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
|
|
- まれあ はにうだ
- 4 years ago
- Views:
Transcription
1 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, Fuuoa Institute of Technology Lax N- N- τ τ τ, 3 Lax L KdV KP oiawa@fitacjp KP Grassmann KP τ Grassmann Plücer KP Plücer KP τ τ 4 Ohta Ohta 5 5
2 64 KdV KdV 4u t uu x + u xxx 7 Lax 9 L + ux, t B 3 + 3u + 3 u x 7 / x L ψ Lψ λ ψ 73 t Bψ λ L t [B, L] BL LB 74 KdV 7 Lax KdV Lax Lax B KdV B, B u + 3 u x, B u u x u xx + 5 u u xxx + 5 uu x 74 B B i u t u x u t 4 u xx + 3 u x u t 6 u xxxx uu xx u x + 5 u3 x B i B B L, B 4 L, [B, L], [B 4, L] B L /, B 3 L 3/, B 5 L 5/ P a j x n j, a x 77 j n microdifferenial operator [B, L] n n af n r r n r a r f n r, n : nn n r + r! 78 a r ax x r x a, a ax n n a a r n r 79 r r n r n 77 n 79 n 78 n r a a a + a 3 + a a a 3 + 3a P a j x n j, Q b x m 7 P Q c i j a j x n j j j b x m a j x n j b x m j r n j a j r b r m+n j r c i m+n i 7 i j++ri n j r a j b r 73 3 c i c a b, c a b + a b + na b, c a b + a b + a b + na b +n a b nn + a b 74
3 65 L / X + f n n 75 n X + f n n + f m m n n m + f n n + f n n + m,n,l n l n f n l f m m n l 76 + u f, f, L / + u u x uxx + + u xxx uu x 4 u B n L n/ [L n/, L] B n : L n/ + L n/ 78 L n/ L n/ + B c n B n L n/ + B c n [B n, L] [L n/, L] + [B c n, L] [B c n, L] [ α n +β n +, +u] 79 α n, β n, u α n L n/ 7 74 [B n, L] + B n, L [B n, L] [B n, L] 79 [B n, L] α n x Lax L t [B n, L] u t α n x 78 B n 75 u x t, t 3, t 5, Lψ λ ψ 7 B n ψ, n, 3, 5, λ L [B n, L], n, 3, 5, 7 KdV KdV hierarchy u t u t3 u t5 u x 4 u xx + 3 u x 6 u xxxx uu xx u x + 5 u3 x 7 B n ψ B m ψ t m B n ψ B m ψ t m B n ψ B m ψ t m B n ψ B m ψ + B n B m t m t m Bn B m + [B n, B m ] ψ t m ψ B n t m B m + [B n, B m ] 73 Zaharov-Shabat Lax 3 KP L u n n 74 n u, u L + u + u L λ Lψ λψ B n ψ, B n : L n + 76
4 66 u, u 3, x t, t, t 3, Lax L [B n, L], n,, 3, 77 B n B B + u B u + 3u 3 + 3u x B u + 4u 3 + 6u x + 4u 4 + 6u 3x + 4u xx + 6u Lax 77 n 77 L [, L] L t x u it 78 u ix, i, 3, i, 3, u i x + t, t, t 3, n 77 u t + u 3 t + u 4 t u + u + u u + u u u xx + u 3x + u 3xx + u 4x + u u x +u 4xx + u 5x u u xx + 4u x u j j,, u t u xx + u 3x u 3 t u 3xx + u 4x + u u x u 4 t u 4xx + u 5x u u xx + 4u x u 3 n 3 77 u t 3 u xxx +3u 3xx +3u 4x +6u u x u 3 t 3 u 3xxx +3u 4xx +3u 5x +6u u 3x +6u x u 3 u 4 t 3 u 4xxx +3u 5xx +3u 6x 3u u 3xx 3u xx u 3 +3u u 4x +9u x u 4 +6u 3 u 3x u, u 3, u 4, u 3, u 4 u 4 u u + u x t 3 x + 3 u x u t 73 Petviashvili KP KP KP hierarchy Zaharov-Shabat 73 Lax 77 L L L + L L [B n, L]L + L[B n, L] B n L LB n L + LB n L LB n B n L L B n [B n, L ] L 3 L L + L L L + L L [B n, L]L + L[B n, L]L + L [B n, L] B n L LB n L +LB n L LB n L+L B n L LB n B n L 3 L 3 B n [B n, L 3 ] L m [B n, L m ] 73 L m Ln t m [B n, L m ] [B m, L n ] 733 L n B n Bn c Bc n L m Ln [L n +B t n, c L m ] [B m, B n Bn] c m [Bn, c B m Bm] c [B m, B n Bn] c [B n, B m ] [B c n, B c m] m, n 734 t y, t 3 t Kadomtsev- Zaharov- Shabat B m B n [B n, B m ] m, n 735 t m m, n 3 KP Bm c Bc n + [B t m, c Bn] c m, n 736 m
