Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull
|
|
|
- かおり もちやま
- 5 years ago
- Views:
Transcription
1 Feynman Encounter with Mathematics 52, [] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull. Sci. Math. vol. 28 (2004) [2] D. Fujiwara and N. Kumano-go, Smooth functional derivatives in Feynman path integral by time slicing approximation. Bull. Sci. Math. vol. 29 (2005) [3] N. Kumano-go and D. Fujiwara, Feynman path integrals and semiclassical approximation. RIMS Kokyuroku Bessatsu B5 (2008)
2 . Introduction 948 Feynman Schrödinger K(T, x, x 0 ) K(T, x, x 0 ) = (i T e i S[γ] D[γ] ) V (T, x) u(t, x) = 0 γ : [0, T ] R d γ(0) = x 0, γ(t ) = x T S[γ] = dγ 2 V (t, γ)dt γ D[γ] 2 dt 0 R. P. Feynman, Rev. Mod. Phys. 20 (948). (T, x) (0, x 0 ) 0 T
3 Feynman Feynman F [γ] F [γ]e i S[γ] D[γ] (DF )[γ][η] 960 Cameron e i S[γ] D[γ] R. H. Cameron, J. of Math. and Phys. Sci. 39(960). Remark limit
4 (DF )[γ][η] Feynman e i S[γ] F [γ]d[γ] Riemann(-Stieltjes) limit γ + η Qγ (DF )[γ][η] 0 0 D. Fujiwara
5 (x, x 0 ) R 2d F [γ] F Remark Schrödinger R 2d F [γ] Fujiwara(979,99) Kitada-H. Kumano-go(98) Yajima(99) N. Kumano-go(995) Fujiwara-Tsuchida(997) W. Ichinose(997) Nelson(964) Cameron-Storvick(983) Wiener Itô(967) Albeverio-Hoegh Krohn(976) Truman(972) Albeverio, Hoegh-Krohn, Mazzucchi, Mathematical theory of Feynman path integrals, 2nd, Springer, Johnson-Lapidus Gill-Zachary(2002) Operational calculus Johnson and Lapidus, The Feynman integral and Feynman s operational calculus, Oxford, T. Ichinose-Tamura(987) 2 Dirac Path Integral, 994.
6 2. Feynman Assumption of S[γ] = T 0 2 dγ dt 2 V (t, γ)dt. V (t, x) : R R d R x α V (t, x) α x V (t, x) C α( + x ) max(2 α,0). Example of F [γ] F ( ) () α x B(t, x) C α( + x ) m t F [γ] = B(t, γ(t)) F, F [γ] F, T Riemann(-Stieltjes) F [γ] = B(t, γ(t))dt F. T (2) x αb(t, x) C α F [γ] = e T T B(t,γ(t))dt F. (3) Z : R R d C d α x Z(t, x) + α x tz(t, x) C α ( + x ) m, t ( x Z) = ( x Z) F [γ] = T T Z(t, γ(t)) dγ(t) F.
7 F [γ] F F F F [γ] F Theorem Smooth algebra F [γ], G[γ] F, η : [0, T ] R d, d d P () F [γ] + G[γ] F, F [γ]g[γ] F. (2) F [γ + η] F, F [P γ] F. (3) (DF )[γ][η] F. Remark γ : [0, T ] R d η : [0, T ] R d (DF )[γ][η] = d dθ θ=0 F [γ + θη].
8 T,0 : T = T J+ > T J > > T > T 0 = 0 [0, T ] t j = T j T j, T,0 = max j J+ t j x = x J+ x J, x J,..., x R d γ T,0 j =, 2,..., J, J + (T j, x j ) (T j, x j ) S[γ T,0 ], F [γ T,0 ] x J+, x J,, x, x 0 S[γ T,0 ] = S T,0 (x J+, x J,..., x, x 0 ), F [γ T,0 ] = F T,0 (x J+, x J,..., x, x 0 ). (T J, x J ) (T 3, x 3 ) (T, x) (T, x ) (0, x 0 ) (T 2, x 2 ) T 0 = 0 T T 2 T 3 T J T = T J+
9 Feynman Theorem 2 Feynman T F [γ] F J+ ( ) e i d/2 S[γ] F [γ]d[γ] lim e i S[γ T,0 ] F [γ T,0 ] T,0 0 2πi t j= j R dj (x, x 0 ) R 2d i.e., well-defined. J j= dx j ( ) Remark F [γ] R d dx j =. J, J.
10 Remark S[γ T,0 ] S T,0 (x J+, x J,, x, x 0 ) Feynman J+ lim T,0 0 j= ( 2πi t j ) d/2 e i S T,0 (x J+,x J,...,x,x 0 ) R dj Nelson S T,0 Trotter v L 2 (R d ) L 2 - J+ ( ) d/2 lim J 2πi T /(J + ) j= R d R d e J+ i j= ( (x j x j ) 2 2T /(J+) E. Nelson, J. Math. Phys. 5 (964). ( J+ (xj x j ) 2 Tj S[γ T,0 ] = V (t, t T j x j + 2t j T j T j T j j= J j= V (x j ) T J+ dx j ) v(x 0 ) J j=0 T ) j t x j )dt T j T j ( ) S[γ T,0 ], F [γ T,0 ] dx j.
11 3. Theorem 3 T m 0, 0 T T T, f(t, x) : R R d C α x f(t, x), α x tf(t, x) x αf(t, x) + α x tf(t, x) C α ( + x ) m e i S[γ]( ) f (T, γ(t )) f (T, γ(t )) D[γ] = e i S[γ]( T T ) ( x f)(t, γ(t)) dγ(t)+ ( t f)(t, γ(t))dt D[γ]. Remark T T T ( x f)(t, γ(t)) dγ(t) γ B(t) (?) T
12 B(t) γ T,0 B(T j ) = x j Itô γ T,0 T T Z(t, B(t)) db(t) j Z(T j, x j ) (x j x j ). Stratonovich γ T,0 T Z(t, B(t)) db(t) ( Tj + T j Z, x ) j + x j T 2 2 j (x j x j ). γ T,0 T T Z(t, γ T,0 (t)) dγ T,0 (t).
13 Itô Riemann (T, x) (0, x 0 ) T 0 = 0 T T T = T J+ (T, x) (0, x 0 ) T 0 = 0 T T T = T J+
14 Proof of Theorem 3 Example Theorem () F 2 [γ] = F [γ] = f (T, γ(t )) f (T, γ(t )) F, T T ( x f)(t, γ(t)) dγ(t) + T γ T,0 Theorem 2 e i S[γ] F [γ]d[γ] lim T,0 0 = lim T,0 0 F [γ T,0 ] = F 2 [γ T,0 ]. ( J+ ) d/2 2πi t j= j J+ ( ) d/2 j= 2πi t j e i S[γ] F 2 [γ]d[γ]. T ( t f)(t, γ(t))dt F. R dj e i R dj e i S[γ T,0 ] F [γ T,0 ] S[γ T,0 ] F2 [γ T,0 ] J j= J j= dx j dx j
15 4. Riemann(-Stieltjes) lim Theorem 4 T m 0, 0 T T T, B(t, x) : [0, T ] R d C x α B(t, x) x αb(t, x) C α( + x ) m T ( ) ( T e i S[γ] B(t, γ(t))d[γ] dt = e i S[γ] T T B(t, γ(t))dt ) D[γ]. Remark limit x αb(t, x) C α e i S[γ]+ i T0 B(τ,γ(τ ))dτ D[γ] ( ) i n T τn τ2 = dτ n dτ n dτ n= e i S[γ] B(τ n, γ(τ n ))B(τ n, γ(τ n )) B(τ, γ(τ ))D[γ].
