2 K = f (x) K[[x]] = r f (x) r D = D (0, r) a D f (x) a D Figure X d : X X R 0 d(x, z) max{d(x, y), d(y, z)} x, y, z X (X, d) clopen 1.1. (X,

Size: px
Start display at page:

Download "2 K = f (x) K[[x]] = r f (x) r D = D (0, r) a D f (x) a D Figure X d : X X R 0 d(x, z) max{d(x, y), d(y, z)} x, y, z X (X, d) clopen 1.1. (X,"

Transcription

1 affinoids analytic reduction Raynaud visualization Zariski-Riemann K. Hensel p A. Ostrowski Q 1930 W. Schöbe 1940 M. Krasner 1

2 2 K = f (x) K[[x]] = r f (x) r D = D (0, r) a D f (x) a D Figure X d : X X R 0 d(x, z) max{d(x, y), d(y, z)} x, y, z X (X, d) clopen 1.1. (X, d) X totally disconnected 2 X

3 3 Q R Q R Q Harvard J. Tate Tate Inventiones Grauert-Remmert 1966 Tate Weierstrass affinoid 1969 Gerritzen-Grauert affinoid R. Kiehl 1967 A B 1972 M. Raynaud 2.2. Tate. Tate Tate Tate curve 3 E : y 2 + xy x 3 + b 2 x + b 3 = 0 ى) = b 3 + b b 2b 3 432b b3 2 0), C E m Z C /Z E(C) an C w q m w x(w) = (1 q m w) 2 2 q m (1 q m ) 2, y(w) = m Z m 1 (q m w) 2 (1 q m w) 3 + q m (1 q m ) 2. q b 2, b 3 n 3 q n b 2 = b 2 (q) = 5 1 q n = 5q(1 + 9q + 28q2 + ), n 1 b 3 = b 3 (q) = n 1 (7n 5 + 5n 3 )q n 12(1 q n ) m 1 = q(1 + 23q + 154q 2 + ).

4 4 Tate C C p C p Q p p- Tate w ( x(w) = (1 w) 2 + q m w (1 q m w) 2 + q m w 1 (1 q m w 1 ) 2 2 q m ) (1 q m ) 2 y(w) = m 1 w 2 (1 w) 3 + m 1 ( q 2m w 2 (1 q m w) 3 q m w 1 (1 q m w 1 ) 3 + q m ) (1 q m ) 2 r 1 w r 2, w q m ε, w 1 q m ε (m Z) R(r 1, r 2, ε) Figure 2. R(r 1, r 2, ε) Tate x = x(w) y = y(w) C w = q m m Z Runge C D D C q Z π: (G m,k ) cl E cl cl Tate q Z C p C K = C p K X X an X cl X an = X

5 5 π underlying sets π π: (G m,k ) an E an Tate Krasner Tate Tate rigidify A 1 k = Spec k[t] ى = {z C z < 1} D 1 K = {z K z 1}. 3. Tate Hilbert 3.1. k k a (X a) Spec k[x] k Spec k[x] k[x]

6 6 K K K V = {a K a 1}, m V = {a K a < 1} K V 1 0 a m V a a- V 0 a m V a a- 1 K = V[ 1 a ] K V[X] a- Zariski V[X] Zar V[X] 1 + av[x] 3.2. D 1 K a (X a) Spm V[X]Zar V K = Spm V[X] Zar [ 1 a ], K Spm X an = X Runge V[X] a- Zariski a- V X = lim n 0 V[X]/a n V[X]. K X = V X [ 1 a ] K X = { n=0 a n X n an K, a n 0 }. K X Tate Tate K X 1 D 1 K K X V[X] Zar [ 1 a ] D 1 K = Spm V[X]Zar [ 1 a ] = Spm K X K X

7 D 1 K = (Spm K X,, O) O U O(U) = U Runge Tate Krasner O O Tate Tate D 1 K K X m V X V X m V X V X a m V V X /m V X k[x] m V X k[x] red: Spm K X m (m V X mod m V X ) Spec k[x] Tate K D 1 K P1 (K) z 1 P 1 (K) = P 1 (V) (x : y) (x : y) P 1 (k) x = (x mod m V ) D 1 K