5 67 77 KP 77 KP Lax Lax Zaharov-Shabat 735 KP Zaharov-Shabat 4 KP W L W LW 737 W + w + w + w LW W W + w + w + w 3 + w LW + w + w + w x + u +w 3 + w x + u w + u 3 + w 4 + w 3x +u w u w x + u 3 w + u j j,, w x + u w x + u w + u w 3x + u w u w x + u 3 w + u 4 u, u 3, w, w, x C + c + c + 74 W W C C W C W C W C C W W W L Lax 77 W L W W L W W W W W W [ W ] W, L 74 [ W ] W B n, L n 74 B n L n + Bn c [ W ] W Bn c, L n 743 W 74 W W B t n, c Bn c W n W n 744 n B c n B n L n, L n W n W 74 W B n W W n, B n W n W + n 745 L W W W x C W W C W W B c nw, n 746 L, B n, B c n W W C 746 W C + W C B t nw c C, n 747 n C C W B c W t nw W C, n 748 n C F n W BnW c W W, n 749 F n 743 [ W ] W W W B c t nw, W n [, F n ] F n / x, n F n x 748 C F n C n 75
6 68 F n, C t, t, t 3, C/ t m C/ t m 75 F m F n t m + [F m, F n ] m, n Bn c Zaharov-Shabat 736 W Bn c W W W W B t nw c n W + W W B t nw c F n n [ ] F n, F m t m W [ B c n, ] Bm c W 75 t m Lax 77 W 745 KP W w, w, 738 W m + w + w + + w m m W m m fx m +w m ++w m fx 754 m f x, f x,, f m x, j,,, m f j xξ j +! ξj x+! ξj x ran m m ξ ξ ξ m Ξ ξ ξ ξ m 756 W m m x! x! Ξ 757 R m Ξ ΞR 757 Ξ {ran m m }/GLm, C 758 GLm, C m Grassmann GMm, Λ expxλ I + xλ + x! Λ + x x /! x 3 /3! x x /! x Hx : expxλξ f f f m f f f m f f f m w, w, w m m f w + m f w ++f w m m f m f m w + m f m w ++f m w m m f m w j j,, m f,, f m
7 69 m f m f f m f m m f m f m w j m f m j f f m f m m j f m f m W m f f m m m f m f m m f m f m W m f f m m f m f m j 76, 763 Hx m Wronsian w j j,, t t, t, 754 f j t, t, f j f j x; t f j x; t, t, 76 H Hx; t Hx; t Hx; t expxλ exp ηt, Λ Ξ 764 ηt, Λ t n Λ n 765 n Λ exp{x+t Λ+t Λ +t 3 Λ 3 + } p n x+t, t, t 3, p n Λ n 766 n ν +ν +ν +3ν 3 + n ν,ν,ν,ν 3, x ν t ν tν ν!ν!ν! 767 p n p p x + t p x + t + t p 3 6 x + t 3 + x + t t + t p n t m p n m, m < p n 769 p n x p n 77 p n 5 Hx; t p p p 3 ξ ξ ξ m p p Hx; t p ξ ξ ξ m ξ ξ ξ m h x; t h x; t hm x; t h x; t h x; t hm x; t 77 h j n x; t h j x; f j x 77 h j n x; t hj x; t n h j x; t x n 773 h j x; t hx; f j x 774 n x n hx; t, n,, W m x; t m h j x; t m + w x; t m + + w m x; th j x; t, j,,, m 776
8 7 w j x; t W m x; t h m h m h h m m h m m h m w j x; t h m h m j h h m m h m m j h m W m x; t h h m h m h m m h m h m m m h h m h m h m m Ohta 5 777, 778 τ w j τ t x t x 6 Lax Lax Lψ λψ B n ψ, B n L n + λ ψ L W W 779 λψ 78 ψ W ψ ψ gt, t 3, e λx 78 ψ + w λ + w λ + gt, t 3, e λx 783 g g λ B n L n + Bn c 779 L n + B c nψ 784 Bn c j,, L j j ju j j ṽ jl L l 786 lj ṽ jl t, t, 784 L n + v n L + v n L + ψ 787 v nj t, t, 779 L j ψ λ j ψ 787 λ n + v n λ + v n λ + ψ 788 log ψ λ n + v n λ + v n λ log ψ λ j t j + t + v j λ j, t : 79 j j log ψ λ 79 v j t, t, 79 ψ exp v j λ j expt + λt + λ t + 79 j exp j v jλ j /λ t t ψ + w λ + w λ + expt +λt +λ t + 79