16 Nelson x j x j (T, x) x j x j (0, x 0 ) T 0 = 0 T j t T j T = T J+ γ T,0 (t) = t T j T j T j x j + T j t T j T j x j (T, x) (0, x 0 ) T 0 = 0 T j t T j T = T J+
17 Proof of Theorem 4 γ T,0 (t) B(t, γ T,0 (t)) t [T, T ] x j Lebesgue T,0 J+ ( ) d/2 J e i S[γ T,0 ] B(t, γ T,0 (t)) dx j 2πi t j R dj j= t [T, T ] Theorem 2 e i S[γ] B(t, γ(t))d[γ] j= J+ ( ) d/2 lim e i S[γ T,0 ] B(t, γ T,0 (t)) T,0 0 2πi t j= j R dj t [T, T ] e i S[γ] B(t, γ(t))d[γ] t [T, T ] Riemann J j= dx j
18 T T T T ( ) e i S[γ] B(t, γ(t))d[γ] dt J+ ( ) d/2 lim e i S[γ T,0 ] B(t, γ T,0 (t)) T,0 0 2πi t j= j R dj T dt lim T T,0 0 T J+ ( ) d/2 = lim e i S[γ T,0 ] B(t, γ T,0 (t)) T,0 0 T 2πi t j R dj j= Fubini J+ ( ) d/2 = lim e i S[γ T,0 ] T,0 0 2πi t j= j R dj ( ) T e i S[γ] B(t, γ(t))dt D[γ]. T T T J j= J j= B(t, γ T,0 (t))dt dx j dt. dx j dt. J j= dx j
19 5. F [γ + η] F [Qγ] Theorem 5 T F [γ] F η : [0, T ] R d e i S[γ+η] F [γ + η]d[γ] = γ(0)=x 0,γ(T )=x γ(0)=x 0 +η(0),γ(t )=x+η(t ) e i S[γ] F [γ]d[γ]. Remark T F [γ] F d d Q e i S[Qγ] F [Qγ]D[γ] = γ(0)=x 0,γ(T )=x γ(0)=qx 0,γ(T )=Qx e i S[γ] F [γ]d[γ].
20 Proof of Theorem 5 Theorem 2 γ T,0 (T j ) = x j e i S[γ+η] F [γ + η]d[γ] γ(0)=x 0,γ(T )=x J+ = lim T,0 0 j= ( 2πi t j ) d/2 R dj e i S[γ T,0 +η] F [γ T,0 + η] T,0 η η(t j ) = y j, j = 0,,..., J, J + J j= dx j. (T, x) (0, x 0 ) η T 0 = 0 T = T J+
21 γ T,0 + η j =, 2,..., J + (T j, x j + y j ) (T j, x j + y j ) J+ ( ) d/2 = lim T,0 0 2πi t j= j e i S T,0 (x J+ +y J+,x J +y J,...,x +y,x 0 +y 0 ) R dj F T,0 (x J+ + y J+, x J + y J,..., x + y, x 0 + y 0 ) x j + y j x j, j =, 2,..., J J+ ( ) d/2 = lim e i T,0 0 2πi t j= j R dj F T,0 (x J+ + y J+, x J,..., x, x 0 + y 0 ) y J+ = η(t ) y 0 = η(0) = e i S[γ] F [γ]d[γ]. γ(0)=x 0 +η(0),γ(t )=x+η(t ) J j= dx j. S T,0 (x J+ +y J+,x J,...,x,x 0 +y 0 ) J j= dx j.
22 6. (DF )[γ][η] Theorem 6 T F [γ] F η(0) = η(t ) = 0 η : [0, T ] R d e i S[γ] (DF )[γ][η]d[γ] = i e i S[γ] (DS)[γ][η]F [γ]d[γ]. Remark F [γ] S[γ] = T 0 dγ 2 V (t, γ)dt 2 dt η(0) = η(t ) = 0 η : [0, T ] R d T ( ) 0 = e i dγ S[γ] dη dt dt ( xv )(t, γ(t))η(t) dtd[γ]. 0
23 Remark T,0 γ η γ(t j ) = x j, η(t j ) = y j, j = 0,,..., J, J + θ R γ + θη j =, 2,..., J, J + (T j, x j + θy j ) (T j, x j + θy j ) F [γ + θη] = F T,0 (x J+ + θy J+, x J + θy J,..., x + θy, x 0 + θy 0 ). γ (T, x) (0, x 0 ) η T 0 = 0 T = T J+
24 (DF )[γ][η] (DF )[γ][η] = d J+ dθ θ=0 F [γ + θη] = ( xj F T,0 )(x J+, x J,..., x, x 0 ) y j. j=0 Remark T F [γ] F η : [0, T ] R d e i S[γ] F [γ + η]d[γ] = + 0 ( θ) L L! L l=0 l! e i S[γ] (D l F )[γ][η] [η]d[γ] e i S[γ] (D L+ F )[γ +θη][η] [η]d[γ]dθ.
25 7. 0 Theorem 7 0 T F [γ] F ( ) e i d/2 ( ) S[γ] F [γ]d[γ] = e i S[γcl ] D(T, x, x 0 ) /2 F [γ cl ] + Υ (, x, x 0 ) 2πi T γ cl γ cl (0) = x 0, γ cl (T ) = x D(T, x, x 0 ) Morette-Van Vleck Υ (, x, x 0 ) C( + x + x 0 ) m. (T, x) (0, x 0 ) γ cl T 0 = 0 T = T J+
26 8. Theorems (),2,7 992 Fujiwara Schrödinger F [γ],, J+ ( ) d/2 J e i S T,0 (x J+,x J,...,x,x 0 ) dx j 2πi t j= j R dj j= ( ) d/2 = e i ( S T,0 (x,x 0) D T,0 2πi T (x, x 0) /2 + Υ (, x, x T,0 0)). (T J, x J ) (T 3, x 3 ) (T, x) (T, x ) (0, x 0 ) (T 2, x 2 ) T 0 = 0 T T 2 T 3 T J T = T J+
27 Remark Remark γ cl S[γ cl ] Hessian T,0 : T = T J+ > T J > > T > T 0 = 0, ( T,TN+, Tn,0) : T = T J+ > > T N+ > T n > > T 0 = 0 D T,0 D ( T,TN+, Tn,0) C(T N+ T n ) 2, Υ T,0 Υ ( T,TN+, Tn,0) C (T N+ T n ) 2, Υ T,0 C. T,0 0 D T,0 D C T,0 T, Υ T,0 Υ C T,0 T ( ), Υ C. 0, 999.