8 8 1 1 D 1 K A1 k = Spec k[x] 8 red 0 Figure 3. D 1 K = Spm K X D 1 K Zariski K X V V X 8 8 red red 0 0 Figure 4. Grothendieck Grothendieck

9 Tate acyclicity. D = D 1 K K X Grothendieck τ D Zariski red Zariski red O D Spf V X admissible blow-up a Tate 3.3 (Tate acyclicity). O 4. D = D 1 K = (Spm K X, τ D, O D ) Tate Tate Grothendieck τ D Gerritzen-Grauert Raynaud 1972 Tate 4.1. Raynaud. K V V m V 0 a m V a K V 1 a V K = Frac(V) = V[ 1 a ] Zariski

10 10 Raynaud 4.1 (Raynaud 1972). rig X X rig { } { } coherent V-formal / coherent rigid schemes of finite type admissible spaces over K blow-ups coherent quasi-compact quasi-separated Raynaud Tate-Raynaud (C) Tate-Raynaud (H) R. Huber adic (Z) Zariski-Riemann (B) Berkovich (Z) (B) Grothendieck Berkovich Berkovich Top ( (C), (H), (Z) ) Top ( (B) ) Berkovich. Berkovich Tate-Raynaud affinoid Tate K X 1, ldots, X n A Spm A Berkovich A M (A) Spm A

11 11 P 1 K Berkovich P 1 tree Tate-Raynaud P 1 K Tate-Raynaud P 1 K Figure 5. Berkovich Berkovich Berkovich Grothendieck Berkovich Berkovich visualization Huber adic Zariski-Riemann

12 12 sober Zariski coherent X X := lim coherent X cofinal Zariski X Raynaud Zariski Raynaud s viewpoint of rigid geometry Zariski s viewpoint of birational geometry Geometry of models Rigid analytic geometry Geometry of formal schemes + Zariski-Riemann space Figure address: kato@math.kyoto-u.ac.jp

[Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2

[Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2 On the action of the Weil group on the l-adic cohomology of rigid spaces over local fields (Yoichi Mieda) Graduate School of Mathematical Sciences, The University of Tokyo 0 l Galois K F F q l q K, F K,

More information

2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) =

2018/10/04 IV/ IV 2/12. A, f, g A. (1) D(0 A ) =, D(1 A ) = Spec(A), D(f) D(g) = D(fg). (2) {f l A l Λ} A I D(I) = l Λ D(f l ). (3) I, J A D(I) D(J) = 2018/10/04 IV/ IV 1/12 2018 IV/ IV 10 04 * 1 : ( A 441 ) yanagida[at]math.nagoya-u.ac.jp https://www.math.nagoya-u.ac.jp/~yanagida 1 I: (ring)., A 0 A, 1 A. (ring homomorphism).. 1.1 A (ideal) I, ( ) I

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

( ) X x, y x y x y X x X x [x] ( ) x X y x y [x] = [y] ( ) x X y y x ( ˆX) X ˆX X x x z x X x ˆX [z x ] X ˆX X ˆX ( ˆX ) (0) X x, y d(x(1), y(1)), d(x

( ) X x, y x y x y X x X x [x] ( ) x X y x y [x] = [y] ( ) x X y y x ( ˆX) X ˆX X x x z x X x ˆX [z x ] X ˆX X ˆX ( ˆX ) (0) X x, y d(x(1), y(1)), d(x Z Z Ẑ 1 1.1 (X, d) X x 1, x 2,, x n, x x n x(n) ( ) X x x ε N N i, j i, j d(x(i), x(j)) < ε ( ) X x x n N N i i d(x(n), x(i)) < 1 n ( ) X x lim n x(n) X x X () X x, y lim n d(x(n), y(n)) = 0 x y x y 1