9 7 779 w j KP KdV Boussinesq Lax l L l L l B l 793 L l B l,, B l ψ λ l ψ,, t l λ l ψ,, w j t l j,,, 797 w j t l, t l, t 3l, l Schrödinger + u ψ λ ψ L B L u 3 u x u 4 4 u xx u u 5 8 u xxx + 3 u u x 799 u u L / L 779 n u + 3 t 3 u x ψ 7 n 5 t 5 B 5 ψ 7 KdV 3-3 B 3 ψ 3 + 3u + 3u 3 + 3u x ψ λ 3 ψ 7 ψ t B ψ + u ψ 73 Boussinesq u t 4 u 3 x 4 u x 74 7 τ 5 τx; t h h m h h m h m h m m 75 τ w j τ τ Hx; t m τt det p p p 3 p p ξ ξ ξ m ξ ξ ξ m ξ ξ ξ m detξ t exp ηt, ΛΞ 76 Ξ t m Ξ t 77
10 7 t x x 76 p n p n t ν +ν +3ν 3 + n ν,ν,ν 3, t ν tν tν 3 3 ν!ν!ν 3! 78 Binet-Cauchy τt l <l < <l m ξ l ξ l ξ l ξ l ξ l m ξ l m p l p l p lm p l p l p lm p l m+ p l m+ p lm m+ ξ m l ξ m l 79 ξ m l m m n < p n l, l,, l m i m ii l m +, l m +,, l m m +, 3, 5, 6, 3, 5, 6 m m 3 3 m , 3, 5, 6, 3, 5, 6 4 φ m 79 p j Schur p l p l p lm p l p l p lm S Y t, 7 p l m+ p l m+ p lm m+ ξ Y ξ l ξ l ξ l ξ l ξ l m ξ l m ξ m l ξ m l ξ m l m 7 Y l, l,, l m 79 τt rowy m S Y tξ Y 7
11 73 m m τtdet p p p p p p p ξ ξ p ξ ξ + p p p p p 3 ξ ξ + p ξ 3 ξ + 3 ξ ξ ξ ξ ξ ξ ξ ξ S φ ξ φ +S ξ +S ξ + 73 ft S Y t ft Y S Y tc Y 74 c Y c Y S Y t ft t 75 S Y t S Y t t Ohta 5 t 7 τt S Y t 74 7 τ ξ Y Plücer m τ 73 4, l < l < ξ ξ l ξ l ξ ξ ξ l ξ l ξ ξ l ξ l ξ ξ l ξ l ξ ξ ξ ξ l ξ l ξ ξ l ξ l ξ 76 3 ξ ξ l ξ ξ ξ l ξ l ξ l ξ ξ ξ l ξ ξ ξ l ξ l ξ l ξ ξ ξ l + 3 ξ ξ ξ l ξ l ξ l ξ 77 l ξ Y, l, l,,,, 3 77 ξ φ ξ ξ ξ + ξ ξ 78, l, l,,,, 4 ξ φ ξ ξ ξ + ξ ξ, 79, l, l,,, 3, 4 ξ φ ξ ξ ξ + ξ ξ, 7, l, l,,, 3, 4 ξ ξ ξ ξ + ξ ξ, 7, l, l,,, 3, 4 ξ ξ ξ ξ + ξ ξ, 7 m < < < m l < l < < l m+ ξ ξ ξ m ξ ξ m l ξ ξ m ξ m m ξ ξ m l ξ m l l ξ ξ m l l ξ l m+ ξ m l m+ l ξ l m+ ξ m l m+ ξ m l 73 m+ δ ξ Y ξ Y 74 i δ m + i j 75 Y,, j, l i, j+,, m Y l,, l i, l i+,, l m+ j < l i < j+ ν l ν 74 ξ Y Plücer ft ft Y S Y ξ Y ξ Y Plücer ft τ 4 ft 76 7 ξ Y τ Plücer τt 76 τt + s detξ t e ηt,λ e ηs,λ Ξ 76
12 74 Ξs : e ηs,λ Ξ 77 w j τ τt + s Y S Y tξ Y s 77 w j τ p j t τ 733 ξ Y s 7 s s, s, s 3, Plücer 77 S Y t S Y Ohta 5 S Y t S Y t t δ Y Y ξ Y s S Y t τt + s t S Y s τt + s t S Y s τs δ {S Y t τt}{s Y t τt} 79 δ τ 78 S φ t τts t τt S t τts t τt + S t τts t τt 73 S φ, S t, S t / + t, S t / t, S t 3 t 3, S t 4 / t t 3 + t, 3,,,,, 3, m l, l 3 p p 3 S p p p p p 3 t 4 / t t 3 + t τ 4 t t t 3 t τ 3 τ t 3 t 3 τ t 3 { + + }{ τ t } τ 73 t t t 6 4D t3 D t +3D t +D 4 t τt τt 73 t x KP τ KP 5 p j t p j t t t w τ τ x w τ τ x τ t w 3 3 τ 6τ x 3 3 τ + τ x t t Lax 739 τ u x log τ 3 u 3 x 3 + log τ x t u x x + log τ t x t 3 x log τ KP τ { τ + p t τ λ ψ τ { τ ν ν! t λ e t +λt +λ t + { exp τ + p t τ λ + ν ν! ν λ t λ t τ t λ, t λ, ψ τt, t, } e t +λt +λ t + t ν } λ τ } τ e t +λt +λ t + e t+λt+λ t τ