28 0 ( ) Υ T,0 Υ C T,0 T? = ( ) F [γ] Υ T,0 Υ C T,0 T ( + x + x 0 )? Remark = (J s).
29 ( Υ T,0 Υ ɛ, 0 ɛ ) ( T,TN+, Tn,0) T,0 0 ɛ ( ) d/2 ɛ = e i ( ) S T,0 D /2 2πi T T,0 + Υ T,0 ɛ ɛ = 0 ( 2πi T ( 2πi T ) d/2 e i S ɛ (D ɛ /2 + Υ ɛ ) ) d/2 e i S ( T,TN+, Tn,0 ) ( D /2 ( T,TN+, Tn,0) + Υ ( T,TN+, Tn,0) Υ T,0 Υ ( T,TN+, Tn,0) = Υ Υ 0 = 0 ( ɛυ ɛ ) dɛ ɛ Υ ɛ C(T N+ T n ) 2 ( + x + x 0 ) Υ ɛ ).
30 Υ ɛ? Υ ɛ D ɛ /2 D ɛ /2 S ɛ S ɛ
31 = = =. γ T,0 F [γ] F [γ] Cauchy T,0 : T = T J+ > T J > > T > T 0 = 0, ( T,TN+, Tn,0) : T = T J+ > > T N+ > T n > > T 0 = 0
32 (T, x) (0, x 0 ) T 0 = 0 T n T N+ T = T J+ Key Lemma x j = T j T n T N+ T n x N+ + T N+ T j T N+ T n x n, j = n,..., N (x N,..., x n ) = (x N,..., x n ) γ T,0 = γ ( T,TN+, Tn,0). (T, x) (0, x 0 ) T 0 = 0 T n T N+ T = T J+
33 J dx j. j= Key Lemma (x N,..., x n ) = (x N,..., x n ) F T,0 (x J+,..., x N+, x N,..., x n, x n,..., x 0 ) = F [γ T,0 ] = F [γ ( T,TN+, Tn,0)] = F ( T,TN+, Tn,0)(x J+,..., x N+, x n,..., x 0 ). J+ n dx j dx j. j=n+ j=
34 Proof of Theorems (),2,7, J. () F [γ] F = ( ) ( ) F [γ] F [γ] (2) T T B(t, γ(t))dt F, B(t, γ(t)) F. ( ) F (3) F [γ], G[γ] F = F [γ] + G[γ], F [γ]g[γ] F. + ()(3) (2)
35 9. F [γ] F Assumption (critical point) l L J + x L,l = (x L, x L,..., x l ) x L,l = x L,l (x L+, x l ) ( xl,l S T,0 )(x J+,L+, x L,l, x l,0) = 0 Definition F F [γ] Assumption F [γ] F
36 Assumption m 0 M A M, X M T,0, 0 = j 0 < j < j < j 2 < j 2 < < j K J +, j K+ = J + α jk+, α jk M () ( K k=0 α j k+ x jk+ α j k x jk )F T,0 (x J+, x J,j K +, x j K,, x js+, x j s+ 2,j s +, x j s, x js, x j s 2,j s +, x j s, x js, x j s 2,j s 2 +, x j s 2,, x j, x j 2,, x 0) A M (X M ) K+ ( + x J+ + x jk + + x js+ + x js + x js + x js + x js + x js x j + x 0 ) m. (2), (3), (4) Remark Theorem (),2,3,4,7
37 Proof F T,0 (, x N+, x N,n, x n, ) = F ( T,TN+, Tn,0)(, x N+, x n, ). T n T N+ T n T N+
38 Assumption m 0, u j 0, J+ j= u j U < M A M, X M T,0, α j M, j = 0,,..., J + k J J+ () ( j=0 α j x j )F T,0 (x J+, x J,..., x, x 0 ) A M (X M ) J+ ( + J+ (2) ( α j x j ) xk F T,0 (x J+, x J,..., x, x 0 ) j=0 A M (X M ) J+ (u k+ + u k )( + J+ j=0 x j ) m. Remark Theorem (),2,3,4,7 J+ j=0 x j ) m,
39 Definition T,0 F T,0 (x J+, x J,..., x, x 0 ) C (R d(j+2) ) γ : [0, T ] R d η l : [0, T ] R d, l =, 2,..., L L L (D L L F )[γ] [η l ] = ( )F [γ + θ l η l ] θ l θ = =θ L =0 l= l= l= γ η 0,l0 η,l η 2,l2 η 3,l3 η j,lj T 0 = 0 T T 2 T 3 T j T j T j+ T = T J+
40 Assumption of F [γ] F m 0, ρ(t), ρ (t) M A M, X M J+ J+ () (D j=0 L j F )[γ] j=0 L j l j = J+ (2) (D + J+ j=0 L j F )[γ][η] j=0 J+ [η j,lj ] A M (X M ) J+ ( + γ ) m L j l j = [η j,lj ] A M (X M ) J+ ( + γ ) m T 0 η(t) d ρ (t) J+ j=0 L j l j = η j,lj j=0 L j l j = η j,lj, T,0, γ : [0, T ] R d, η : [0, T ] R d L j = 0,,..., M, [T j, T j+ ] η j,lj : [0, T ] R d, l j =, 2,..., L j γ = max 0 t T γ(t) Remark Theorem,2,3,4,5,6,7
41
42 0. Theorem (),2,7 J+ ( ) d/2 J ( ) lim e i S[γ T,0 ] F [γ T,0 ] dx j, T,0 0 2πi t j= j R dj j=, J F T,0 (x J+, x J,..., x, x 0 ) = F [γ T,0 ] Step. H. Kumano-go-Taniguchi ( ) J C C C Step 2. Fujiwara ( ) J C Step 3. ( ) T,0 0
43 . H. Kumano-go-Taniguchi H. Kumano-go, Pseudo differential operators, MIT Press, p.360 Fourier J+ j= = ( ) d/2 2πi t j ( ) d/2 e i S T,0 (x,x 0) q T,0 (, x, x 0). 2πi T R dj e i S T,0 (x J+,x J,...,x,x 0 ) F T,0 (x J+, x J,..., x, x 0 ) Lemma KT m 0 M A M, X M α j M, j =, 2,..., J, J + J+ ( j=0 α j x j )F T,0 (x J+, x J,..., x, x 0 ) A M (X M ) J+ ( + J+ J C J q T,0 (, x J+, x 0 ) C J ( + x J+ + x 0 ) m. j=0 J j= x j ) m. Remark x αb(t, x) C α F [γ] = e T B(t,γ(t))dt 0 J+ T j Tj j= T B(t,γ Tj,T F T,0 = e j j (t,x j,x j ))dt = J+ j= e T B(t,γ Tj,T j j (t,x j,x j ))dt. dx j
44 Proof of Lemma KT D. Fujiwara, N. Kumanogo, K. Taniguchi, Funkcial. Ekvac. 40 (997) m = 0 ω j (x j, x j ) 2πi t j+ ( = 2πi T j+ S T,0 (x J+, x J,..., x, x 0 ) = J+ j= (x j x j ) 2 2t j t j 0 (/t j ) ( ) d/2 (x i j+ x j ) 2 ( ) 2t d/2 (x i j x 0 ) 2 e j+ 2T e j 2πi T j ) d/2 (x i j+ x 0 ) 2 ( 2T e j+ 2π ) d e i S (x,x 0 ) i (x x 0 ) 2 T,0 2T q T,0 (, x, x 0 ) = R d e ( J Φ = (x j T j x j+ t j+ x 0 )ξ j T j+ T j+ j= 2π J+ j= ω j (x j, x j ). i ( t j+ T j ξ 2 2T j+ j +(x j T j x T j+ t j+ x j+ T 0 )ξ j ) j+ dξj J j= ) dj R 2dJ e i Φ F T,0 t j+ T j 2T j+ ξ 2 j J+ j= J j= dx j dξ j, ω j (x j, x j ).
