January 16, (a) (b) 1. (a) Villani f : R R f 2 f 0 x, y R t [0, 1] f((1 t)x + ty) (1 t)f(x) + tf(y) f 2 f 0 x, y R t [0, 1] f((1 t)x + ty) (1 t

Size: px
Start display at page:

Download "January 16, (a) (b) 1. (a) Villani f : R R f 2 f 0 x, y R t [0, 1] f((1 t)x + ty) (1 t)f(x) + tf(y) f 2 f 0 x, y R t [0, 1] f((1 t)x + ty) (1 t"

Transcription

1 January 16, Villani f : R R f 2 f 0 x, y R t [0, 1] f((1 t)x + ty) (1 t)f(x) + tf(y) f 2 f 0 x, y R t [0, 1] f((1 t)x + ty) (1 t)f(x) + tf(y) (simple) (general) (stable) f((1 t)x + ty) (1 t)f(x) + tf(y) 3 f((1 t)x + ty) (1 t)f(x) + tf(y) (useful) (simple, general, stable, useful Villani 2. 1

2 3. f, g C f g C f g S 4. (Φ (L) ) 1 (Φ (L) ) 1 (x) = 2 n Φ (L) (2 1 x) L p, 1 p < S (i) (ii) f N 7. B s 0 p 0 q 0 = F s 0 p 0 q 0 a k f = a k γ (2k ) k=1 f B s0 p 0q 0 \ F s0 p 0q 0 8. f Fp s0 0q 0 \ Bp s0 0q 0 (3.75) Ψ 9. (3.77) (3.78) (3.80) {h (j) } j N0 Ψ (F s pq ) q > 1 (3.78) q = 1 {φ j (D)F (j) } j N Lp (l 1 ) = φ j (D)F (j) j=1 L p = sup φ j (D)F (j) G G L p j=1 L 1 = sup sup φ j (D)F (j) (x)a j (x)g(x) dx G L p a j L, a j L R n j=1 2

3 φ j (D) {φ j (D)F (j) } j N L p (l 1 ) = sup sup G L p a j L, a j L R n j=1 F (j) (x)φ j (D)[a j G](x) dx {φ j (D)F (j) } j N Lp (l 1 ) sup sup G L p a j L, a j L sup sup G L p a j L, a j L {F (j) } j N L p (l 1 ) 10. R n j=1 R n j=1 F (j) (x)m[a j G](x) dx F (j) (x)mg(x) dx S 0 Z S 11. [f], Φ([f]) 12. (3.101) f, τ φ j (D)f [f] 13. L = 0 Φ(f) = φ j (D)f S 14. j= L n + L 15. c l(q l,2 j ) 2l(Q k,2 j 1) 3

4 16. non-smooth 17. L br L b {R j } j=1 L b Rj L b, R j L b = lim L b Rj b L j 18. (1 ψ(d))f(x) h(x) dx 19. x P sup 2 j (c(p ) y) n η f(y) y R n η,q inf M (η) [χ 10nP f](x) x P + 2 jn( 1 q 1 η ) l(p ) n η sup k Z n ( 20. f(y) q dy (l(p )k+p ) φ j = ψ j (φ j 1 + φ j + φ j+1 ) ψ S 21. ) 1 q 4

5 F s q Ḃ s f S B(2 j ) f(x) sup m Z n 22. ( 1 Q jm Q jm f(y) q dy T q ({F j } j A ) G q ({G j } j A ) q = 23. O λ 24. O O = {F > λ} ) 1 q µ{(y, t) R n+1 + : F (y, t) > λ} 3 n µ Carleson O 25. u( ; t) M s pq f M s pq 26. M 0 21 M 0 p1 27. supp(ψ) B(2 r ) r < r supp(ψ) B(2 r ) r = ρ

6 R 29. Λ νm 2 ν(s n p ) inf M ( η0 2 ) [φν (D)f](y) y Q νm i. ν 1 y Q νm 1 ( φν m ) (D)f M ( η 0 2 ) [φν (1 + 2 ν y 2 ν m ) 2n/η0 2 ν (D)f](y) ( Λ νm = 2 (s n p )ν m ) φ ν (D)f 2 ν = (1 + 2 ν y 2 ν m ) 2n/η 0 2 (s n p )ν 1 ( m ) φ (1 + 2 ν y 2 ν m ) 2n/η ν (D)f 0 2 ν (1 + n) 2n/η0 2 (s n p )ν 1 (1 + 2 ν y 2 ν m ) 2n/η0 φν (D)f 2 (s n p )ν M ( η 0 2 ) [φν (D)f](y) y Λ νm 2 ν(s n p ) inf y Q νm M ( η0 2 ) [φν (D)f](y) ii. ν = z C 1 ρ 1, ρ 2, ρ 3, ρ 4 ( m 2 ν ) ρ 1 (l) = s q q l s l, ρ 2 (l) = p p l q q l, ρ 3 (l) = 1 p p l, ρ 4 (l) = q q l, l = 0, 1 z C 1 ρ 1, ρ 2, ρ 3, ρ 4 ρ 1 (l) = s q q l s l, ρ 2 (l) = p p l q q l, ρ 3 (l) = 1 p p l, ρ 4 (l) = q q l, l = 0, 1 6

7 31. (5.126) f F s pq f = λ νmaνm ν N 0 m Z n λ = {λ νm} ν N0 λ fpq f F s pq, α a νm 2 ν(s n/p)+ α ν a νm = 2 ν(s n/p) a νm, λ νm = 2 ν(s n/p) λ νm f = ν N 0 m Z n λ νm a νm { } λ fpq = 2 νn/p λ νmχ Qνm m Z ν=0 { } = 2 νs λ νm χ Qνm m Z ν=0 L p (l q ) L p (l q ) 32. (Key theorems in function spaces) 33. B M (R n ) B N (R n ) 34. (5.150) I ψ B 1 (R n )+ ψ 1 B 1 (R n ) B 1 (R n ) B N (R n ) 7

8 35. (5.158) (5.158) f S S B11 n λβ νm, (βqu) νm f = λ β νm(βqu) νm (1) m Z n β N n 0 ν N 0 B n 11 Bn 11 B 0 1 BUC (5.158) f S 36. φ(x) = (x, x n ω(x )) ω D(R n 1 ) 37. sup j N 0 (m j+1 m j ) < 0 lim j m j = 38. y β η α e iy η dy dη = (2π) n ( i) α α!δ αβ R 2n 39. (6.119) Θ 40. f g, f, g, S(R n ) S(R 2n ) T 41. T [φ(τ )] L T [(1 κ B )φ(τ )] K 8

9 43. A 44. l 1 Ψ l ( y)φ k ( y) = Φ k ( y) β [(1 + ξ 2 ) σ (ψ j+3 ψ j 1 )] L 1 2 j(2σ+n β ) M > 0 F 1 [(1 + ξ 2 ) σ (ψ j+3 ψ j 1 )](x) 2 j(2σ+n M) x M F 1 [(1 + ξ 2 ) σ (ψ j+3 ψ j 1 )](x) dx R n ( ) n+1 2 j(2σ+n) 1 min 1, R 2 j dx x n 2 2jσ 9