13 75 ft 74 Schur ft KP τ Plücer 4 KP Grassmann 6 Ξt e ηt,λ Ξ 738 KP Grassmann e ηt,λ KP {expηt, Λ} Grassmann Zaharov, V E & Shabat, A B: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problemi, Funct Anal Appl Kadomtsev, B B & Petviashvili, V I: On the stability of solitary waves in wealy dispersing media, Sov Phys Dolady : I : : 96 4 Sato, M & Sato, Y: Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, Nonlinear Partial Differential Equations in Applied Science: proceedings of the US-Japan seminar, ed Fujita, H, Lax, P D & Strang, G Kinouniya/North- Holland, Toyo, :, : 99 Sato, M: Soliton equation as dynamical systems on a infinite dimensional Grassmann manifolds, RIMS Koyurou Kyoto University : No8 984, 4 :, No Ohta, Y, Satsuma, J, Taahashi, D & Toihiro, T: An elementary introduction to Sato theory, Progr Theor Phys Suppl No : No : KP UC No ,, :, 993, 7 9 Lax, P D: Integrals of nonlinear equations of evolution and solitary waves, Comm Pure Appl Math
t, x (4) 3 u(t, x) + 6u(t, x) u(t, x) + u(t, x) = 0 t x x3 ( u x = u x (4) u t + 6uu x + u xxx = 0 ) ( ): ( ) (2) Riccati ( ) ( ) ( ) 2 (1) : f
: ( ) 2008 5 31 1 f(t) t (1) d 2 f(t) + f(t) = 0 dt2 f(t) = sin t f(t) = cos t (1) 1 (2) d dt f(t) + f(t)2 = 0 (1) (2) t (c ) (3) 2 2 u(t, x) c2 u(t, x) = 0 t2 x2 1 (1) (1) 1 t, x (4) 3 u(t, x) + 6u(t,
More information2 1 1 (1) 1 (2) (3) Lax : (4) Bäcklund : (5) (6) 1.1 d 2 q n dt 2 = e q n 1 q n e q n q n+1 (1.1) 1 m q n n ( ) r n = q n q n 1 r ϕ(r) ϕ (r)
( ( (3 Lax : (4 Bäcklud : (5 (6 d q = e q q e q q + ( m q ( r = q q r ϕ(r ϕ (r 0 5 0 q q q + 5 3 4 5 m d q = ϕ (r + ϕ (r + ( Hooke ϕ(r = κr (κ > 0 ( d q = κ(q q + κ(q + q = κ(q + + q q (3 ϕ(r = a b e br
More information²ÄÀÑʬΥ»¶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
More information九州大学学術情報リポジトリ Kyushu University Institutional Repository ソリトンの二次元相互作用について 及川, 正行九州大学応用力学研究所 辻, 英一九州大学応用力学研究所 Oikawa, Masayuki Research Institute for A
九州大学学術情報リポジトリ Kyushu University Institutional Repository ソリトンの二次元相互作用について 及川, 正行九州大学応用力学研究所 辻, 英一九州大学応用力学研究所 Oikawa, Masayuki Research Institute for Applied Mechanics, Kyushu University Tsuji, Hidekazu
More information201711grade1ouyou.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 informationtakei.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
More informationRelaxation scheme of Besse t t n = n t, u n = u(t n ) (n = 0, 1,,...)., t u(t) = F (u(t)) (1). (1), u n+1 u n t = F (u n ) u n+1 = u n + tf (u n )., t
RIMS 011 5 3 7 relaxation sheme of Besse splitting method Scilab Scilab http://www.scilab.org/ Google Scilab Scilab Mathieu Colin Mathieu Colin 1 Relaxation scheme of Besse t t n = n t, u n = u(t n ) (n
More informationII 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 information4. 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 informationBlack-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