45 M j = i( ξ j Φ) ξj + ξj Φ, N 2 j = i( x j Φ) xj, j =, 2,..., J + xj Φ 2 M j e i Φ = e i Φ, N j e i Φ = e i Φ. ( ) dj J ( ) e i dj Φ F T,0 dx j dξ j = e i Φ F 2π R 2dJ 2π R 2dJ T,0 j= J j= dx j dξ j, F T,0 = (N J )d+ (N 2 )d+ (N )d+ (M J )d+ (M 2 )d+ (M )d+ F T,0 M j, N j M j, N j
46 J J C J ξj Φ = x j T j x j+ t j+ x 0 t j+t j ξ j, T j+ T j+ T j+ M j = a j (x j+, ξ j, x j, x 0 ) ξj + a 0 j (x j+, ξ j, x j, x 0 ), J 4d xj+, ξj, xj M j N j+, M j, N j M j F T,0 J C J M j = C ( + ξj Φ 2 ) /2, N j = C ( + xj Φ 2 ) /2.
47 z j = ξj Φ, ζ j = xj Φ, j =, 2,..., J J F T,0 (C ) J ( + j= z j 2 ) (d+)/2 ( + ζ j 2 ), (d+)/2 det (x J,..., x, ξ J,..., ξ ) (z J,..., z, ζ J,..., ζ ) (C ) J. (x J,..., x, ξ J,..., ξ ) (z J,..., z, ζ J,..., ζ ) ( ) dj J e i Φ F 2π R 2dJ T,0 dx j dξ j j= ( ) dj = e i Φ F 2π R 2dJ T,0 det (x J,..., x, ξ J,..., ξ ) J (z J,..., z, ζ J,..., ζ ) dz j dζ j. (z J,..., z, ζ J,..., ζ ) J C J j=
48 2. Fujiwara D. Fujiwara, Nagoya Math. J. 24 (99). γ T,0 (T j, x j ) (T j, x j ) S[γ T,0 ] = S T,0 (x J+, x J,..., x, x 0 ), F [γ T,0 ] = F T,0 (x J+, x J,..., x, x 0 ). (T J, x J ) (T 3, x 3 ) (T, x) (T, x ) (0, x 0 ) (T 2, x 2 ) T 0 = 0 T T 2 T 3 T J T = T J+ Remark γ T,0 (T, x) (0, x 0 ) S[γ T,0 ] = S T,0 (x, x 0 ), F [γ T,0 ] = F T,0 (x, x 0 ).
49 ( (xj,...,x )S T,0 )(x J+, x J,..., x, x 0) = 0 (x J,..., x ) J+ j= D T,0 (x J+, x 0 ) = t d j det( 2 (x J,...,x ) S T,0 )(x J+, x J,..., x, x 0) T J+ J+ ( ) d/2 J e i S T,0 (x J+,x J,...,x,x 0 ) F T,0 (x J+, x J,..., x, x 0 ) dx j 2πi t j= j R dj j= ( ) d/2 ( ) = e i S T,0(x,x 0 ) D T,0 (x, x 0 ) /2 F T,0 (x, x 0 ) + Υ T,0 (, x, x 0 ). 2πi T Lemma F m 0 M A M, X M T,0 α j M, j = 0,,..., J, J + J+ ( j=0 α j x j )F T,0 (x J+, x J,..., x, x 0 ) A M (X M ) J+ ( + J+ j=0 x j ) m. J C Υ T,0 (, x, x 0 ) CT ( + x J+ + x 0 ) m.
50 Remark J = 0,, 2 Lemma F T = T > T 0 = 0 J = 0 α x α 0 x 0 F T,0 (x, x 0 ) A M (X M ) ( + x + x 0 ) m. T = T 2 > T > T 0 = 0 J = α 2 x 2 α x α 0 x 0 F T,T,0(x 2, x, x 0 ) A M (X M ) 2 ( + x 2 + x + x 0 ) m. T = T 3 > T 2 > T > T 0 = 0 J = 2 α 3 x 3 α 2 x 2 α x α 0 x 0 F T,T2,T,0(x 3, x 2, x, x 0 ) A M (X M ) 3 ( + x 3 + x 2 + x + x 0 ) m. Remark x αb(t, x) C α F [γ] = e T B(t,γ(t))dt 0 F T,0 = F [γ T,0 ] = Tj J+ j= e T B(t,γ Tj,T j j (t,x j,x j ))dt.