More information

x ( ) x dx = ax

x ( ) x dx = ax x ( ) x dx = ax 1 dx = a x log x = at + c x(t) = e at C (C = e c ) a > 0 t a < 0 t 0 (at + b ) h dx = lim x(t + h) x(t) h 0 h x(t + h) x(t) h x(t) t x(t + h) x(t) ax(t) h x(t + h) x(t) + ahx(t) 0, h, 2h,

More information

Gmech08.dvi

Gmech08.dvi 63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)

More information

koji07-02.dvi

koji07-02.dvi 007 I II III 1,, 3, 4, 5, 6, 7 5 4 1 ε-n 1 ε-n ε-n ε-n. {a } =1 a ε N N a a N= a a

More information

1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1

1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1 1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2

More information

1 2

1 2 1 2 4 3 5 6 8 7 9 10 12 11 0120-889-376 r 14 13 16 15 0120-0889-24 17 18 19 0120-8740-16 20 22 21 24 23 26 25 28 27 30 29 32 31 34 33 36 35 38 37 40 39 42 41 44 43 46 45 48 47 50 49 52 51 54 53 56 55 58

More information

3 5 6 7 7 8 9 5 7 9 4 5 6 6 7 8 8 8 9 9 3 3 3 3 8 46 4 49 57 43 65 6 7 7 948 97 974 98 99 993 996 998 999 999 4 749 7 77 44 77 55 3 36 5 5 4 48 7 a s d f g h a s d f g h a s d f g h a s d f g h j 83 83

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

01_教職員.indd

01_教職員.indd T. A. H. A. K. A. R. I. K. O. S. O. Y. O. M. K. Y. K. G. K. R. S. A. S. M. S. R. S. M. S. I. S. T. S. K.T. R. T. R. T. S. T. S. T. A. T. A. D. T. N. N. N. Y. N. S. N. S. H. R. H. W. H. T. H. K. M. K. M.

More information

°ÌÁê¿ô³ØII

°ÌÁê¿ô³ØII July 14, 2007 Brouwer f f(x) = x x f(z) = 0 2 f : S 2 R 2 f(x) = f( x) x S 2 3 3 2 - - - 1. X x X U(x) U(x) x U = {U(x) x X} X 1. U(x) A U(x) x 2. A U(x), A B B U(x) 3. A, B U(x) A B U(x) 4. A U(x),

More information

2010 IA ε-n I 1, 2, 3, 4, 5, 6, 7, 8, ε-n 1 ε-n ε-n? {a n } n=1 1 {a n } n=1 a a {a n } n=1 ε ε N N n a n a < ε

2010 IA ε-n I 1, 2, 3, 4, 5, 6, 7, 8, ε-n 1 ε-n ε-n? {a n } n=1 1 {a n } n=1 a a {a n } n=1 ε ε N N n a n a < ε 00 IA ε-n I,, 3, 4, 5, 6, 7, 8, 9 4 6 ε-n ε-n ε-n? {a } = {a } = a a {a } = ε ε N N a a < ε ε-n ε ε N a a < ε N ε ε N ε N N ε N [ > N = a a < ε] ε > 0 N N N ε N N ε N N ε a = lim a = 0 ε-n 3 ε N 0 < ε

More information

untitled

untitled 20 7 1 22 7 1 1 2 3 7 8 9 10 11 13 14 15 17 18 19 21 22 - 1 - - 2 - - 3 - - 4 - 50 200 50 200-5 - 50 200 50 200 50 200 - 6 - - 7 - () - 8 - (XY) - 9 - 112-10 - - 11 - - 12 - - 13 - - 14 - - 15 - - 16 -

More information

untitled

untitled 19 1 19 19 3 8 1 19 1 61 2 479 1965 64 1237 148 1272 58 183 X 1 X 2 12 2 15 A B 5 18 B 29 X 1 12 10 31 A 1 58 Y B 14 1 25 3 31 1 5 5 15 Y B 1 232 Y B 1 4235 14 11 8 5350 2409 X 1 15 10 10 B Y Y 2 X 1 X

More information

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) 2017 12 9 4 1 30 4 10 3 1 30 3 30 2 1 30 2 50 1 1 30 2 10 (1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10) (1) i 23 c 23 0 1 2 3 4 5 6 7 8 9 a b d e f g h i (2) 23 23 (3) 23 ( 23 ) 23 x 1 x 2 23 x