More informationu = 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 information2011de.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 information9. 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)
More informationhttp://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................
More informationFeynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull
Feynman Encounter with Mathematics 52, 200 9 [] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull. Sci. Math. vol. 28 (2004) 97 25. [2] D. Fujiwara and
More information1 8, : 8.1 1, 2 z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = n i=1 a ii x 2 i + i<j 2a ij x i x j = ( x, A x), f =
1 8, : 8.1 1, z = ax + by + c ax by + z c = a b +1 x y z c = 0, (0, 0, c), n = ( a, b, 1). f = a ii x i + i
More information: , 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 informationShunsuke Kobayashi 1 [6] [11] [7] u t = D 2 u 1 x 2 + f(u, v) + s L u(t, x)dx, L x (0.L), t > 0, Neumann 0 v t = D 2 v 2 + g(u, v), x (0, L), t > 0. x
Shunsuke Kobayashi [6] [] [7] u t = D 2 u x 2 + fu, v + s L ut, xdx, L x 0.L, t > 0, Neumann 0 v t = D 2 v 2 + gu, v, x 0, L, t > 0. x2 u u v t, 0 = t, L = 0, x x. v t, 0 = t, L = 0.2 x x ut, x R vt, x
More informationKorteweg-de Vries
Korteweg-de Vries 2011 03 29 ,.,.,.,, Korteweg-de Vries,. 1 1 3 1.1 K-dV........................ 3 1.2.............................. 4 2 K-dV 5 2.1............................. 5 2.2..............................
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. 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( ) ) ) ) 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 informationQMI13a.dvi
I (2013 (MAEDA, Atsutaka) 25 10 15 [ I I [] ( ) 0. (a) (b) Plank Compton de Broglie Bohr 1. (a) Einstein- de Broglie (b) (c) 1 (d) 2. Schrödinger (a) Schrödinger (b) Schrödinger (c) (d) 3. (a) (b) (c)
More informationuntitled
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 informationAnderson ( ) Anderson / 14
Anderson 2008 12 ( ) Anderson 2008 12 1 / 14 Anderson ( ) Anderson 2008 12 2 / 14 Anderson P.W.Anderson 1958 ( ) Anderson 2008 12 3 / 14 Anderson tight binding Anderson tight binding Z d u (x) = V i u
More informationW u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)
3 215 4 27 1 1 u u(x, t) u tt a 2 u xx, a > (1) D : {(x, t) : x, t } u (, t), u (, t), t (2) u(x, ) f(x), u(x, ) t 2, x (3) u(x, t) X(x)T (t) u (1) 1 T (t) a 2 T (t) X (x) X(x) α (2) T (t) αa 2 T (t) (4)
More information7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
More information2 (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,
More informationI ( ) 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 informationDecember 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..........................