51 Proof of Lemma F m = 0 x, x 2,..., x J x (M F T,0 ) (R F T,0 ) ( ) d/2 ( ) d/2 e i S T,0 F T,0 (..., x 2, x, x 0 )dx 2πi t 2 2πi t R ( ) d d/2 = e i S ( T,T2,0) (M F 2πi T )(..., x T,0 2, x 0 ) 2 ( ) d/2 + e i S ( T,T2,0) (R F 2πi T )(..., x T,0 2, x 0 ). 2
52 (M F T,0 ) x 2 (M F T,0 )(..., x 2, x 0 ) = D T2,T,0(x 2, x 0 ) /2 F T,0 (..., x 2, x, x 0) = D (x T2,0 2, x 0 ) /2 F ( T,T2,0)(..., x 2, x 0 ), ( T,T2, 0) : T = T J+ > T J > > T 2 > T 0 = 0 (R F T,0 ) x 2 (t 2 ) (R F T,0 ) C(t 2 ). (T, x) (0, x 0 ) (T 2, x 2 ) T 0 = 0 T T 2 T = T J+
53 (M F T,0 ) x 2 x 2 ( ) d/2 ( ) d/2 e i S ( T,T2,0) (M F T,0 )(..., x 3, x 2, x 0 )dx 2 2πi t 3 2πi T 2 R ( ) d d/2 = e i S ( T,T3,0) (M 2 M F 2πi T )(..., x T,0 3, x 0 ) 3 ( ) d/2 i + e S ( T,T3,0) (R 2 M F T,0 )(..., x 3, x 0 ). 2πi T 3
54 (M 2 M F T,0 ) x 3 (M 2 M F T,0 )(..., x 3, x 0 ) = D T3,T 2,0(x 3, x 0 ) /2 D T2,0 (x 2, x 0) /2 F ( T,T2,0)(..., x 3, x 2, x 0). = D (x T3,0 3, x 0 ) /2 F ( T,T3,0)(..., x 3, x 0 ), ( T,T3, 0) : T = T J+ > T J > > T 3 > T 0 = 0 (R 2 M F T,0 ) x 3 (t 3 ) (R 2 M F T,0 ) C(t 3 ). (T 3, x 3 ) (T, x) (0, x 0 ) T 0 = 0 T 2 T 3 T = T J+
55 (M 2 M F T,0 ) x 3 x 3 Theorem 7 (M J M J... M F T,0 ) = D T,0 (x, x 0 ) /2 F T,0 (x, x 0 ) = D T,0 (x, x 0 ) /2 F [γ T,0 ]. (T, x) (0, x 0 ) γ T,0 T 0 = 0 T = T J+
56 (R F T,0 ) x 2 x 2 x 3 ( ) d/2( ) d/2( ) d/2 e i S ( T,T2,0) (R F T,0 )(..., x 4, x 3, x 2, x 0 )dx 3 2πi t 4 2πi t 3 2πi T 2 R ( ) d d/2 ( ) d/2 = e i S ( T,T4,T 2,0) (M 3 R F 2πi (t 4 + t 3 ) 2πi T )(..., x T,0 4, x 2, x 0 ) 2 ( ) d/2 ( ) d/2 + e i S ( T,T4,T 2,0) (R 3 R F 2πi (t 4 + t 3 ) 2πi T )(..., x T,0 4, x 2, x 0 ). 2 (R 3 R F T,0 ) C(t 4 )C(t 2 ). (M 3 R F T,0 ) x 4 x 4 (R 3 R F T,0 ) x 4 x 4 x 5
57 Fujiwara x j x j+ x j+ x j+ x j+ x j+2
58 q T,0 (, x J+, x 0 ) = q 0 (x J+, x 0 ) + qjk,jk,...,j (x J+, x 0 ). q 0 (x J+, x 0 ) = D T,0 (x, x 0 ) /2 F T,0 (x, x 0 ) Theorem 7 0 = j 0 < j < j < j 2 < j 2 < < j K < j K J +, (j K, j K,..., j ) q jk,j K,...,j (x J+, x 0 ) ( ) d/2 e i S T,0(x J+,x 0 ) q jk,j 2πi T K,...,j (x J+, x 0 ) = K+ k= RdK e i S (T,T jk,...,t j,0) b jk,j K,...,j (x J+, x jk,..., x j, x 0 ) ( 2πi (T jk T jk ) K dx jk, k= ) d/2
59 b jk,j K,...,j (x J+, x jk,..., x j, x 0 ) = (Q J Q 3 Q 2 Q F T,0 )(x J+, x jk,..., x j, x 0 ). Q j = Identity if j = j K, j K,..., j R j if j = j K, j K,..., j M j otherwise (t jk ) b jk,j K,...,j (x J+, x jk,..., x j, x 0 ) C K ( K k= ) ( t jk ).
60 H. Kumano-go-Taniguchi q jk,j K,...,j (x J+, x 0 ) (C ) K ( K q jk,j K,...,j (x J+, x 0 ) k= ) ( t jk ). Υ T,0 (, x J+, x 0 ) = qjk,jk,...,j (x J+, x 0 ) J+ t j = T, 0 < < j= Υ T,0 (, x J+, x 0 ) K ) ((C ) K ( t jk ) ( J+ k= ) ( + C t j ) j= (C )T
61 3. m 0, u j 0, J+ j= u j U < M A M, X M T,0, α j M, j = 0,,..., J + k J J+ () ( j=0 α j x j )F T,0 (x J+, x J,..., x, x 0 ) A M (X M ) J+ ( + J+ (2) ( α j x j ) xk F T,0 (x J+, x J,..., x, x 0 ) j=0 A M (X M ) J+ (u k+ + u k )( + J+ j=0 x j ) m. J+ j=0 x j ) m, Remark x αb(t, x) C α F [γ] = e T B(t,γ(t))dt 0 xk F T,0 = Tj J+ j= e T B(t,γ Tj,T j j (t,x j,x j ))dt ( T k+ T k xk B(t, γ Tk+,T k (x k+, x k ))dt + T k T k xk B(t, γ Tk,T k (x k, x k ))dt).
62 J+ ( ) d/2 J ( ) e i S[γ T,0 ] F [γ T,0 ] dx j 2πi t j= j R dj j= ( ) d/2 = e i S T,0(x,x 0 ) q T,0 (, x, x 0 ) 2πi T ( ) d/2 ( ) = e i S T,0(x,x 0 ) D T,0 (x, x 0 ) /2 F T,0 (x, x 0 ) + Υ T,0 (, x, x 0 ) 2πi T Lemma F. Υ T,0 (, x, x 0 ) CT (T + U)( + x + x 0 ) m, q T,0 (, x, x 0 ) C ( + x + x 0 ) m. Lemma F m = 0
63 (T, x) (0, x 0 ) T 0 = 0 T = T J+ (T, x) (0, x 0 ) T 0 = 0 T = T J+ CT (T + U)
64 (T, x) (0, x 0 ) T 0 = 0 T = T J+ C T,0 ( ) Cauchy T,0 : T = T J+ > T J > > T > T 0 = 0, ( T,TN+, Tn,0) : T = T J+ > > T N+ > T n > > T 0 = 0,
65 J+ j= = ( ) d/2 e i S[γ T,0 ] F [γ T,0 ] 2πi t j ( ) d/2 e i S T,0(x,x 0 ) q T,0 (, x, x 0 ), 2πi T J j= dx j J+ ( j=n+2 = ( 2πi T 2πi t j ) d/2 ( 2πi (T N+ T n ) ) n j= ( 2πi t j ) d/2 i e S[γ ( T,TN+, Tn,0 )] F [γ ( T,TN+, Tn,0)] ) d/2 e i S T,0(x,x 0 ) q ( T,TN+, Tn,0)(, x, x 0 ). J j=n+ n dx j j= dx j
66 (T n, x n ) (T, x) (0, x 0 ) (T N+, x N+ ) T 0 = 0 T n T N+ T = T J+ (T n, x n ) (T, x) (0, x 0 ) (T N+, x N+ ) T 0 = 0 T n T N+ T = T J+ C C(T N+ T N )(T N+ T n + U N+ U n ) C
67 q T,0 (, x, x 0 ) q ( T,TN+, Tn,0)(, x, x 0 ) C(T N+ T n )(T N+ T n + U N+ U n )( + x + x 0 ) m. Theorem 2 Feynman q(t,, x, x 0 ) q T,0 (, x, x 0 ) q(t,, x, x 0 ) C T,0 (U + T )( + x + x 0 ) m, ( ) T,0 0 R 2d
68 ( Fujiwara ) Fujiwara D. Fujiwara and N. Kumano-go, J. Math. Soc. Japan Vol 58, No. 3 (2006). Lemma γ T,T2,T,0 (0, x 0 ), (T, x ), (T 2, x 2 ), (T, x) ( ) q(t ) q(t, x, x 0 ) lim x D(T, x, x 0 ) /2 x F [γ 2 =γ T,T2,T T2,0] cl (T 2 ) T (x, x 0 ) R 2d x =γ cl (T ). (T, x) x (0, x 0 ) x 2 γ cl T 0 = 0 T T 2 T = T J+
69 Theorem 8 Lemma q(t) ( ) e i d/2 S[γ] F [γ]d[γ] = e i S[γcl ] D(T, x, x 0 ) /2 2πi T ( F [γ cl ] + i T ) D(t, γ cl (t), x 0 ) /2 q(t)dt + 2 Υ (, x, x 0 ) 2 0 Υ (, x, x 0 ) C( + x + x 0 ) m. Remark F [γ] i 2 T 0 D(t, γ cl (t), x 0 ) /2 ( y D(t, y, x 0 ) /2 ) y=γ cl (t) dt Schrödinger Birkhoff
C 1 -path x t x 1 (f(x u), dx u ) rough path analyi p-variation (1 < p < 2) rough path 2 Introduction f(x) = (fj i(x)) 1 i n,1 j d (x R d ) (n, d) Cb
Rough path analyi 1 x t ( t 1) R d path f(x) = t (f 1 (x),, f d (x)) R d R d - C x t 1 (f(x u), dx u ) Stieltje path x t p-variation norm (1 < p < 2) x p := { up D } 1/p N x ti x ti 1 p (D = { = t <
43433 8 3 . Stochastic exponentials...................................... 3. Girsanov s theorem......................................... 4 On the martingale property of stochastic exponentials 5. Gronwall