10 3. ( 1) j+1 j=1 ξ j ( 1) j+1 j=1 ξ 2j 4. (1 ) s 1 2 (1 ) s (1.34) W m 2 = W m 2 := 6. m m 7. Ff(ξ) = F 1 f( ξ) Ff = F 1 f( ) U S φ S U S φ U ε 1, ε 2,..., ε L > 0 10

11 ε 1, ε 2,..., ε L (0, 1) j = 1, 2,..., k j = 1, 2,..., L N 1, N 2,..., N L N α 1, α 2,..., α L N n n 0, β 1, β 2,..., β L N (2.8) L {ψ S : p αl,β l (ψ φ) < ε l } U l= j = 1, 2,..., k j = 1, 2,..., L α j + β j α j + β j φ (N;α) (x) := e N x α φ(x) φ (N;α) := e N α φ 17. φ y (x) := φ(x y) φ y := φ( y) 11

12 18. sup p N (φ t ) t [0,1] n sup p N (φ y ) y [0,1] n φ y = φ( y) 19. (2.13) 2 4 N 1 2 N (2.19) τ S 21. α f S φ S 22. ψ η 23. (2.30) φ S 24. Θ j (ξ) := Θ(2 j ξ) Θ j := Θ(2 j ) 12

13 25. ψ(ξ) := η( ξ ), φ(ξ) := η( ξ ) η(2 ξ ) ψ := η( ), φ := η( ) η(2 ) (2.38) p N (φ) = p N (φ) = α N 0 n α N = α N 0 n α N α N 0 n α N = α N 0 n α N sup x N α φ(x) x R n sup α φ(x) (1 + x 2 ) N x R n ( ) sup x N α φ(x) x R n ( ) sup α φ(x) (1 + x 2 ) N x R n

14 dx dξ 31. j = 1, 2,..., n 32. ix ξ ix ξ 33. E(x) := exp ( 1 2 x 2), E t (x) := E(tx) E := exp ( 1 2 2), E t := E(t ) 34. FE t (ξ) = 1 ( ) ξ t n E (2π) n 2 t FE t = 1 ( ) t n E (2π) n 2 t 35. Fφ(x)Fψ(x) dx Fφ(ξ)Fψ(ξ) dξ 36. (2π) n 2 (2π) n F (x) := m Z n f(x 2πm) 14

15 F := m Z n f( 2πm) (2.90) 40. Φ (L) (x) := vol(s n 1 ) 1 κ (L+1) ( x ), Ψ (L) (x) := vol(s n 1 ) 1 ψ (L) ( x ) Φ (L) := vol(s n 1 ) 1 κ (L+1) ( ), Ψ (L) := vol(s n 1 ) 1 ψ (L) ( ) 41. γ L γ = L 42. α L α = L 43. L p S 0 < p < 1 L p S 44. sup λ B λ < X = R B λ = ( λ, λ), λ (0, ) 15

16 45. (2.112) f(x) dx f(y) dy 46. max(1, p) min(1, p) 47. n n (p 1 ) p n n (p 1 ) p (p < 1 ) 0 (p > 1 ) 49. min(1, p) η = 2 min(1, p) η := f p... f p h n p (2.158) (x z) ( z) 16

17 f(x) dx f(y) dy f(x) M dyadic f(x) f(x) M dyadic f(x) R jn f(x) dx n 2 jn R n f(y) dy 56. T g(x) dx R n T g(x) dx = R n 57. λ λ > K(x) := F 1 [ψ(2 j )m](x) K := j= j= F 1 [ψ(2 j )m] 17

18 59. x R n \ {0} 60. m j (ξ) = ψ(2 j ξ)m(ξ) m j = ψ j m 61. ψ(2 j ) ψ(2 j ξ) 62. m(d)f j (x) m(d)f(x) m(d)f j (x) m(d)f(x) Ψ χ B(4)\B(3) χ B(3.9)\B(3.1) Ψ 2 χ B(4)\B(3) 64. χ B(2) Ψ χ B(3) χ B(2.9) Ψ χ B(3) 65. f = f := 66. φ j (D)f = 2 jn a j F 1 [φ j ] 18

19 φ j (D)f = 2 jn a j F 1 [φ j Ψ j+1 ] 67. min(p, q) < ψ j+2 (2 j+2 ) (ψ j+2 (2 j+2 ) ψ j 3 (2 j+2 )) 69. ψ j+2 (2 j+2 ) (ψ j+2 (2 j+2 ) ψ j 3 (2 j+2 )) 70. ψ j+2 (2 j+2 x) (ψ j+2 (2 j+2 x) ψ j 3 (2 j+2 x)) 71. F (x) p 0 p 1 f F s 0 p 0 F (x) p 0 p 1 f 1 p 0 p 1 F s 0 p f a := a j F 1 [κ j ] f a := j=1 2 j(n+s n/p) a j F 1 [κ j ] j=

20 (3.1) 74. (3.51) F s pq (3.1) F s pq 75. (3.51) N A s pq B s n/p A s pq B 0 1 BUC 77. (3.1) 78. (3.51) N (3.52) supp(f N G N ) B(2 N+4 ) supp(f(f N G N )) B(2 N+4 ) BUC 20

21 C c C c 82. (3.56) κ η 83. (3.56) χ B(2.1)\B(1.9) κ χ B(2.2)\B(1.8) χ Q(2.1)\Q(1.9) η 1 χ Q(2.2)\Q(1.8) 84. F 1 η(2 k ) F 1 [η(2 k )] 85. e i2k x 1 F 1 [η( 2 l e 1 )] F 1 [η( 2 l e 1 )]( 2 l e 1 ) 86. F 1 (α (k) ) α (k) 87. F 1 (β (k) ) β (k) 88. (3.61) (3.62) (3.61) (3.62) 21

22 89. (3.61), (3.62) F 1 (γ (k) ) γ (k) 90. F 1 (δ (k) ) δ (k) 91. (3.64) χ B(1) ψ χ B(3/2) χ Q(11/10) ψ χ Q(3/2) 92. η ρ η (3.68) (η) (ρ) η F 1 [δ (k) ] δ (k) 95. F (A s pq) Ψ (A s pq)

23 φ j (D)F (j) 2 js φ j (D)F (j) 97. (3.77) F (F s pq ) Ψ (F s pq ) 98. (3.79) L p (l q ) L p (l q ) 99. (3.80) φ j (D)h (j) 2 js φ j (D)h (j) 100. (3.80) g F s pq {h (j) } j N0 Lp (l q ) Ψ (F s pq ) < g F s p q {h(j) } j N0 L p (l q ) Ψ (F s pq ) < 101. (3.80) F (0) = ψ(d)h (0), F (j) = φ j (D)h (j) F (0) := ψ(d)η, F (j) := 2 js φ j (D)η 102. h (j) φ j (D)η 2 js h (j) φ j (D)η 103. φ j (D)ηh (j) 2 js φ j (D)h (j) 23

24 104. τ j (x) = τ(2 j x) τ j = τ(2 j ) 105. S 0 S 0 S 106. (3.97) lim J J j= J F 1 [ 2s Ff] F 1 [ 2s φ j Ff] 107. f A s pq Φ S 0 p N (f) f Ȧs pq 108. F s pq Ḃs p min(p,q) p N (Φ) Φ Ȧs pq F s pq Ḃs p max(p,q) j 2(n+1) 2 j 2(n+1) 24