More informationp = mv p x > h/4π λ = h p m v Ψ 2 Ψ
II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π
More informationMathematical Logic I 12 Contents I Zorn
Mathematical Logic I 12 Contents I 2 1 3 1.1............................. 3 1.2.......................... 5 1.3 Zorn.................. 5 2 6 2.1.............................. 6 2.2..............................
More information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 informationI 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 information211 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 informationChebyshev 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
More information25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
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
[ ] 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 information1 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)
More information1. 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
More information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
More informationSAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
More informationC p (.2 C p [[T ]] Bernoull B n,χ C p p q p 2 q = p p = 2 q = 4 ω Techmüller a Z p ω(a a ( mod q φ(q ω(a Z p a pz p ω(a = 0 Z p φ Euler Techmüller ω Q
p- L- [Iwa] [Iwa2] -Leopoldt [KL] p- L-. Kummer Remann ζ(s Bernoull B n (. ζ( n = B n n, ( n Z p a = Kummer [Kum] ( Kummer p m n 0 ( mod p m n a m n ( mod (p p a ( p m B m m ( pn B n n ( mod pa Z p Kummer
More informationn 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 informationII 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 informationx () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
More informationIntroduction 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........................................
More information2 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,,,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 informationQMI_09.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 3.1.2 σ τ 2 2 ux, t) = ux, t) 3.1) 2 x2 ux, t) σ τ 2 u/ 2 m p E E = p2 3.2) E ν ω E = hν = hω. 3.3) k p k = p h. 3.4) 26 3 hω = E = p2 = h2 k 2 ψkx ωt) ψ 3.5) h
More informationQMI_10.dvi
25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 σ τ x u u x t ux, t) u 3.1 t x P ux, t) Q θ P Q Δx x + Δx Q P ux + Δx, t) Q θ P u+δu x u x σ τ P x) Q x+δx) P Q x 3.1: θ P θ Q P Q equation of motion P τ Q τ σδx
More information20 6 4 1 4 1.1 1.................................... 4 1.1.1.................................... 4 1.1.2 1................................ 5 1.2................................... 7 1.2.1....................................
More informationZ: 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 informationV(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)
More informationsakigake1.dvi
(Zin ARAI) arai@cris.hokudai.ac.jp http://www.cris.hokudai.ac.jp/arai/ 1 dynamical systems ( mechanics ) dynamical systems 3 G X Ψ:G X X, (g, x) Ψ(g, x) =:Ψ g (x) Ψ id (x) =x, Ψ gh (x) =Ψ h (Ψ g (x)) (
More information( ) ( 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 informationτ 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 informationnewmain.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 information2S III IV K A4 12:00-13:30 Cafe David 1 2 TA 1 appointment Cafe David K2-2S04-00 : C
2S III IV K200 : April 16, 2004 Version : 1.1 TA M2 TA 1 10 2 n 1 ɛ-δ 5 15 20 20 45 K2-2S04-00 : C 2S III IV K200 60 60 74 75 89 90 1 email 3 4 30 A4 12:00-13:30 Cafe David 1 2 TA 1 email appointment Cafe
More informationx, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y)
x, y x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 1 1977 x 3 y xy 3 x 2 y + xy 2 x 3 + y 3 = 15 xy (x y) (x + y) xy (x y) (x y) ( x 2 + xy + y 2) = 15 (x y) ( x 2 y + xy 2 x 2 2xy y 2) = 15 (x y) (x + y) (xy