More information

, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,,

, 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p, p 3,..., p n p, p,..., p n N, 3,,,, 6,,3,4,, 3 4 8 6 6................................. 6.................................. , 3, 6 = 3, 3,,,, 3,, 9, 3, 9, 3, 3, 4, 43, 4, 3, 9, 6, 6,, 0 p, p, p 3,..., p n N = p p p 3 p n + N p n N p p p,

More information

( ) ( ) ( ) i (i = 1, 2,, n) x( ) log(a i x + 1) a i > 0 t i (> 0) T i x i z n z = log(a i x i + 1) i=1 i t i ( ) x i t i (i = 1, 2, n) T n x i T i=1 z = n log(a i x i + 1) i=1 x i t i (i = 1, 2,, n) n

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

平成 22 年度 ( 第 32 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 22 月年 58 日開催月 2 日 ) V := {(x,y) x n + y n 1 = 0}, W := {(x,y,z) x 3 yz = x 2 y z 2

平成 22 年度 ( 第 32 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 22 月年 58 日開催月 2 日 ) V := {(x,y) x n + y n 1 = 0}, W := {(x,y,z) x 3 yz = x 2 y z 2 3 90 2006 1. V := {(x,y) x n + y n 1 = 0}, W := {(x,y,z) x 3 yz = x 2 y z 2 = xz y 2 = 0} V (x,y) n = 1 n = 2 (x,y) V n = 1 n = 2 (3/5,4/5),(5/13,12/13)... n 3 V (0,±1),(±1,0) ( ) n 3 x n + y n = z n,

More information

Noether [M2] l ([Sa]) ) ) ) ) ) ( 1, 2) ) ( 3) K F = F q O K K l q K Spa(K, O K ) adc adc [Hu1], [Hu2], [Hu3] K A Spa(A, A ) Sp A A B X A X B = X Spec

Noether [M2] l ([Sa]) ) ) ) ) ) ( 1, 2) ) ( 3) K F = F q O K K l q K Spa(K, O K ) adc adc [Hu1], [Hu2], [Hu3] K A Spa(A, A ) Sp A A B X A X B = X Spec l Wel (Yoch Meda) Graduate School of Mathematcal Scences, The Unversty of Tokyo 0 Galos ([M1], [M2]) Galos Langlands ([Ca]) K F F q l q K, F K, F Fr q Gal(F /F ) F Frobenus q Fr q Fr q Gal(F /F ) φ: Gal(K/K)

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

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)

More information

,.,, L p L p loc,, 3., L p L p loc, Lp L p loc.,.,,.,.,.,, L p, 1 p, L p,. d 1, R d d. E R d. (E, M E, µ)., L p = L p (E). 1 p, E f(x), f(x) p d

,.,, L p L p loc,, 3., L p L p loc, Lp L p loc.,.,,.,.,.,, L p, 1 p, L p,. d 1, R d d. E R d. (E, M E, µ)., L p = L p (E). 1 p, E f(x), f(x) p d 1 L p L p loc, L p L p loc, Lp L p loc,., 1 p.,. L p L p., L 1, L 1., L p, L p. L 1., L 1 L 1. L p L p loc L p., L 2 L 2 loc,.,. L p L p loc L p., L p L p loc., L p L p loc 1 ,.,, L p L p loc,, 3., L p

More information

2

2 () () 980-8578 Tel: 022-795-6092 Fax: 022-795-6096 email: 1. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 17 46 47 4.1.1

More information

1 X X T T X (topology) T X (open set) (X, T ) (topological space) ( ) T1 T, X T T2 T T T3 T T ( ) ( ) T1 X T2 T3 1 X T = {, X} X (X, T ) indiscrete sp