25 (s+2n+2)j f Ȧs pq n+1 τ 1 2 (n+1)j f Ḃ n 1 n+1 τ f Ȧs pq p n+1 (τ) (τ S 0 ) f Ḃ n 1 p 2n+2(τ) (τ S 0 ) 112. s > 0 s R 113. s > 0 s R 114. (3.104) (3.104) L > s n p (3.105) 116. (3.105) A s pq Ḃs σp max(1,p) A s pq Ḃs n/p 117. (3.105) (3.105) 25

26 f Ḃs, L > s 118. (3.108) (3.108) α φ j (D)f, τ φ j (D)f 119. (3.109) (3.109) R n α φ j (D)f, τ 2 α j φ j (D)f 120. (3.109) (3.109) 1 j= α φ j (D)f, τ 121. (3.110) (3.110) L s n p, q (3.112) (3.112) s < n p 123. j 0 j ( ) 1 s < n min p, 1 1 j= 26 2 (n+ α )j dx 2 j x (n+1) 2 α j φ j (D)f 2 ( α s)j f Ḃs s,l f Ḃs

27 s < n p 125. ( ) 1 s = n min p, 1, q 1 s = n p, q (3.122) (ω ) 2 ω n (1) A min(a, A + s) > (2) min(1, p) n A min(a, A + s) > min(1, p, q) 129. η 0 < η < min(1, p, q), Aη > n 130. (3.141) L = N A + 1 L = [A + 1] 131. (3.142) 27

28 2(j l)n+jn 2 j x A 2(j l)a+jn 2 j x A 132. (3.143) 2 (j l)n+jn 2 (j l)a+jn 133. (3.145) 2 (j l)n+jn 2 (j l)a+jn 134. (3.146) 2 (j l)n+jn 2 (j l)a+jn (j l)n+(l m)n+ln 2 (j l)a+(l m)a+ln (j m)n+ln 2 (j m)a+ln (j m)n+mn 2 (j m)a+mn (j m)n+mn 2 (j m)a+mn 28

29 139. (3.147) 2 (j m)n+mn 2 (j m)a+mn 140. N 1 Nη > n 141. (3.148) 2 (j m)nη+mn 2 (j m)aη+mn 142. (3.149) 2 (j m)(nη n)+mn 2 (j m)(aη n)+mn 143. (3.151) 2 (j l)(nη n)+jn 2 (j l)(a f η n)+jn (m l)nη 2 (m l)aη (m j)(nη n)+mn 2 (m j)(aη n)+mn

30 2 (m j)(nη n)+mn 2 (m j)(aη n)+mn (m j)(nη n)+mn 2 (m j)(aη n)+mn 148. (3.152) 2 (m l)((n+s)η n)+mn 2 (m l)((a+s)η n)+mn (1) L > σ p + s (2) L > σ pq + s 151. (3.160) 0 / supp(η) 0 / supp(fη) 152. sup y R n y N α [ y 2N Fφ(y)] f 1 sup y R n α [ y 2N Fφ(y)] 153. P M 1 P M 30

31 154. p, q > x y 156. ( L Φ) f(x) ( L Φ) l f(x) l= (4.2) L p 159. L 2 [0, 1] L 2 [0, 1) 160. l p L p [0, 1)

32 (3) (4.2) a j = 0 l L p [0, 1] L p [0, 1) 163. (4.14) N r j (t)φ j (D)f j=1 N r j (t)φ j (D)f L p j= fg fg 165. fg fg 166. sup fg fg g S, g p =1 sup fg p f F 0 p2 g S, g p =

33 s > 0, 1 < p, 0 < q s > 0, 1 p, 0 < q < p, 0 < q, s > 0 1 p, 0 < q, s > (4.21) φ j (D)f φ k (D)f 171. knr kn jnr 2 jn jnr 2 jn 174. (4.51) sup{l N : 1000Q l 1000Q j } <. j N sup {l N : 1000Q l 1000Q j } <. j N 33

34 175. Q(x) Q j 176. (4.54) ( x ψ (j) c(qj ) (x) := ψ l(q j ) ( ψ (j) c(qj ) := ψ l(q j ) ), Ψ(x) := ), Ψ := 177. l=1 l=1 sup {l N : supp(φ (l) ) 1000Q l } <. j N sup {l N : supp(φ (j) ) 1000Q l } <. j N 178. m(d)f(x) = lim R m(d)ψ(r 1 D)f(x) m(d)f(x) = m(d)ψ(r 1 D)f(x) 179. κ F N, ψ (l) (x), φ (j) (x) := ψ(j) (x) Ψ(x) ψ (l), φ (j) := ψ(j) Ψ ψ(r 1 D)f H p = M[ψ(R 1 D)f] p = sup κ j ψ(r 1 D)f. j Z ψ(r 1 D)f H p sup ψ(2 j D)ψ(R 1 D)f. j Z 180. ψ(r 1 D)f H p sup κ F N, j Z 34 κ j f p p p = Mf p = f H p.

35 ψ(r 1 D)f H p Mf p = f H p φ (j) (x) := φ (j) ( x) φ (j) := φ (j) ( ) 182. { m(d) f } 1 > λ λ 2 m(d) f(x) 2 dx R ( ) n e t 1 F (x) F, exp x 2 (x R n ) (4πt) n 4t F S (R n ) M 0 F (x) sup e t F (x) j Z { M 0 [m(d) f] } 1 > λ λ 2 m(d) f(x) 2 dx R n 183. { m(d) f } 1 > λ λ 2 f(x) 2 dx 1 R λ n { M 0 [m(d) f] } 1 > λ λ 2 f(x) 2 dx R n 1 λ 2 Mf(x) 2 dx + Ω R n \Ω 184. (4.62) Ψ (j) x (y) φ (j) (x y) ( F 1 m(d)(y) 1 I (j) R n \Ω Mf(x) dx + Ω ) F 1 m(d)(x z)φ (j) (z) dz R n Ψ (j) x ( F 1 m(d)(x z j + y) 1 I (j) (y) ) F 1 m(x z)φ (j) (z) dz R n φ (j) (z j y) 35

36 185. (4.63) Ψ (j) x f(x) Ψ (j) x f(z j ) 186. Ψ (j) x (y) = 1 ( [ F 1 m(d)(y) F 1 m(d)(x z) ] ) φ (j) (z) dz I (j) R n φ (j) (x y) Ψ (j) x (y) = 1 ( [ F 1 m(x z I (j) j + y) F 1 m(x z) ] ) φ (j) (z) dz R n φ (j) (z j y) 187. ( 1 [ F 1 m(d)(y) F 1 m(d)(z) ] ) φ (j) (x z) dz I (j) R n φ (j) (x y) ( 1 [ F 1 m(d)(x z I (j) j + y) F 1 m(d)(z) ] ) φ (j) (x z) dz R n φ (j) (z j y) 188. (4.64) α F 1 m(d)(w) α F 1 m(w) 189. Ψ (j) x Ψ (j) x f(x) f(z j ) l(q j ) n+1 Mf(x) x c(q j ) n+1 l(q j ) n+1 x c(q j ) n+1 Mf(z j) 36