,. 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,
takei.dvi
0 Newton Leibniz ( ) α1 ( ) αn (1) a α1,...,α n (x) u(x) = f(x) x 1 x n α 1 + +α n m 1957 Hans Lewy Lewy 1970 1 1.1 Example 1.1. (2) d 2 u dx 2 Q(x)u = f(x), u(0) = a, 1 du (0) = b. dx Q(x), f(x) x = 0
Grushin 2MA16039T
Grushin 2MA1639T 3 2 2 R d Borel α i k (x, bi (x, 1 i d, 1 k N d N α R d b α = α(x := (αk(x i 1 i d, 1 k N b = b(x := (b i (x 1 i d X = (X t t x R d dx t = α(x t db t + b(x t dt ( 3 u t = Au + V u, u(,
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
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
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
Akito Tsuboi June 22, T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1
Akito Tsuboi June 22, 2006 1 T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1 1. X, Y, Z,... 2. A, B (A), (A) (B), (A) (B), (A) (B) Exercise 2 1. (X) (Y ) 2. ((X) (Y )) (Z) 3. (((X) (Y )) (Z)) Exercise
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
II Brown Brown
II 16 12 5 1 Brown 3 1.1..................................... 3 1.2 Brown............................... 5 1.3................................... 8 1.4 Markov.................................... 1 1.5
Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)
![Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)](/thumbs/94/118373946.jpg "Z[i] Z[i] π 4,1 (x) π 4,3 (x) 1 x (x ) 2 log x π m,a (x) 1 x ϕ(m) log x 1.1 ( ). π(x) x (a, m) = 1 π m,a (x) x modm a 1 π m,a (x) 1 ϕ(m) π(x)")
3 3 22 Z[i] Z[i] π 4, (x) π 4,3 (x) x (x ) 2 log x π m,a (x) x ϕ(m) log x. ( ). π(x) x (a, m) = π m,a (x) x modm a π m,a (x) ϕ(m) π(x) ϕ(m) x log x ϕ(m) m f(x) g(x) (x α) lim f(x)/g(x) = x α mod m (a,
grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
2 9 2 5 2.2.3 grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = g () g () (3) grad φ(p ) p grad φ φ (P, φ(p )) y (, y) = (ξ(t), η(t)) ( ) ξ (t) (t) := η (t) grad f(ξ(t), η(t)) (t) g(t) := f(ξ(t), η(t))
Trapezoidal Rule θ = 1/ x n x n 1 t = 1 [f(t n 1, x n 1 ) + f(t n, x n )] (6) 1. dx dt = f(t, x), x(t 0) = x 0 (7) t [t 0, t 1 ] f t [t 0, t 1 ], x x
![Trapezoidal Rule θ = 1/ x n x n 1 t = 1 [f(t n 1, x n 1 ) + f(t n, x n )] (6) 1. dx dt = f(t, x), x(t 0) = x 0 (7) t [t 0, t 1 ] f t [t 0, t 1 ], x x](/thumbs/92/107866474.jpg "Trapezoidal Rule θ = 1/ x n x n 1 t = 1 [f(t n 1, x n 1 ) + f(t n, x n )] (6) 1. dx dt = f(t, x), x(t 0) = x 0 (7) t [t 0, t 1 ] f t [t 0, t 1 ], x x")
University of Hyogo 8 8 1 d x(t) =f(t, x(t)), dt (1) x(t 0 ) =x 0 () t n = t 0 + n t x x n n x n x 0 x i i = 0,..., n 1 x n x(t) 1 1.1 1 1 1 0 θ 1 θ x n x n 1 t = θf(t n 1, x n 1 ) + (1 θ)f(t n, x n )
(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)
Chebyshev Schrödinger Heisenberg H = 1 2m p2 + V (x), m = 1, h = 1 1/36 1 V (x) = { 0 (0 < x < L) (otherwise) ψ n (x) = 2 L sin (n + 1)π x L, n = 0, 1, 2,... Feynman K (a, b; T ) = e i EnT/ h ψ n (a)ψ
Lebesgue Fubini L p Banach, Hilbert Höld
II (Analysis II) Lebesgue (Applications of Lebesgue Integral Theory) 1 (Seiji HIABA) 1 ( ),,, ( ) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................
Hilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2
![Hilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2](/thumbs/91/105692760.jpg "Hilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2")
hara@math.kyushu-u.ac.jp 1 1 1.1............................................... 2 1.2............................................. 3 2 3 3 5 3.1............................................. 6 3.2...................................
(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 :.
December 28, 2018
e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................
z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy
f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =
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
v_-3_+2_1.eps
I 9-9 (3) 9 9, x, x (t)+a(t)x (t)+b(t)x(t) = f(t) (9), a(t), b(t), f(t),,, f(t),, a(t), b(t),,, x (t)+ax (t)+bx(t) = (9),, x (t)+ax (t)+bx(t) = f(t) (93), b(t),, b(t) 9 x (t), x (t), x (t)+a(t)x (t)+b(t)x(t)
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,
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 (](/thumbs/97/133036028.jpg "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
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)
Black-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W
![Black-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W](/thumbs/97/132191371.jpg "Black-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W")
003 7 14 Black-Scholes [1] Nelson [] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-Wu Nelson e-mail: takatoshi-tasaki@nifty.com kabutaro@mocha.freemail.ne.jp
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](/thumbs/103/158001152.jpg "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..
9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P
9 (Finite Element Method; FEM) 9. 9. P(0) P(x) u(x) (a) P(L) f P(0) P(x) (b) 9. P(L) 9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L)
2011 8 26 3 I 5 1 7 1.1 Markov................................ 7 2 Gau 13 2.1.................................. 13 2.2............................... 18 2.3............................ 23 3 Gau (Le vy
( ) ) ) ) 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)
7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a
9 203 6 7 WWW http://www.math.meiji.ac.jp/~mk/lectue/tahensuu-203/ 2 8 8 7. 7 7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa,
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.........................