1 X X T T X (topology) T X (open set) (X, T ) (topological space) ( ) T1 T, X T T2 T T T3 T T ( ) ( ) T1 X T2 T3 1 X T = {, X} X (X, T ) indiscrete sp 1 X X T T X (topology) T X (open set) (X, T ) (topological space) ( ) T1 T, X T T2 T T T3 T T ( ) ( ) T1 X T2 T3 1 X T = {, X} X (X, T ) indiscrete space T1 T2 =, X = X, X X = X T3 =, X =, X X = X 2 X

More information

1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe

1.2 (Kleppe, cf. [6]). C S 3 P 3 3 S 3. χ(p 3, I C (3)) 1 C, C P 3 ( ) 3 S 3( S 3 S 3 ). V 3 del Pezzo (cf. 2.1), S V, del Pezzo 1.1, V 3 del Pe 3 del Pezzo (Hirokazu Nasu) 1 [10]. 3 V C C, V Hilbert scheme Hilb V [C]. C V C S V S. C S S V, C V. Hilbert schemes Hilb V Hilb S [S] [C] ( χ(s, N S/V ) χ(c, N C/S )), Hilb V [C] (generically non-reduced)

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

(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0

(1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e ) e OE z 1 1 e E xy e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 (1) 3 A B E e AE = e AB OE = OA + e AB = (1 35 e 0 1 15 ) e OE z 1 1 e E xy 5 1 1 5 e = 0 e = 5 OE = ( 2 0 0) E ( 2 0 0) (2) 3 E P Q k EQ = k EP E y 0 Q y P y k 2 M N M( 1 0 0) N(1 0 0) 4 P Q M N C EP

More information

π, R { 2, 0, 3} , ( R),. R, [ 1, 1] = {x R 1 x 1} 1 0 1, [ 1, 1],, 1 0 1,, ( 1, 1) = {x R 1 < x < 1} [ 1, 1] 1 1, ( 1, 1), 1, 1, R A 1

π, R { 2, 0, 3} , ( R),. R, [ 1, 1] = {x R 1 x 1} 1 0 1, [ 1, 1],, 1 0 1,, ( 1, 1) = {x R 1 < x < 1} [ 1, 1] 1 1, ( 1, 1), 1, 1, R A 1 sup inf (ε-δ 4) 2018 1 9 ε-δ,,,, sup inf,,,,,, 1 1 2 3 3 4 4 6 5 7 6 10 6.1............................................. 11 6.2............................... 13 1 R R 5 4 3 2 1 0 1 2 3 4 5 π( R) 2 1 0

More information

29

29 9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n

More information

() 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)

() 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 ()

More information

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

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

More information

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx

4 4 4 a b c d a b A c d A a da ad bce O E O n A n O ad bc a d n A n O 5 {a n } S n a k n a n + k S n a a n+ S n n S n n log x x {xy } x, y x + y 7 fx 4 4 5 4 I II III A B C, 5 7 I II A B,, 8, 9 I II A B O A,, Bb, b, Cc, c, c b c b b c c c OA BC P BC OP BC P AP BC n f n x xn e x! e n! n f n x f n x f n x f k x k 4 e > f n x dx k k! fx sin x cos x tan

More information

16, dim V V U, V U V (c) ϕ P V, p, q ϕ = p/q, q(p ) 0, ϕ P (regular), P ϕ ϕ dom ϕ, ϕ ϕ P ϕ ϕ(p ) = p(p )/q(p ), k (= A 1 ) ϕ(p ) = 0 P ϕ (zero) 1.3, P

16, dim V V U, V U V (c) ϕ P V, p, q ϕ = p/q, q(p ) 0, ϕ P (regular), P ϕ ϕ dom ϕ, ϕ ϕ P ϕ ϕ(p ) = p(p )/q(p ), k (= A 1 ) ϕ(p ) = 0 P ϕ (zero) 1.3, P 15, pp.15-60 Riemann-Roch 1 1.1 (a) k A n = A n (k) = {(x 1,, x n ) x 1,, x n k} (affine space) A 1, A 2 k[x] = k[x 1,, X n ] n k- X = (X 1,, X n ) A n f(x) k[x], x A n, f x f(x) k[x] I, V = V (I) = {x