37 190. Mf(x) p dx R n Mf(x) p dx R n 191. { m(d)[f f] > λ } Ω + { m(d)[f f] > λ } Ω Ω + 1 m(d)[f λ f](x) dx Ω R n \Ω { M 0 [m(d)[f f]] > λ } Ω + { M 0 [m(d)[f f]] > λ } Ω c Ω + 1 M 0 [m(d)[f λ f](x)] dx R n \Ω Ω M 192. { m(d)f > 2λ } { m(d) f) > λ } + { m(d)[f f] > λ } 1 λ 2 f(x) 2 dx + { Mf > λ } {Mf λ} 1 λ 2 Mf(x) 2 dx + { Mf > λ } {Mf λ} { M 0 [m(d)f] > 2λ } { M 0 [m(d) f)] > λ } + { M 0 [m(d)[f f]] > λ } 1 λ 2 f(x) 2 dx + { Mf > λ } {Mf λ} 1 λ 2 Mf(x) 2 dx + { Mf > λ } {Mf λ} 37

38 M 193. m(d)f p p = M 0 [m(d)f] p p = p 0 p λ p 1 { m(d)f > λ } dλ = p 2 p λ p 1 { m(d)f > 2λ } dλ 0 λ p 3 min{mf(x) 2, λ 2 } dλ dx R n 0 Mf(x) p dx f p H p R n 0 λ p 1 { M 0 [m(d)f] > λ } dλ = p 2 p λ p 1 { M 0 [m(d)f] > 2λ } dλ 0 λ p 3 min{mf(x) 2, λ 2 } dλ dx R n 0 Mf(x) p dx f p H p R n M 194. x R n \ 2Q 195. Q p Q 1 p (4.84) ( ) x ψ (j) c(qj ) (x) := ψ, φ (j) ψ (j) (x) := l(q j ) χ R n \Ω + k N ψ(k) 38

39 ( ) ψ (j) c(qj ) := ψ, φ (j) ψ (j) := l(q j ) χ Rn \Ω + k N ψ(k) 197. (4.90) α, β N 0 n 198. (4.94) p (j) (x) = f, e(j,k) φ(j) e (j,k) (x) I (j) k K j p (j) = f, e(j,k) φ(j) e (j,k) I (j) k K j 199. p (j) (x) p (j) 200. (4.95) x R n Q j Q j 202. Φ j,t (x) = Φ j,t (x) := 203. (4.109) M[e (j,k) φ (j) ] l(q j ) n, f, e (j,k) φ (j) Mf(x 0 ) 39

40 M[e (j,k) φ (j) ] 1, f, e (j,k) φ (j) l(q j ) n Mf(x 0 ) 204. (4.118) 1 ( ) t n φ b (j) (x 0 ) t Mf(z l(q j ) n+l+1 Mf(z j ) j) (l(q j ) + x 0 c(q j ) ) n+l+1 1 ( ) t n φ b (j) (x 0 ) t l(q j ) n+l+1 Mf(z j ) (l(q j ) + x 0 c(q j ) ) n+l+1 λl(q j ) n+l+1 (l(q j ) + x 0 c(q j ) ) n+l (x) (4.121) f φ (l) f, 1 ( ) t n φ t l J j,0 f φ (l) f, 1 ( ) x t n φ t l J j,0 ( ) sup x N x zj + t x x R n α φ 1 φ (l) (z j t x) t l J j,0 ( ) sup x N x zj + t v v R n α φ 1 φ (l) (z j t v) t l J j,

41 b (j) b 209. D = {f X p : Mf 1, f < } D = {f H p : Mf 1 < } Mf 1 < f L H p Y p Y p H p 211. (4.133) f H p f Y p f Y p f H p 212. (4.135) g 2 J 1 g 2 J (4.141) ( ) A j,k = 1 φ (l) l,2 φ (k) j 1 l,2 f p (k) j 1 2 φ (k) j j 1 A j,k = + l=1 ( 1 l=1 l=1 φ (l) l,2 j 1 p (l) 2 j φ (l) 2 j φ (k) 2 j 1 + ) l= (p, ) (1, ) φ (k) l,2 j 1 f p (k) 2 j 1 φ (k) 2 j 1 p (j,k,l) φ (l) 2 j l=1 p (l) 2 j φ (l) 2 j φ (k) 2 j 1 41

42 ( ψ ) sup ψ F N, 0<t 1 t n f t p 1 ( ) sup ψ t n f t p ψ F N, 0<t κ(td)f κ(td)f(x) 217. (4.167) p 218. (4.167) p 219. V m (x) V m (x) := χ m+[0,1] n(x)ψ(d)f(x) V m V m := χ m+[0,1] nψ(d)f dx 42

43 222. Q λ} λ} 223. λ > θ θ > ψ S ψ S 226. (2π) n 2 (2π) n R n 228. j 2 j Q 2 h BMO = j 2 j h BMO j 2 jn Q 2 h BMO = j 2 jn h BMO

44 φ j (D)f φ j (D)f(x) 230. ( ) 1 q f(x) q dx (l(p )k+p ) ( ) 1 q f(y) q dy (l(p )k+p ) 231. f(y) f(y) min w B(x,δ) min w B(y,δ) f(w) + 2δ f(w) + δ 232. sup x y n η y R n sup x y n η y R n 233. sup w B(x,δ) sup w B(y,δ) f(w), x y δ 1 f(w), δ 1 min f(w) + 2δ sup x y n η w B(x,δ) y R n min f(w) + δ sup x y n η w B(y,δ) y R n sup f(w) w B(x,δ) sup f(w) w B(y,δ) sup x y n η min f(w) + 2δ sup x w n η f(w) y R n w B(x,δ) w R n sup x y n η min f(w) + δ sup x w n η f(w) y R n w B(y,δ) w R n 234. P

45 ( x y n ( x y n f(w) η dw B(y,δ) 236. ( x y n ( x y n 237. ) 1 η χ 10nP (w) f(w) η dw B(y,δ) f(x + w) η dw B(y x,δ) ) 1 η ) 1 η χ 10nP (x + w) f(x + w) η dw B(y x,δ) P ( x y n ( x y n 239. ( 1 (1 + x y ) n ( 1 (1 + x y ) n f(x + w) η dw B(1+ x y ) ) 1 η ) 1 η χ 10nP (x + w) f(x + w) η dw B(1+ x y ) B(1+ x y ) B(1+ x y ) 240. min f(w) w B(y,δ) f(x + w) η dw ) 1 η ) 1 η χ 10nP (x + w) f(x + w) η dw ) 1 η 45

46 inf f(w) w B(y,δ) 241. k k Z n \ [ 1, 1] n 242. k Z n k Z n \ [ 1, 1] n 243. k = 0 Q 0m P \ P m m Q jm T q (G j )(x) q dx 2 inf{g q (G)(x) q : x Q jm } Q jm T q (G j )(x) q dx 2 inf{g q (G j )(x) q : x Q jm } 246. (4.219) 2 1/q G q ({F k } k A )} 2 1/q G q ({F k } k A )(x)}

47 2 T q ({F j } j= )(x) R n 2 T q ({F j } j= )(x) R n m Z n j=j(x) m Z n j=j(x) Qj(x)m G j (y) q Qjm G j (y) q dy Q j(x)m dy Q jm 1 q 1 q dx dy 2 T q ({F j } j A )(x) G j (y) R n m Z n j=j(x) Qjm q dy Q jm 2 T q ({F j } j A )(x)t q ({G j } j A [j(x), ) )(x) dx R n 2 1+1/q R n T q ({F j } j A )(x)g q ({G j } j A )(x) dx 1 q dx 248. F s 1q F q s F s 1q F s q F1q s F q s F 1q s F q s 249. (4.222) j= j= φ j φ j B(y,2 j ) B(x,2 n j )