(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law
I (Radom Walks ad Percolatios) 3 4 7 ( -2 ) (Preface),.,,,...,,.,,,,.,.,,.,,. (,.) (Basic of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios, Meas).............................
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
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
²ÄÀÑʬΥ»¶ÈóÀþ·¿¥·¥å¥ì¡¼¥Ç¥£¥ó¥¬¡¼ÊýÄø¼°¤ÎÁ²¶á²òÀÏ Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation
Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation ( ) ( ) 2016 12 17 1. Schrödinger focusing NLS iu t + u xx +2 u 2 u = 0 u(x, t) =2ηe 2iξx 4i(ξ2 η 2 )t+i(ψ 0 +π/2) sech(2ηx
all.dvi
I 1 Density Matrix 1.1 ( (Observable) Ô :ensemble ensemble average) Ô en =Tr ˆρ en Ô ˆρ en Tr  n, n =, 1,, Tr  = n n  n Tr  I w j j ( j =, 1,, ) ˆρ en j w j j ˆρ en = j w j j j Ô en = j w j j Ô j emsemble
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
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
163 KdV KP Lax pair L, B L L L 1/2 W 1 LW = ( / x W t 1, t 2, t 3, ψ t n ψ/ t n = B nψ (KdV B n = L n/2 KP B n = L n KdV KP Lax W Lax τ KP L ψ τ τ Cha
63 KdV KP Lax pair L, B L L L / W LW / x W t, t, t 3, ψ t n / B nψ KdV B n L n/ KP B n L n KdV KP Lax W Lax τ KP L ψ τ τ Chapter 7 An Introduction to the Sato Theory Masayui OIKAWA, Faculty of Engneering,
I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( )
I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( ) 1 (Introduction) 1 1.1... 1 1.2 Riemann Lebesgue... 2 2 (Measurable sets and Measures) 4 2.1 σ-... 4 2.2 Borel... 5 2.3...
(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi
I (Basics of Probability Theory ad Radom Walks) 25 4 5 ( 4 ) (Preface),.,,,.,,,...,,.,.,,.,,. (,.) (Basics of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios,
(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](/thumbs/67/56612994.jpg "(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 ]
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(
July 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i
July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac
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....
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..................................
III III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). T
III III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). Theorem 1.3 (Lebesgue ) lim n f n = f µ-a.e. g L 1 (µ)
S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt
S I. x yx y y, y,. F x, y, y, y,, y n http://ayapin.film.s.dendai.ac.jp/~matuda n /TeX/lecture.html PDF PS yx.................................... 3.3.................... 9.4................5..............
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
1 4 1 ( ) ( ) ( ) ( ) () 1 4 2
7 1995, 2017 7 21 1 2 2 3 3 4 4 6 (1).................................... 6 (2)..................................... 6 (3) t................. 9 5 11 (1)......................................... 11 (2)
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
Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math))
Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math)) 2001 1 e-mail:s00x0427@ip.media.kyoto-u.ac.jp 1 1 Van der Pol 1 1 2 2 Bergers 2 KdV 2 1 5 1.1........................................
ver.1 / c /(13)
1 -- 11 1 c 2010 1/(13) 1 -- 11 -- 1 1--1 1--1--1 2009 3 t R x R n 1 ẋ = f(t, x) f = ( f 1,, f n ) f x(t) = ϕ(x 0, t) x(0) = x 0 n f f t 1--1--2 2009 3 q = (q 1,..., q m ), p = (p 1,..., p m ) x = (q,
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
untitled
3 3. (stochastic differential equations) { dx(t) =f(t, X)dt + G(t, X)dW (t), t [,T], (3.) X( )=X X(t) : [,T] R d (d ) f(t, X) : [,T] R d R d (drift term) G(t, X) : [,T] R d R d m (diffusion term) W (t)
IA hara@math.kyushu-u.ac.jp Last updated: January,......................................................................................................................................................................................
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](/thumbs/94/118569863.jpg "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
1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) / 25
.. IV 2012 10 4 ( ) 2012 10 4 1 / 25 1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) 2012 10 4 2 / 25 1. Ω ε B ε t
main.dvi
SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1
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](/thumbs/101/149159646.jpg "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
1 R n (x (k) = (x (k) 1,, x(k) n )) k 1 lim k,l x(k) x (l) = 0 (x (k) ) 1.1. (i) R n U U, r > 0, r () U (ii) R n F F F (iii) R n S S S = { R n ; r > 0
III 2018 11 7 1 2 2 3 3 6 4 8 5 10 ϵ-δ http://www.mth.ngoy-u.c.jp/ ymgmi/teching/set2018.pdf http://www.mth.ngoy-u.c.jp/ ymgmi/teching/rel2018.pdf n x = (x 1,, x n ) n R n x 0 = (0,, 0) x = (x 1 ) 2 +
A bound of the number of reduced Arakelov divisors of a number field (joint work with Ryusuke Yoshimitsu) Takao Watanabe Department of Mathematics Osa
A bound of the number of reduced Arakelov divisors of a number field (joint work with Ryusuke Yoshimitsu) Takao Watanabe Department of Mathematics Osaka University , Schoof Algorithmic Number Theory, MSRI
??
( ) 2014 2014 1/119 = (ISS) ISS ISS ISS iss-clf iss-clf ISS = (ISS) FB 2014 2/119 = (ISS) ISS ISS ISS iss-clf iss-clf ISS R + : 0 K: γ: R + R + K γ γ(0) = 0 K : γ: R + R + K γ K γ(r) (r ) FB K K K K R
08-Note2-web
r(t) t r(t) O v(t) = dr(t) dt a(t) = dv(t) dt = d2 r(t) dt 2 r(t), v(t), a(t) t dr(t) dt r(t) =(x(t),y(t),z(t)) = d 2 r(t) dt 2 = ( dx(t) dt ( d 2 x(t) dt 2, dy(t), dz(t) dt dt ), d2 y(t) dt 2, d2 z(t)
P F ext 1: F ext P F ext (Count Rumford, ) H 2 O H 2 O 2 F ext F ext N 2 O 2 2
1 1 2 2 2 1 1 P F ext 1: F ext P F ext (Count Rumford, 1753 1814) 0 100 H 2 O H 2 O 2 F ext F ext N 2 O 2 2 P F S F = P S (1) ( 1 ) F ext x W ext W ext = F ext x (2) F ext P S W ext = P S x (3) S x V V
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
V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
z f(z) f(z) x, y, u, v, r, θ r > 0 z = x + iy, f = u + iv C γ D f(z) f(z) D f(z) f(z) z, Rm z, z 1.1 z = x + iy = re iθ = r (cos θ + i sin θ) z = x iy
z fz fz x, y, u, v, r, θ r > z = x + iy, f = u + iv γ D fz fz D fz fz z, Rm z, z. z = x + iy = re iθ = r cos θ + i sin θ z = x iy = re iθ = r cos θ i sin θ x = z + z = Re z, y = z z = Im z i r = z = z
2 R U, U Hausdorff, R. R. S R = (S, A) (closed), (open). (complete projective smooth algebraic curve) (cf. 2). 1., ( ).,. countable ( 2 ) ,,.,,
15, pp.1-13 1 1.1,. 1.1. C ( ) f = u + iv, (, u, v f ). 1 1. f f x = i f x u x = v y, u y = v x.., u, v u = v = 0 (, f = 2 f x + 2 f )., 2 y2 u = 0. u, u. 1,. 1.2. S, A S. (i) A φ S U φ C. (ii) φ A U φ
( ) 1 1.1? ( ) ( ) ( ) 1.1(a) T m ( ) 1.1(a) T g ( ) T g T g 500 74% ( ) T K ( 1.1(b) 15 T g T g 10 13 T g T g T g [ ] A ( ) exp (1.1) T T 0 Vogel-Fulcher T 0 T 0 T K T K Ortho-Terphenil (OTP) SiO 2 (1.1)
= M + M + M + M M + =.,. f = < ρ, > ρ ρ. ρ f. = ρ = = ± = log 4 = = = ± f = k k ρ. k
7 b f n f} d = b f n f d,. 5,. [ ] ɛ >, n ɛ + + n < ɛ. m. n m log + < n m. n lim sin kπ sin kπ } k π sin = n n n. k= 4 f, y = r + s, y = rs f rs = f + r + sf y + rsf yy + f y. f = f =, f = sin. 5 f f =.