More information

A

A A05-132 2010 2 11 1 1 3 1.1.......................................... 3 1.2..................................... 3 1.3..................................... 3 2 4 2.1............................... 4 2.2

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

,. 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

(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

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { ( 5 5.1 [ ] ) d f(t) + a d f(t) + bf(t) : f(t) 1 dt dt ) u(x, t) c u(x, t) : u(x, t) t x : ( ) ) 1 : y + ay, : y + ay + by : ( ) 1 ) : y + ay, : yy + ay 3 ( ): ( ) ) : y + ay, : y + ay b [],,, [ ] au xx

More information

Microsoft Word - 触ってみよう、Maximaに2.doc

Microsoft Word - 触ってみよう、Maximaに2.doc i i e! ( x +1) 2 3 ( 2x + 3)! ( x + 1) 3 ( a + b) 5 2 2 2 2! 3! 5! 7 2 x! 3x! 1 = 0 ",! " >!!! # 2x + 4y = 30 "! x + y = 12 sin x lim x!0 x x n! # $ & 1 lim 1 + ('% " n 1 1 lim lim x!+0 x x"!0 x log x

More information

2010 II / y = e x y = log x = log e x 2. ( e x ) = e x 3. ( ) log x = 1 x 1.2 Warming Up 1 u = log a M a u = M a 0

2010 II / y = e x y = log x = log e x 2. ( e x ) = e x 3. ( ) log x = 1 x 1.2 Warming Up 1 u = log a M a u = M a 0 2010 II 6 10.11.15/ 10.11.11 1 1 5.6 1.1 1. y = e x y = log x = log e x 2. e x ) = e x 3. ) log x = 1 x 1.2 Warming Up 1 u = log a M a u = M a 0 log a 1 a 1 log a a a r+s log a M + log a N 1 0 a 1 a r

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

Ⅱ 防災計画の概要

Ⅱ 防災計画の概要 12 ( 10 304 3 16 14 25 3 ) ( ) 1 3 1 18 12 2 ( ) ( ) 18 12 ( ) ( ( ) ( ) ( ) ( ) 1 ( ) 3 ( ) ( ) ( ) 4 5 1 2 3 ( ) 4 1 1 6 7 3 3 2 3 3 ( ) 1 AM PM 8 9 8 30 8 50 8 50 9 00 9 00 9 50 10 00 11 00 11 00 12

More information

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B 9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A

More information

Z: Q: R: C:

Z: Q: R: C: 0 Z: Q: R: C: 3 4 4 4................................ 4 4.................................. 7 5 3 5...................... 3 5......................... 40 5.3 snz) z)........................... 4 6 46 x

More information

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

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,

More information

日本数学会・2011年度年会(早稲田大学)・総合講演

日本数学会・2011年度年会(早稲田大学)・総合講演 日本数学会 2011 年度年会 ( 早稲田大学 ) 総合講演 2011 年度日本数学会春季賞受賞記念講演 MSJMEETING-2011-0 ( ) 1. p>0 p C ( ) p p 0 smooth l (l p ) p p André, Christol, Mebkhout, Kedlaya K 0 O K K k O K k p>0 K K : K R 0 p = p 1 Γ := K k

More information

i

i i 3 4 4 7 5 6 3 ( ).. () 3 () (3) (4) /. 3. 4/3 7. /e 8. a > a, a = /, > a >. () a >, a =, > a > () a > b, a = b, a < b. c c n a n + b n + c n 3c n..... () /3 () + (3) / (4) /4 (5) m > n, a b >, m > n,

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

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n ( 3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc

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

2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( (

2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( ( (. x y y x f y = f(x y x y = y(x y x y dx = d dx y(x = y (x = f (x y = y(x x ( (differential equation ( + y 2 dx + xy = 0 dx = xy + y 2 2 2 x y 2 F (x, y = xy + y 2 y = y(x x x xy(x = F (x, y(x + y(x 2