48 L F 1q s L ( F 1q) s 252. S S js φ j (D)f(x) q dx P 1 P P j= log 2 l(p ) P j= log 2 l(p ) 2 js φ j (D)f(x) q dx 254. m P 2 js φ j (D)f q m P j= log 2 l(p ) j= log 2 l(p ) 255. a j C (Int(P )) 2 js φ j (D)f q j= log 2 l(p ) P j= log 2 l(p ) 1 q 1/q a j (x) q = χ 2P (x) a j (x) q dx = P 48

49 256. (4.225) j= log 2 l(p ) k N P j= log 2 l(p ) φ j (D)a j (x) q χ 2P (x) φ j (D)a j (x) q dx P q = 1 1 < q < φ j (D)a j (x) Ma j (x) k N 257. x 2 k+1 P \ 2 k P x 2 k+1 P \ 2 k P, L > n 258. φ j (D)a j (x) 2jn a j (y)f 1 φ(2 j (x y)) dy P P φ j (D)a j (x) 2 jn a j (y)f 1 φ(2 j (x y)) dy 259. (2 j l(p ) 1 ) n (2 k+j l(p )) L 2 jn (2 k+j l(p )) L 260. (4.226) P P P a j (y) dy φ j (D)a j (x) q 2 kql P q 1 φ j (D)a j (x) q 2 kql P q a j (y) dy P P a j (y) q dy a j (y) q dy 49

50 261. (4.227), (4.228) 1/q 262. B (0, r(b)) Int(conv((c(B), r(b)) B {0})) conv B (0, r(b)) Int(conv((c(B), r(b)) B {0})) 265. N µ B j N O = B j, B j j=1 3 n µ Carleson, B j O = j=1 j=1 B j 266. Ô M j=1 ˆB j m j=1 ˆ 3B j 50

51 N Ô M ˆB j 3Bˆ ι(j) j=1 j= B j B ι(j) 268. = sup κ(td)f(y) κ(td)f(x) Mf(x) (y,t) Γ(x) sup κ(td)f(y) Mf(x) (y,t) Γ(x) 271. exp(it D 2 )f exp( it D 2 )f iξ k 2itξ k 273. (4.238) 51

52 τ( k) 1 k Z n Q 0m 3Q νm 275. κ > κ n n min(1, p, q) 276. = (5.19) κ > n 277. (5.35) m νm (x) m νm 278. λ jm = 0 λ jm 0 (c) ν N 0 ν= (5.61) 52

53 281. L - L (5.67) = c = 283. (5.67) (5.68) ρ 284. (5.67) a > a > 1 a 285. N (5.68) (a N) ( ) 2n N η

54 288. Λ νm 2 ν(s n p ) inf M ( η0 2 ) [φν (D)f](y) y Q νm ν = < s 0 < s < s 1 <. < s 1 < s < s 0 < f S f S (5.104), (5.107) 292. G F (z) = e K(z θ) F (z) G F (z) = e δz2 δθ 2 +K(z θ) F (z) 293. = inf F e δ inf F Λ 0m inf y Q 0m M ( η0 2 ) [τ(d)f](y) 294. (5.111), (5.112) 295. M 1 θ M 1 θ inf F 54

55 296. (5.126) 2 νs 2 νs 297. (5.126) Λ ν (x) Λ ν 298. l = 0, 1 F (l + i t) F s l p l q l F (l + it) F s l p l q l f [F s 0 p 0 q 0,F s 1 p 1 q 1 ] θ l = 0, 1 F (l + i t) F s l p l q l F (l + it) s F l f p l q F s l pq 299. φ(x) = ψ(x) ψ(2x) φ(ξ) = ψ(ξ) ψ(2ξ) 300. ψ(d)g p0 + 2 js φ j (D)g L p 0 (lq ) ψ(d)g p2 + 2 js φ j (D)g L p 2 (lq ) i I i = 1, 2,, I ψ 55

56 λ β,i ν, m := (βqu) i ν, m := λ β,i ν, m := (βqu) i ν, m := { λ β ν,θi ν( m) m Mi ν 0 { (βqu) ν,θ ν i ( m) ψ m M ν i 0. { λ β ν,θi ν( m) 0 { (βqu) ν,θ ν i ( m) ψ m ι ν i (M ν i ) m ι ν i (M i ν) I ψ B1 (R n ) + ψ 1 B1 (R n ) I ψ,ψ f i,β F s pq (R n ) ψ {λ β,i νm } ν N 0, m Z n f pq (R n ) ψ λ β fpq (R n ) f i,β F s pq (R n ) ψ 2 (r+ε) β {λ β,i νm } ν N 0, m Z n f pq(r n ) ψ 2 (r+ε) β λ β fpq(r n ) 305. δ = ρ r ε 306. λ β = λ β := 307. (5.159) v j := β N n 1 0 ν N 0 m Z n 1 β λ νm (2L + j)!2 ν(2l+j) L [((β, 2L + j)qu) ν(m,0)] 56

57 v j := β N n 1 0 ν N 0 m Z n 1 β λ νm (2L + j)!2 ν(2l+j) ((β, 2L + j)qu) ν(m,0) 308. (5.162) 3 δ jl (2L + j)! δ jl (2L + j)! β N n 1 0 β N n 1 0 λ β ν N 0 m Z n 1 λ β ν N 0 m Z n (5.167) E 1 (x) = λ ν1 mχ ν1 m(x) m Z n E 1 λ ν1 mχ ν1 m m Z n 310. Ẽ1(x) = m Z n λ ν1 mκ(2 ν1 x m) Ẽ1 m Z n λ ν1 mκ(2 ν1 m) 311. κ(2 ν 2 x m) κ(2 ν 2 m) 312. κ(2 ν l x m) κ(2 ν l m) 313. κ νm Tr R n 1 2L+j x n νm Tr R n[ 2L+l x n χ [ 1,1] κ χ [ 2,2] κ : R R. (x 2L j n (β qu) νm ) (x 2L+j n (β qu) νm )] 57

58 314. (5.174) R m0 (x) = ρ(x 10am am 0 ) m Z n R m0 := ρ( 10am am 0 ) m Z n 315. (J σ g 1 ) R n = (J σ g 2 ) R n (J σ g 1 ) R n + = (J σ g 2 ) R n (6.16) δ 0l M (n + 1)N 318. (6.19) (6.21) (βqu) νm ((βqu) νm ) 319. (6.23), (6.27) lim ϵ 0 Tr R n + Ext N [f(, n + ϵ) R n + ] lim ϵ 0 Tr R next N [f(, n + ϵ) R n + ] 320. (6.24) ( 1 δ s (n 1) q )+ 1 δ > 0 58