More informationxia2.dvi
Journal of Differential Equations 96 (992), 70-84 Melnikov method and transversal homoclinic points in the restricted three-body problem Zhihong Xia Department of Mathematics, Harvard University Cambridge,
More information2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
More informationA11 (1993,1994) 29 A12 (1994) 29 A13 Trefethen and Bau Numerical Linear Algebra (1997) 29 A14 (1999) 30 A15 (2003) 30 A16 (2004) 30 A17 (2007) 30 A18
2013 8 29y, 2016 10 29 1 2 2 Jordan 3 21 3 3 Jordan (1) 3 31 Jordan 4 32 Jordan 4 33 Jordan 6 34 Jordan 8 35 9 4 Jordan (2) 10 41 x 11 42 x 12 43 16 44 19 441 19 442 20 443 25 45 25 5 Jordan 26 A 26 A1
More information漸化式のすべてのパターンを解説しましたー高校数学の達人・河見賢司のサイト
https://www.hmg-gen.com/tuusin.html https://www.hmg-gen.com/tuusin1.html 1 2 OK 3 4 {a n } (1) a 1 = 1, a n+1 a n = 2 (2) a 1 = 3, a n+1 a n = 2n a n a n+1 a n = ( ) a n+1 a n = ( ) a n+1 a n {a n } 1,
More information1 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 information1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji
8 4 2018 6 2018 6 7 1 (Contents) 1. 2 2. (1) 22 3. 31 1. Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji SETO 22 3. Editorial Comments Tadashi
More informationKENZOU Karman) x
KENZO 8 8 31 8 1 3 4 5 6 Karman) 7 3 8 x 8 1 1.1.............................. 3 1............................................. 5 1.3................................... 5 1.4 /.........................
More information13 0 1 1 4 11 4 12 5 13 6 2 10 21 10 22 14 3 20 31 20 32 25 33 28 4 31 41 32 42 34 43 38 5 41 51 41 52 43 53 54 6 57 61 57 62 60 70 0 Gauss a, b, c x, y f(x, y) = ax 2 + bxy + cy 2 = x y a b/2 b/2 c x
More informationGelfand 3 L 2 () ix M : ϕ(x) ixϕ(x) M : σ(m) = i (λ M) λ (L 2 () ) ( 0 ) L 2 () ϕ, ψ L 2 () ((λ M) ϕ, ψ) ((λ M) ϕ, ψ) = λ ix ϕ(x)ψ(x)dx. λ /(λ ix) ϕ,
A spectral theory of linear operators on Gelfand triplets MI (Institute of Mathematics for Industry, Kyushu University) (Hayato CHIBA) chiba@imi.kyushu-u.ac.jp Dec 2, 20 du dt = Tu. (.) u X T X X T 0 X
More informationPart () () 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 information24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x
24 I 1.1.. ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x 1 (t), x 2 (t),, x n (t)) ( ) ( ), γ : (i) x 1 (t),
More informationTitle 最適年金の理論 Author(s) 藤井, 隆雄 ; 林, 史明 ; 入谷, 純 ; 小黒, 一正 Citation Issue Date Type Technical Report Text Version publisher URL
Title 最適年金の理論 Author(s) 藤井, 隆雄 ; 林, 史明 ; 入谷, 純 ; 小黒, 一正 Citation Issue 2012-06 Date Type Technical Report Text Version publisher URL http://hdl.handle.net/10086/23085 Right Hitotsubashi University Repository
More informationMacdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona
Macdonald, 2015.9.1 9.2.,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdonald,, q., Heckman Opdam q,, Macdonald., 1 ,,. Macdonald,
More information1 Abstract 2 3 n a ax 2 + bx + c = 0 (a 0) (1) ( x + b ) 2 = b2 4ac 2a 4a 2 D = b 2 4ac > 0 (1) 2 D = 0 D < 0 x + b 2a = ± b2 4ac 2a b ± b 2
1 Abstract n 1 1.1 a ax + bx + c = 0 (a 0) (1) ( x + b ) = b 4ac a 4a D = b 4ac > 0 (1) D = 0 D < 0 x + b a = ± b 4ac a b ± b 4ac a b a b ± 4ac b i a D (1) ax + bx + c D 0 () () (015 8 1 ) 1. D = b 4ac
More informationNetR36_CD01-CD24_190512A.indd
36 Vol. 63 0 553 29 0 7 51 5129 5130 51 51 5133 5134 5158 9 515898 555 515911 5 515912 30 515915 5 515916 5 60-600017 24 600018 24 600019 24 600020 24 600070 24 6000 24 600072 24 6000 24 600129 24 600181
More information30
3 ............................................2 2...........................................2....................................2.2...................................2.3..............................