9 Feb 2008 NOGUCHI (UT) HDVT 9 Feb / 33
9 Feb 2008 NOGUCHI (UT) HDVT 9 Feb 2008 1 / 33 1 NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33 1 Green-Griffiths (1972) NOGUCHI (UT) HDVT 9 Feb 2008 2 / 33 1 Green-Griffiths (1972) X f : C X f (C) X NOGUCHI (UT)
untitled
Lie L ( Introduction L Rankin-Selberg, Hecke L (,,, Rankin, Selberg L (GL( GL( L, L. Rankin-Selberg, Fourier, (=Fourier (= Basic identity.,,.,, L.,,,,., ( Lie G (=G, G.., 5, Sp(, R,. L., GL(n, R Whittaker
[1][2] [3] *1 Defnton 1.1. W () = σ 2 dt [2] Defnton 1.2. W (t ) Defnton 1.3. W () = E[W (t)] = Cov[W (t), W (s)] = E[W (t)w (s)] = σ 2 mn{s, t} Propo
![[1][2] [3] *1 Defnton 1.1. W () = σ 2 dt [2] Defnton 1.2. W (t ) Defnton 1.3. W () = E[W (t)] = Cov[W (t), W (s)] = E[W (t)w (s)] = σ 2 mn{s, t} Propo](/thumbs/93/114156950.jpg "[1][2] [3] *1 Defnton 1.1. W () = σ 2 dt [2] Defnton 1.2. W (t ) Defnton 1.3. W () = E[W (t)] = Cov[W (t), W (s)] = E[W (t)w (s)] = σ 2 mn{s, t} Propo")
@phykm 218 7 12 [2] [2] [1] ([4] ) 1 Ω = 2 N {Π n =1 A { 1, 1} N n N, A {{ 1, 1}, { 1}, {1}, }} B : Ω { 1, 1} P (Π n =1 A 2 N ) = 2 #{ A={ 1},{1}} X = j=1 B j B X +k X V[X ] = 1 ( ) 1 1 dt dx W (t) = t/dt
chap9.dvi
9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j =
,,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,,
![,,,17,,, ( ),, E Q [S T F t ] < S t, t [, T ],,,,,,,,](/thumbs/91/105754403.jpg ",,,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.........................................
第5章 偏微分方程式の境界値問題
October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ
φ s i = m j=1 f x j ξ j s i (1)? φ i = φ s i f j = f x j x ji = ξ j s i (1) φ 1 φ 2. φ n = m j=1 f jx j1 m j=1 f jx j2. m
2009 10 6 23 7.5 7.5.1 7.2.5 φ s i m j1 x j ξ j s i (1)? φ i φ s i f j x j x ji ξ j s i (1) φ 1 φ 2. φ n m j1 f jx j1 m j1 f jx j2. m j1 f jx jn x 11 x 21 x m1 x 12 x 22 x m2...... m j1 x j1f j m j1 x
1 a b = max{a, b}, a b = mi{a, b} a 1 a 2 a a 1 a = max{a 1,... a }, a 1 a = mi{a 1,... a }. A sup A, if A A A A A sup A sup A = + A if A = ± y = arct
27 6 2 1 2 2 5 3 8 4 13 5 16 6 19 7 23 8 27 N Z = {, ±1, ±2,... }, R =, R + = [, + ), R = [, ], C =. a b = max{a, b}, a b = mi{a, b}, a a, a a. f : X R [a < f < b] = {x X; a < f(x) < b}. X [f] = [f ],
2 1 x 2 x 2 = RT 3πηaN A t (1.2) R/N A N A N A = N A m n(z) = n exp ( ) m gz k B T (1.3) z n z = m = m ρgv k B = erg K 1 R =
1 1 1.1 1827 *1 195 *2 x 2 t x 2 = 2Dt D RT D = RT N A 1 6πaη (1.1) D N A a η 198 *3 ( a =.212µ) *1 Robert Brown (1773-1858. *2 Albert Einstein (1879-1955 *3 Jean Baptiste Perrin (187-1942 2 1 x 2 x 2
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............................................
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=
φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)
φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x
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
9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =
5 5. 5.. A II f() f() F () f() F () = f() C (F () + C) = F () = f() F () + C f() F () G() f() G () = F () 39 G() = F () + C C f() F () f() F () + C C f() f() d f() f() C f() f() F () = f() f() f() d =
II 1 II 2012 II Gauss-Bonnet II
II 1 II 212 II Gauss-Bonnet II 1 1 1.1......................................... 1 1.2............................................ 2 1.3.................................. 3 1.4.............................................
2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i
1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x,
4................................. 4................................. 4 6................................. 6................................. 9.................................................... 3..3..........................
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
25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
untitled
2 : n =1, 2,, 10000 0.5125 0.51 0.5075 0.505 0.5025 0.5 0.4975 0.495 0 2000 4000 6000 8000 10000 2 weak law of large numbers 1. X 1,X 2,,X n 2. µ = E(X i ),i=1, 2,,n 3. σi 2 = V (X i ) σ 2,i=1, 2,,n ɛ>0
A
A 2563 15 4 21 1 3 1.1................................................ 3 1.2............................................. 3 2 3 2.1......................................... 3 2.2............................................
L(A) l(a)
4 3 214 6 214 6 13 1 3 1.1................................................ 3 1.2................................... 5 1.3 L(A) l(a)............................................. 6 1.4.......................................
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)
Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu
![Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu](/thumbs/93/114422882.jpg "Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu")
rigidity 2014.9.1-2014.9.2 Fuchs 1 Introduction y + p(x)y + q(x)y = 0, y 2 p(x), q(x) p(x) q(x) Fuchs 19 Fuchs 83 Gauss Fuchs rigid rigid rigid 7 1970 1996 Nicholas Katz Rigid local systems [6] Fuchs Katz
30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
III Kepler ( )
III 9 8 3....................................... 3.2 Kepler ( ).......................... 0 2 3 2.................................. 3 2.2......................................... 7 3 9 3..........................................
2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c
GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,