0, 1] [v j (v j c j )/u B j, v j ] [u j 1, u j ] u j > 0 j u j 0 j u 0 F G ˆF j Ĝj

40 Ĝ j (ĉ) = G(ĉ) ˆF j (ˆv) = F (ˆv + (1 u)) ˆω(ˆp) = uˆp 1ˆp ˆΓ 1 α (ˆΦ(uˆv))d ˆF (uˆv) 1 ˆF (uˆp) uˆp p ˆω β [ 1 lim ˆF ] (uˆv) u 0 (uˆv) ˆf(uˆv) = lim v 1 [ 1 1 ˆF ] (uˆv) u 0 ˆv u f(uˆv) β 1 u 0 ˆΦ(uˆv) ˆv 1 ˆv u β [ 1 F (v ) f(v ) ] = 1 β. 1 u ˆΓ α (uĉ) u 0 ( ĉ 1 + α ) σ 1 1 ˆΓ α (ux) u 0 u x 1 + α/σ, ˆ F (k) = ˆF ( ˆR 1 (k)), ˆF (p) = 1 ˆF (1) (p) 1 δ ˆF (1) (p), ˆF(1) (p) = ˆR j (p) = uj p π B ˆF (p) B B=0 ˆΦ(v)d ˆF (1) (v) 1 ˆF. (1) (p) ˆR 1 j ˆR 1 j ˆR j

41 ˆR j ˆω ˆR ˆR ˆR j (p) ˆR j (p) = uj p ˆΦ(v)d ˆF (1) (v) 1 ˆF (1) (p) = 1 p+1 u j (Φ(y) (1 u j ))f (1) (y)dy 1 F (1) (p + 1 u j ) j u j ˆR j ˆω ω ω(p) = ω(p) ˆω ω(p) v p ω(p) = p [Γ 1 α ( Φ(v)) + δv (Γ 1 α ( Φ(v)))]dF (v), 1 F (p) V (c) = Γ 1 α A := v p Γ 1 α (v) Γ 1 α c (v) (1 F ( Φ 1 (Γ α (y))))dy B := v p Γ 1 α ( Φ(v))dF (v) ( Φ(p)) (1 F ( Φ 1 (Γ α (y))))dydf (v) ω(p) ω(p) = p (B + δa)/(1 F (p)) Φ(p) = Φ(R(p)) B B A A = = Γ 1 α Γ 1 α (p) Γ 1 α (v) Γ 1 α (p) (v) Φ(Γα(y)) p df (v)(1 F ( Φ 1 (Γ α (y))))dy (F ( Φ(Γ α (y))) F (p))(1 F ( Φ 1 (Γ α (y))))dy. A ˆω F G

42 Γ 1 α (x) v Γ 1 α (x) = Γ 1 α (v) + [Γ 1 α (v)] (x v) + [Γ 1 α (v)] (x v) n v γ n := [Γ 1 α (v)] (n) φ := Φ(v) p ω(p) =E[Γ 1 α (φ) v p] γ = n n! E[(φ v)n v p] = n=0 n=0 n=3 [Γ 1 α (v)] (n) n! (x v) n. γ n n! {(E[φ v p] v)n (E[φ v p] v) n + E[(φ v) n v p]} =Γ 1 α (E[φ v p]) + =Γ 1 α (p) + n=0 n=0 γ n n! {E[(φ v)n v p] (E[φ v p] v) n } γ n n! {E[(φ v)n v p] (E[φ v p] v) n }, E[Φ(v) v p] = p n = 0 n = 1 n = 2 [φ v v p] = E[(φ v) 2 v p] (E[φ v p] v) 2 t t 1 δ δ δ

43 c t v b [v, v] v b = v t = (vb t)b b=1 t b = 1,.., B v = (v t ) t=0 Q t S (v t, c) t Q t b (v t, c) b M t S (v t, c) b M t b (v t, c) (v t, c) B Q t b(v t, c) Q t S(v t, c) b=1 t (v t, c) t Q t S (v t, c) t+1 (1 δ)(1 Q t S (v t, c)) Q t B(v t, c) = (Q t 1(v t, c),.., Q t B (v t, c)) M t B(v t, c) = (M t 1(v t, c),.., M t B (v t, c)) (v, c) Q S (v, c) = ( Q t S(v t, c) ) t=0 Q B (v, c) = ( Q t B(v t, c) ) t=0 M S (v, c) = ( M t S(v t, c) ) t=0 M B (v, c) = ( M t B(v t, c) ) t=0. Q M {Q S (v, c), Q B (v, c)} {M S (v, c), M B (v, c)} (v, c) Q, M Q

44 t c [ ] t 1 q S (c) := E v Q t S(v t, c) δ(1 Q τ S(v τ, c)), t=0 τ=0 [ ] t 1 m S (c) := E v MS(v t t, c) δ(1 Q τ S(v τ, c)). t=0 τ=0 c W S (c) = m S (c) q S (c)c, c B t 1 W I (c) = E v Mb(v t t, c) δ(1 Q τ S(v τ, c)) m S (c), t=0 b=1 τ=0 W i (c) i = I, S max E c[αw I (c) + (1 α)(w I (c) + W S (c))] Q,M