59 321. f(, n + ϵ) = Ext N f(, n + ϵ) f(, n + ϵ) = Ext N [f(, n + ϵ) R n + ] 322. ξ m δ α +ρ β ξ m+δ β ρ α 323. α, β N 0 n 324. j 1 a k=0 a k j 1 a(x, ξ) a k (x, ξ) k= γ 2 γ k 326. γ ξ [φ(2 k ξ)] x β α γ ξ [a k (x, ξ)] ξ mj+δ β ρ α 2 γ ξ m k m j+ρ γ χ B(2)\B(1) (2 k ξ) ξ m j+δ β ρ α 2 γ +m k m j +ρ γ χ B(2)\B(1) (2 k ξ) 2 m k m j ξ mj+δ β ρ α χ B(2)\B(1) (2 k ξ) 59

60 ξ (mj+δ β ρ α ) γ ξ [φ(2 k ξ)] x β α γ ξ [a k (x, ξ)] 2 γ k ξ m j+m k +ρ γ χ B(2)\B(1) (2 k ξ) 2 (m k m j +(ρ 1) γ )k χ B(2)\B(1) (2 k ξ) 2 (m k m j )k a S m0 ρδ 328. min(µ j (α, β), m j ) > k min(µ j (α, β), m j ρ β + δ α ) > k 329. m j > k m j ρ β + δ α > k 330. ξ m ξ m j ρ β +δ α 331. A(x, ξ) γ

61 = = = i α γ α,β t α γ ( α γ ) R 2n y β γ e iy η x α+β γ ξ α [a(x + t y, ξ + t η)] dy dη = ( ) (it) α + β γ t γ α γ γ α,β γ min(α,β) γ min(α,β) x α+β γ R 2n C α,β,γ t α γ α+β γ ξ a(x + t y, ξ + t η)]e iy η dy dη y β γ e iy η x α+β γ ξ α [a(x + t y, ξ + t η)] dy dη R 2n C α,β,γ t α+β γ t γ ( ) α γ x α+β γ R 2n α+β γ ξ [a(x + t y, ξ + t η)]e iy η dy dη α+β+γ x α+β γ x α+β+γ ξ a α+β γ ξ a 334. a(x, η + ξ) y 2L (1 η ) L y 2L (1 η) L a(x, η + ξ) 335. ( ) α β 61

62 336. α α β β ( α α α α,β β ) ( ) β β c α,β,α β α x β ξ R 2n α R 2n x β ξ a(x, η + ξ) α α x a(x, η+ξ) α α x dy dη β β ξ b(y + x, ξ) e iy η dy dη β β ξ b(y+x, ξ) e iy η α x β ξ a Sm 1+δ α ρ β α x β ξ a Sm 1 δ α +ρ β 338. x α α β β ξ a S m 1+δ α α ρ β β x α α β β ξ a S m 1 δ α α +ρ β β 339. ξ m1+m2 (ρ δ)( α + β )+2δL ξ m 1+m 2 +(ρ δ)( α + β )+2δL 340. R 2n R 2n 341. dt 62

63 342. N = [s + 1] + N 0 N = [s + 1] + + [σ pq + 1] N 0 (d) s + 2N > σ pq jM α +m+δl β 2 (2M α +m+δl β )j m+δl 2 (m+δl)j 345. Ff N (x) Ff N (ξ) exp( x n 1 + ξ 2 iξ y ) exp( x n 1 + ξ 2 iξ y ) 348. N+1 j=1 N+1 j=1 Ef 0 (x, jx) λ j Ef 0 (x, jx n ) 63

64 φ S(R n ) 351. φ 2 φ T a f 2 a BMO f φ j (D)a 2 ψ j 4 (D)f 2 a BMO f 1 j= T a f T a f(x) 354. T a 1, A T a 1, A 355. Ta,J 1, A T a,j 1, A 64

65 356. T k = lim r 0 T k,r T k f = lim r 0 T k,r f L. Jorgen J. Löfström D. Edmund D. E. Edmunds 359. B. X. Wang and C. Huang 360. Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The case q < Journal d Analyse Mathématique 363. Some observations of Besov and Lizorkin-Triebel spaces Some observations on Besov and Lizorkin-Triebel spaces 364. Pseudo-differential Pseudodifferential

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

I 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 information

n 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 information

III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F

III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ

More information

実解析的方法とはどのようなものか

実解析的方法とはどのようなものか (1) ENCOUNTER with MATHEMATICS 2001 10 26 (2) 1807 J. B. J. Fourier 1 2π f(x) f(x) = n= c n (f) = 1 2π c n (f)e inx (1) π π f(t)e int dt Fourier 2 R f(x) f(x) = F[f](ξ)= 1 2π F [f](ξ)e ixξ dξ f(t)e iξx

More information

201711grade1ouyou.pdf

201711grade1ouyou.pdf 2017 11 26 1 2 52 3 12 13 22 23 32 33 42 3 5 3 4 90 5 6 A 1 2 Web Web 3 4 1 2... 5 6 7 7 44 8 9 1 2 3 1 p p >2 2 A 1 2 0.6 0.4 0.52... (a) 0.6 0.4...... B 1 2 0.8-0.2 0.52..... (b) 0.6 0.52.... 1 A B 2

More information

Part () () Γ Part ,

Part () () 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 information

B ver B

B 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 information

2011de.dvi

2011de.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 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.   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 information

20 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

20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................

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.   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 information

2019 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 information

1 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 = 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 information

基礎数学I

基礎数学I I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............

More information

Note.tex 2008/09/19( )

Note.tex 2008/09/19( ) 1 20 9 19 2 1 5 1.1........................ 5 1.2............................. 8 2 9 2.1............................. 9 2.2.............................. 10 3 13 3.1.............................. 13 3.2..................................

More information

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y [ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)

More information

newmain.dvi

newmain.dvi 数論 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/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published

More information

f(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

f(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 information

: , 2.0, 3.0, 2.0, (%) ( 2.

: , 2.0, 3.0, 2.0, (%) ( 2. 2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................

More information

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

II 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 information

X G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2

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

) ] [ 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 information

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

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 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 information

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.

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

More information

meiji_resume_1.PDF

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

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10% 1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n

More information

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 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

More information

2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =

More information

211 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

Morse ( ) 2014

Morse ( ) 2014 Morse ( ) 2014 1 1 Morse 1 1.1 Morse................................ 1 1.2 Morse.............................. 7 2 12 2.1....................... 12 2.2.................. 13 2.3 Smale..............................

More information

30

30 3 ............................................2 2...........................................2....................................2.2...................................2.3..............................

More information

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7

More information

u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3

u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3 2 2 1 5 5 Schrödinger i u t + u = λ u 2 u. u = u(t, x 1,..., x d ) : R R d C λ i = 1 := 2 + + 2 x 2 1 x 2 d d Euclid Laplace Schrödinger 3 1 1.1 N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3,... } Q

More information

6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m f 4

6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m f 4 35-8585 7 8 1 I I 1 1.1 6kg 1m P σ σ P 1 l l λ λ l 1.m 1 6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m

More information

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A

..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A .. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.