More informationad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign(
I n n A AX = I, YA = I () n XY A () X = IX = (YA)X = Y(AX) = YI = Y X Y () XY A A AB AB BA (AB)(B A ) = A(BB )A = AA = I (BA)(A B ) = B(AA )B = BB = I (AB) = B A (BA) = A B A B A = B = 5 5 A B AB BA A
More informationSO(2)
TOP URL http://amonphys.web.fc2.com/ 1 12 3 12.1.................................. 3 12.2.......................... 4 12.3............................. 5 12.4 SO(2).................................. 6
More informationgr09.dvi
.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
More informationver Web
ver201723 Web 1 4 11 4 12 5 13 7 2 9 21 9 22 10 23 10 24 11 3 13 31 n 13 32 15 33 21 34 25 35 (1) 27 4 30 41 30 42 32 43 36 44 (2) 38 45 45 46 45 5 46 51 46 52 48 53 49 54 51 55 54 56 58 57 (3) 61 2 3
More informationI
I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............
More informationX 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 information1 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 informationNote.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 information43433 8 3 . Stochastic exponentials...................................... 3. Girsanov s theorem......................................... 4 On the martingale property of stochastic exponentials 5. Gronwall
More informationJanuary 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
January 16, 2017 1 1. 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)
More informationx (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( ) Loewner SLE 13 February
( ) Loewner SLE 3 February 00 G. F. Lawler, Conformally Invariant Processes in the Plane, (American Mathematical Society, 005)., Summer School 009 (009 8 7-9 ) . d- (BES d ) d B t = (Bt, B t,, Bd t ) (d
More information8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) Carathéodory 10.3 Fubini 1 Introduction 1 (1) (2) {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a
% 100% 1 Introduction 2 (100%) 2.1 2.2 2.3 3 (100%) 3.1 3.2 σ- 4 (100%) 4.1 4.2 5 (100%) 5.1 5.2 5.3 6 (100%) 7 (40%) 8 Fubini (90%) 2007.11.5 1 8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory
More information( ) ( )
20 21 2 8 1 2 2 3 21 3 22 3 23 4 24 5 25 5 26 6 27 8 28 ( ) 9 3 10 31 10 32 ( ) 12 4 13 41 0 13 42 14 43 0 15 44 17 5 18 6 18 1 1 2 2 1 2 1 0 2 0 3 0 4 0 2 2 21 t (x(t) y(t)) 2 x(t) y(t) γ(t) (x(t) y(t))
More informationφ 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
More information2 0 B B B B - B B - B - - B (1.0.6) 0 1 p /p p {0} (1.0.7) B m n ϕ : B ϕ(m) n ϕ 1 (n) = m /m B/n 1.1. (1.1.1) a a n > 0 x n a x r(a) a r(r(a)) = r(a)
1 0 1. 1.0. (1.0.1) - (1.0.2), B ϕ : B resp. B- M a m = ϕ(a) m (resp. m a = m ϕ(a)) resp. - M - B- resp. - M [ϕ] L - u : L M [ϕ] a x L u(a x) = ϕ(a) u(x) ϕ- L M (ϕ, u) u (, L) (B, M) - L (, L) (1.0.3)
More information1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)? (2) () f(x)? b lim a f n (x)dx = b
1 Introduction 2 2.1 2.2 2.3 3 3.1 3.2 σ- 4 4.1 4.2 5 5.1 5.2 5.3 6 7 8. Fubini,,. 1 1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)?
More informationé éτ Γ ζ ä
é éτ Γ ζ ä é é éτ Γ ζ ä \\ é \ No.16ME-S1 Reports of RIAM Symposium No.16ME-S1 Physics and Mathematical Structures of Nonlinear Waves Proceedings of a symposium held at Chikushi Campus, Kyushu Universiy,
More informationS 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....
More informationall.dvi
5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0
More informationTrapezoidal 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 )
More informationphs.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