45 ω α q S (c) c m S (c) = q S (c)c + c c q S (x)dc + W S (c) q S (c) c W S (c) 0 W S (c) αw S (c) W S (c) = 0 α [0, 1] t p t Q t b (v t, c) = 1 v b = max{v t } v b p t Q t i(v t, c) = 0 i = 1,.., B Q, M t p t ˆQ, ˆM t (v t, c) Q t b (v t, c) = 1 b b

46 t ˆQ, ˆM t ˆQ, ˆM ˆQ, ˆM W I (c) W S (c) ˆQ, ˆM Q p(c) = (p t (c)) t=0 c t p t (c) k t := R(p t ) t t p(c) k(c) = (k t (c)) t=0 k = (k t ) t=0 q t (k) := ( 1 F (1) (R 1 (k t )) ) t 1 δf (1) (R 1 (k τ )) δ t t=0 q t(k)k t k t=0 q t(k) k = τ=0 t=0 q t(k)k t t=0 q t(k) k 1 δ

47 k k ( t=0 q t(k)k t )/( t=0 q t(k)) t=0 q t(k) k [v, v] { T } 1 F T (k) := max q t (k) (k t) T t=0 t=0 T t=0 q t(k)k t T t=0 q t(k) 1 F (k) := lim 1 F T (k). T { T } (kt (k)) T t=0 max (kt) T t=0 t=0 q t(k) t=0 = k, T t=0 qt(k)kt T t=0 qt(k) kt (k) = k t ( T t 1 ) (1 1 F (k) = lim δf (1) (R 1 (k)) F(1)(R 1 (k)) ) T = lim T τ=0 1 δ T +1 F (1) (R 1 (k)) T +1 ( 1 F(1) (R 1 (k)) ) 1 δf (1) (R 1 (k)) = 1 F (1)(R 1 (k)) 1 δf (1) (R 1 (k)) = 1 F (R 1 (k)). = k c k W I (c) = k(c)(1 F (k(c))) m S (c) W S (c) = m S (c) q S (c)c W S (c) = 0 c αw I (c) + (1 α)(w I (c) + W S (c)) = k(1 F (k)) cq S (c) α c c q S (x)dx.

48 q S (k) = 1 F (k) max k(c) c c [k(c) Γ α (c)] (1 F (k(c)))g(c)dc Γ α (c) := c + α G(c) g(c). Γ(c) Γ α (c) k 0 = f(k(c)) [ Φ(k(c)) Γ α (c) ], k(c) = Φ 1 (Γ α (c)) Φ(v) Φ(k) Γ α (c). k (c) := Φ 1 (Γ α (c)) F (k) = F (R 1 (k)) Φ(k) = Φ(R 1 (k)) v R 1 (k (c)) Φ Φ(v) Γ α (c) δ = (δ t ) t=0 F = (F t ) t=0 δ t t F t t t Φ t (v) = v 1 Ft(v) f t(v) v δ τ = δ T δ τ = δ τ T δ τ = 0 τ > T π t B t F (1),t(v)

49 F (2),t (v) t p t t R t (p) = v p Φt(v)dF (1),t(v) 1 F (1),t (p) k t=0 q t(k) q t (k) := ( 1 F (1),t (R 1 t (k t )) ) t 1 τ=0 δ τ F (1),τ (R 1 τ (k τ )). 1 F (k) := lim T 1 F T (k) 1 F T (k) T t=1 q t(k) ( T t=0 q t(k)k t )/( T t=0 q t(k)) = k k(c) ω(k) G ω t (p) ω F 1 F (v u(v v)) lim 1 u 0 1 F (v u(v v)) = 1 ( ) β v v =: F (v), v v β v u(v v)

50 F v u(v v) v F a n > 0 b n n = 1, 2,... lim [F (a nx + b n )] n = F (x) n x F F n (max{x 1, X 2,..., X n } b n )/a n n F f(v) > 0 v (v, v) F

51 1 F (v u(v v)) lim 1 u 0 1 F (v u(v v)) = 1 ( ) β v v, v v β F v u(v v) v f(v) > 0 v (v, v) F F F [ ] d 1 F (v) lim = β, v v dv f(v) β F (v) = 1 ( 1 + ) 1/ξ ξ(v µ). σ

52 ξ < 0 v = µ v = µ σ/ξ β = 1/ξ ξ 0 µ lim ξ 0 F (v) = 1 e (v µ)/σ ξ > 0 σ = µξ F (v) = 1 (µ/v) 1/ξ ξ > 0 ( 1 ) 1 F (u + x) F 1 F (u) u(x) u 0, F u [(1 F (v))/f(v)] = ξ ξ < 0 f(v)/(1 F (v)) ξ < 1 Φ (v) > 0

Discovery Guide. Beautiful, mysterious woman pursued by gunmen. Sounds like a spy story...

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