More information

.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

.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

More information

t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z

t = h x z z = h z = t (x, z) (v x (x, z, t), v z (x, z, t)) ρ v x x + v z z = 0 (1) 2-2. (v x, v z ) φ(x, z, t) v x = φ x, v z I 1 m 2 l k 2 x = 0 x 1 x 1 2 x 2 g x x 2 x 1 m k m 1-1. L x 1, x 2, ẋ 1, ẋ 2 ẋ 1 x = 0 1-2. 2 Q = x 1 + x 2 2 q = x 2 x 1 l L Q, q, Q, q M = 2m µ = m 2 1-3. Q q 1-4. 2 x 2 = h 1 x 1 t = 0 2 1 t x 1 (t)

More information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

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

More information

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

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 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 information

1 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

1 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 information

構造と連続体の力学基礎

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 information

2 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) =

2 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 information

y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a =

y π π O π x 9 s94.5 y dy dx. y = x + 3 y = x logx + 9 s9.6 z z x, z y. z = xy + y 3 z = sinx y 9 s x dx π x cos xdx 9 s93.8 a, fx = e x ax,. a = [ ] 9 IC. dx = 3x 4y dt dy dt = x y u xt = expλt u yt λ u u t = u u u + u = xt yt 6 3. u = x, y, z = x + y + z u u 9 s9 grad u ux, y, z = c c : grad u = u x i + u y j + u k i, j, k z x, y, z grad u v =

More information

No 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 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 information

( )

( ) 7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................

More information

I ( ) 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 ( ) 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 information

v er.1/ c /(21)

v er.1/ c /(21) 12 -- 1 1 2009 1 17 1-1 1-2 1-3 1-4 2 2 2 1-5 1 1-6 1 1-7 1-1 1-2 1-3 1-4 1-5 1-6 1-7 c 2011 1/(21) 12 -- 1 -- 1 1--1 1--1--1 1 2009 1 n n α { n } α α { n } lim n = α, n α n n ε n > N n α < ε N {1, 1,

More information

: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j =

: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j = 72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(

More information

untitled

untitled 1 kaiseki1.lec(tex) 19951228 19960131;0204 14;16 26;0329; 0410;0506;22;0603-05;08;20;0707;09;11-22;24-28;30;0807;12-24;27;28; 19970104(σ,F = µ);0212( ); 0429(σ- A n ); 1221( ); 20000529;30(L p ); 20050323(

More information

DVIOUT

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)

More information

r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B

r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B 1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n

More information

Z: Q: R: C: sin 6 5 ζ a, b

Z: 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 information

I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10

I, II 1, 2 ɛ-δ 100 A = A 4 : 6 = max{ A, } A A 10 1 2007.4.13. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 0. 1. 1. 2. 3. 2. ɛ-δ 1. ɛ-n

More information

untitled

untitled 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 information

IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................

More information

d 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 )

d 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

(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)

(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

4................................. 4................................. 4 6................................. 6................................. 9.................................................... 3..3..........................

More information

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s ... x, y z = x + iy x z y z x = Rez, y = Imz z = x + iy x iy z z () z + z = (z + z )() z z = (z z )(3) z z = ( z z )(4)z z = z z = x + y z = x + iy ()Rez = (z + z), Imz = (z z) i () z z z + z z + z.. z

More information

A S hara/lectures/lectures-j.html ϵ-n 1 ϵ-n lim n a n = α n a n α 2 lim a n = 0 1 n a k n n k= ϵ

A S hara/lectures/lectures-j.html ϵ-n 1 ϵ-n lim n a n = α n a n α 2 lim a n = 0 1 n a k n n k= ϵ A S1-20 http://www2.mth.kyushu-u.c.jp/ hr/lectures/lectures-j.html 1 1 1.1 ϵ-n 1 ϵ-n lim n n = α n n α 2 lim n = 0 1 n k n n k=1 0 1.1.7 ϵ-n 1.1.1 n α n n α lim n n = α ϵ N(ϵ) n > N(ϵ) n α < ϵ (1.1.1)

More information

D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j

D = [a, b] [c, d] D ij P ij (ξ ij, η ij ) f S(f,, {P ij }) S(f,, {P ij }) = = k m i=1 j=1 m n f(ξ ij, η ij )(x i x i 1 )(y j y j 1 ) = i=1 j 6 6.. [, b] [, d] ij P ij ξ ij, η ij f Sf,, {P ij } Sf,, {P ij } k m i j m fξ ij, η ij i i j j i j i m i j k i i j j m i i j j k i i j j kb d {P ij } lim Sf,, {P ij} kb d f, k [, b] [, d] f, d kb d 6..

More information

B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 (

B [ 0.1 ] x > 0 x 6= 1 f(x) µ 1 1 xn 1 + sin sin x 1 x 1 f(x) := lim. n x n (1) lim inf f(x) (2) lim sup f(x) x 1 0 x 1 0 ( . 28 4 14 [.1 ] x > x 6= 1 f(x) µ 1 1 xn 1 + sin + 2 + sin x 1 x 1 f(x) := lim. 1 + x n (1) lim inf f(x) (2) lim sup f(x) x 1 x 1 (3) lim inf x 1+ f(x) (4) lim sup f(x) x 1+ [.2 ] [, 1] Ω æ x (1) (2) nx(1

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.

() 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 information

τ τ

τ τ 1 1 1.1 1.1.1 τ τ 2 1 1.1.2 1.1 1.1 µ ν M φ ν end ξ µ ν end ψ ψ = µ + ν end φ ν = 1 2 (µφ + ν end) ξ = ν (µ + ν end ) + 1 1.1 3 6.18 a b 1.2 a b 1.1.3 1.1.3.1 f R{A f } A f 1 B R{AB f 1 } COOH A OH B 1.3

More information

Riemann-Stieltjes Poland S. Lojasiewicz [1] An introduction to the theory of real functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,,. Riemann-S

Riemann-Stieltjes Poland S. Lojasiewicz [1] An introduction to the theory of real functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,,. Riemann-S Riemnn-Stieltjes Polnd S. Lojsiewicz [1] An introduction to the theory of rel functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,, Riemnn-Stieltjes 1 2 2 5 3 6 4 Jordn 13 5 Riemnn-Stieltjes 15 6 Riemnn-Stieltjes

More information

.1 1,... ( )

.1 1,... ( ) 1 δ( ε )δ 2 f(b) f(a) slope f (c) = f(b) f(a) b a a c b 1 213 3 21. 2 [e-mail] nobuo@math.kyoto-u.ac.jp, [URL] http://www.math.kyoto-u.ac.jp/ nobuo 1 .1 1,... ( ) 2.1....................................

More information

8 i, III,,,, III,, :!,,,, :!,,,,, 4:!,,,,,,!,,,, OK! 5:!,,,,,,,,,, OK 6:!, 0, 3:!,,,,! 7:!,,,,,, ii,,,,,, ( ),, :, ( ), ( ), :... : 3 ( )...,, () : ( )..., :,,, ( ), (,,, ),, (ϵ δ ), ( ), (ˆ ˆ;),,,,,,!,,,,.,,

More information

1. A0 A B A0 A : A1,...,A5 B : B1,...,B

1. A0 A B A0 A : A1,...,A5 B : B1,...,B 1. A0 A B A0 A : A1,...,A5 B : B1,...,B12 2. 3. 4. 5. A0 A, B Z Z m, n Z m n m, n A m, n B m=n (1) A, B (2) A B = A B = Z/ π : Z Z/ (3) A B Z/ (4) Z/ A, B (5) f : Z Z f(n) = n f = g π g : Z/ Z A, B (6)

More information

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz 1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x

More information

量子力学 問題

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 information

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )

More information

,,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,,

,,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,, 14 5 1 ,,,17,,,194 1 4 ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,, 1 4 1.1........................................ 4 5.1........................................ 5.........................................