More information

SC210301 Ł\†EŒÚ M-KL.ec6

SC210301 Ł\†EŒÚ M-KL.ec6 30 36 01 02 07 08 05 95 11 94 11 97 13 91 13 9T 14 15 15 96 16 BE 16 BF 16 BG 17 CL 17 00 17 17 17 1 180 28 28 180 2 180 181 60 180 180 90 32 180 30 15 29 29 30 14 3 15 30 29 29 14 30 14 19 19 30 30 22

More information

untitled

untitled F(r)=QE(r) Q ρ( r ') 3 E= ke 3 ( r r ') d r ' V r r' () Er () = Fr Q E E x y Ez ρ = x y z ε E E z y Ex E E z y Ex =, =, = y z z x x y A A x y Az diva = x y z A A z y A A x A z y Ax rot A =,, y z z x x

More information

73

73 73 74 ( u w + bw) d = Ɣ t tw dɣ u = N u + N u + N 3 u 3 + N 4 u 4 + [K ] {u = {F 75 u δu L σ (L) σ dx σ + dσ x δu b δu + d(δu) ALW W = L b δu dv + Aσ (L)δu(L) δu = (= ) W = A L b δu dx + Aσ (L)δu(L) Aσ

More information

?

? 240-8501 79-2 Email: nakamoto@ynu.ac.jp 1 3 1.1...................................... 3 1.2?................................. 6 1.3..................................... 8 1.4.......................................

More information

i

i 009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3

More information

(, Goo Ishikawa, Go-o Ishikawa) ( ) 1

(, Goo Ishikawa, Go-o Ishikawa) ( ) 1 (, Goo Ishikawa, Go-o Ishikawa) ( ) 1 ( ) ( ) ( ) G7( ) ( ) ( ) () ( ) BD = 1 DC CE EA AF FB 0 0 BD DC CE EA AF FB =1 ( ) 2 (geometry) ( ) ( ) 3 (?) (Topology) ( ) DNA ( ) 4 ( ) ( ) 5 ( ) H. 1 : 1+ 5 2

More information

i 1 1 1.1.......................................... 1 1.1.1......................................... 1 1.1.2...................................... 1 1.1.3....................................... 2 1.1.4......................................

More information

Chap11.dvi

Chap11.dvi . () x 3 + dx () (x )(x ) dx + sin x sin x( + cos x) dx () x 3 3 x + + 3 x + 3 x x + x 3 + dx 3 x + dx 6 x x x + dx + 3 log x + 6 log x x + + 3 rctn ( ) dx x + 3 4 ( x 3 ) + C x () t x t tn x dx x. t x

More information

k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+

k + (1/2) S k+(1/2) (Γ 0 (N)) N p Hecke T k+(1/2) (p 2 ) S k+1/2 (Γ 0 (N)) M > 0 2k, M S 2k (Γ 0 (M)) Hecke T 2k (p) (p M) 1.1 ( ). k 2 M N M N f S k+ 1 SL 2 (R) γ(z) = az + b cz + d ( ) a b z h, γ = SL c d 2 (R) h 4 N Γ 0 (N) {( ) } a b Γ 0 (N) = SL c d 2 (Z) c 0 mod N θ(z) θ(z) = q n2 q = e 2π 1z, z h n Z Γ 0 (4) j(γ, z) ( ) a b θ(γ(z)) = j(γ, z)θ(z)

More information

²ÄÀÑʬΥ»¶ÈóÀþ·¿¥·¥å¥ì¡¼¥Ç¥£¥ó¥¬¡¼ÊýÄø¼°¤ÎÁ²¶á²òÀÏ Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation

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

More information

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C

18 ( ) I II III A B C(100 ) 1, 2, 3, 5 I II A B (100 ) 1, 2, 3 I II A B (80 ) 6 8 I II III A B C(80 ) 1 n (1 + x) n (1) n C 1 + n C 8 ( ) 8 5 4 I II III A B C( ),,, 5 I II A B ( ),, I II A B (8 ) 6 8 I II III A B C(8 ) n ( + x) n () n C + n C + + n C n = 7 n () 7 9 C : y = x x A(, 6) () A C () C P AP Q () () () 4 A(,, ) B(,, ) C(,,

More information

Gmech08.dvi

Gmech08.dvi 145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2

More information