More information

ohpr.dvi

ohpr.dvi 2003/12/04 TASK PAF A. Fukuyama et al., Comp. Phys. Rep. 4(1986) 137 A. Fukuyama et al., Nucl. Fusion 26(1986) 151 TASK/WM MHD ψ θ ϕ ψ θ e 1 = ψ, e 2 = θ, e 3 = ϕ ϕ E = E 1 e 1 + E 2 e 2 + E 3 e 3 J :

More information

Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence

Hanbury-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 information

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou

(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou (Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :.

More information

ii 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

ii 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

1 (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

1 (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 information

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )

( ) ) ) ) 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 information

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g(

.3. (x, x = (, u = = 4 (, x x = 4 x, x 0 x = 0 x = 4 x.4. ( z + z = 8 z, z 0 (z, z = (0, 8, (,, (8, 0 3 (0, 8, (,, (8, 0 z = z 4 z (g f(x = g( 06 5.. ( y = x x y 5 y 5 = (x y = x + ( y = x + y = x y.. ( Y = C + I = 50 + 0.5Y + 50 r r = 00 0.5Y ( L = M Y r = 00 r = 0.5Y 50 (3 00 0.5Y = 0.5Y 50 Y = 50, r = 5 .3. (x, x = (, u = = 4 (, x x = 4 x,

More information

/02/18

/02/18 3 09/0/8 i III,,,, III,?,,,,,,,,,,,,,,,,,,,,?,?,,,,,,,,,,,,,,!!!,? 3,,,, ii,,,!,,,, OK! :!,,,, :!,,,,,, 3:!,, 4:!,,,, 5:!,,! 7:!,,,,, 8:!,! 9:!,,,,,,,,, ( ),, :, ( ), ( ), 6:!,,, :... : 3 ( )... iii,,

More information

³ÎΨÏÀ

³ÎΨÏÀ 2017 12 12 Makoto Nakashima 2017 12 12 1 / 22 2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22 . (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22 . (,, L p

More information

(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

(1.2) 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

More information

( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1

( ) ( 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 information

i 18 2H 2 + O 2 2H 2 + ( ) 3K

i 18 2H 2 + O 2 2H 2 + ( ) 3K i 18 2H 2 + O 2 2H 2 + ( ) 3K ii 1 1 1.1.................................. 1 1.2........................................ 3 1.3......................................... 3 1.4....................................

More information

(ii) (iii) z a = z a =2 z a =6 sin z z a dz. cosh z z a dz. e z dz. (, a b > 6.) (z a)(z b) 52.. (a) dz, ( a = /6.), (b) z =6 az (c) z a =2 53. f n (z

(ii) (iii) z a = z a =2 z a =6 sin z z a dz. cosh z z a dz. e z dz. (, a b > 6.) (z a)(z b) 52.. (a) dz, ( a = /6.), (b) z =6 az (c) z a =2 53. f n (z B 4 24 7 9 ( ) :,..,,.,. 4 4. f(z): D C: D a C, 2πi C f(z) dz = f(a). z a a C, ( ). (ii), a D, a U a,r D f. f(z) = A n (z a) n, z U a,r, n= A n := 2πi C f(ζ) dζ, n =,,..., (ζ a) n+, C a D. (iii) U a,r

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

IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + (

IA 2013 : :10722 : 2 : :2 :761 :1 (23-27) : : ( / ) (1 /, ) / e.g. (Taylar ) e x = 1 + x + x xn n! +... sin x = x x3 6 + x5 x2n+1 + ( IA 2013 : :10722 : 2 : :2 :761 :1 23-27) : : 1 1.1 / ) 1 /, ) / e.g. Taylar ) e x = 1 + x + x2 2 +... + xn n! +... sin x = x x3 6 + x5 x2n+1 + 1)n 5! 2n + 1)! 2 2.1 = 1 e.g. 0 = 0.00..., π = 3.14..., 1

More information

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a =

simx simxdx, cosxdx, sixdx 6.3 px m m + pxfxdx = pxf x p xf xdx = pxf x p xf x + p xf xdx 7.4 a m.5 fx simxdx 8 fx fx simxdx = πb m 9 a fxdx = πa a = II 6 ishimori@phys.titech.ac.jp 6.. 5.4.. f Rx = f Lx = fx fx + lim = lim x x + x x f c = f x + x < c < x x x + lim x x fx fx x x = lim x x f c = f x x < c < x cosmx cosxdx = {cosm x + cosm + x} dx = [

More information

DVIOUT-fujin

DVIOUT-fujin 2005 Limit Distribution of Quantum Walks and Weyl Equation 2006 3 2 1 2 2 4 2.1...................... 4 2.2......................... 5 2.3..................... 6 3 8 3.1........... 8 3.2..........................

More information

,,,,., = (),, (1) (4) :,,,, (1),. (2),, =. (3),,. (4),,,,.. (1) (3), (4).,,., () : = , ( ) : = F 1 + F 2 + F 3 + ( ) : = i Fj j=1 2

,,,,., = (),, (1) (4) :,,,, (1),. (2),, =. (3),,. (4),,,,.. (1) (3), (4).,,., () : = , ( ) : = F 1 + F 2 + F 3 + ( ) : = i Fj j=1 2 6 2 6.1 2 2, 2 5.2 R 2, 2 (R 2, B, µ)., R 2,,., 1, 2, 3,., 1, 2, 3,,. () : = 1 + 2 + 3 + (6.1.1).,,, 1 ,,,,., = (),, (1) (4) :,,,, (1),. (2),, =. (3),,. (4),,,,.. (1) (3), (4).,,., () : = 1 + 2 + 3 +,

More information

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc 013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8

More information

ii 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.

ii 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 information

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

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

More information

1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1.

1. 1 BASIC PC BASIC BASIC BASIC Fortran WS PC (1.3) 1 + x 1 x = x = (1.1) 1 + x = (1.2) 1 + x 1 = (1. Section Title Pages Id 1 3 7239 2 4 7239 3 10 7239 4 8 7244 5 13 7276 6 14 7338 7 8 7338 8 7 7445 9 11 7580 10 10 7590 11 8 7580 12 6 7395 13 z 11 7746 14 13 7753 15 7 7859 16 8 7942 17 8 Id URL http://km.int.oyo.co.jp/showdocumentdetailspage.jsp?documentid=

More information

phs.dvi

phs.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 information

( )/2 hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1

( )/2   hara/lectures/lectures-j.html 2, {H} {T } S = {H, T } {(H, H), (H, T )} {(H, T ), (T, T )} {(H, H), (T, T )} {1 ( )/2 http://www2.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html 1 2011 ( )/2 2 2011 4 1 2 1.1 1 2 1 2 3 4 5 1.1.1 sample space S S = {H, T } H T T H S = {(H, H), (H, T ), (T, H), (T, T )} (T, H) S

More information

x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n

x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n 1, R f : R R,.,, b R < b, f(x) [, b] f(x)dx,, [, b] f(x) x ( ) ( 1 ). y y f(x) f(x)dx b x 1: f(x)dx, [, b] f(x) x ( ).,,,,,., f(x)dx,,,, f(x)dx. 1.1 Riemnn,, [, b] f(x) x., x 0 < x 1 < x 2 < < x n 1

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