On a branched Zp-cover of Q-homology 3-spheres
|
|
- たみえ はしかわ
- 7 years ago
- Views:
Transcription
1 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 On a branched Zp -cover of Q-homology 3-spheres 植木 潤 九州大学大学院数理学府 D2 December 23, 2014 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres
2 Z p Z p Arithmetic topology p Z p := lim n Z/p n Z p Z p = { n a n p n a n Z} Z Z p Z Z p R[T ] R[[T ]] n mod p n : Z p Z/p n Z
3 Z p Arithmetic topology K S 3 X = S 3 \ K τ : π 1 (X ) Z Z-cover X X n Z Z/nZ τ τ n Ker τ n < π 1 (X ) n X n X K (Fox ) K n M n M Γ := Gal(X X ) = Deck(X X ) = t, Λ := Z[Γ] Λ H 1 (X ) H 1 (X ) Λ = K,1 (t) H 1 (M n ) K,1 (t)
4 Arithmetic topology Z p Arithmetic topology F /k QHS 3 N M primes S = {p 1,..., p s } L = K 1... K s π 1 (Spec O k \ S) π 1 (M \ L) (Artin ) Hurwicz 1 = Cl(k) = Gal(kab ur 1(M) ab = H 1 (M) = Gal(Mab unbr M) #Cl(k) < #H 1 (M) < ( M QHS 3 ) iff Alexander-Fox Z p k /k Z X X Alexander -Mazur π ab
5 Z p Arithmetic topology 1 Z p Arithmetic topology 2 Z p Z p Λ Γ X 3 4 GL 1 Z p GL 1 5 Appendix: 2014 April
6 Z p Z p Z p Λ Γ X k k /k Z p def Gal(k /k) = Z p. k /k Z p (in C) p n k n /k k = k 1 k 2... k n... k = n k n. Q 1 p Q(ζ p ) Z p p Z p Z p n 0 Z p k /k n0
7 Z p Z p Z p Λ Γ X Cl(k) p-part Cl(k) (p) Theorem ( ) k /k Z p p n k n λ, µ, ν n 0 n > n 0 #Cl(k n ) (p) = p λn+µpn +ν. λ, µ, ν [Iwasawa 1959] On Γ-extensions of algebraic number fields
8 Z p Z p Λ Γ X Z p Z p -cover p n A n Definition L M QHS 3 p n {h n : M n M} L M QHS 3 Z p -cover n h n h n+1 subcover h n,n+1 : M n+1 M n h n+1 = h n h n,n+1 Remark. X = M \ L {Z p -cover br. over L} 1:1 { τ : π 1 (X ) Z p } up to isom.
9 Z p Z p Z p Λ Γ X L M QHS 3 X := M \ L τ : π 1 (X ) Z p τ : π 1 (X ) Z p 1:1 Z p -cover br. over L (up to isom) τ : π 1 (X ) Z τ mod p n : π 1 (X ) Z/p n Z τ : π 1 (X ) Z p L Z p -cover Z-cover L = K 1 K 2 S 3 µ i H 1 (X ) K i p 1 mod 4 1 Z p τ : µ 1 1, µ 2 1 Z-cover Z p -cover
10 Z p Z p Z p Λ Γ X Theorem ( U. ) L QHS 3 Z p -cover {h n : M n M} λ, µ, ν n 0 n > n 0 #H 1 (M n ) (p) = p λn+µpn +ν. λ, µ, ν [Hillman, Matei, Morishita 2006] for K S 3 [Kadokami, Mizusawa 2008] for L M, Z-cover
11 Λ 1 Z p Z p Λ Γ X Z p Γ = t Λ := Z p [[ t ]] = Z p [[T ]]; t 1 + T Lemma (Washington, Chapter 13) E Λ (1) E Λ iff E/(p, T ) (2) E Λ E Λ r i Λ/(f e i i ) j Λ/(p m j ), r, e i, m k Z, f i Z p [T ], f i Weierstrass E E def hom f : E E such that Ker(f ), Coker(f ) <
12 Λ 2 Z p Z p Λ Γ X Lemma ( ( ) 2.45, 2.50) E Λ (1) E/(t pn 1) char Λ (E) iff (t pn 1) (2) (1) λ, µ, ν n 0 #(E/(t pn 1)) = p λn+µpn +ν, n > n 0 (3) t pn 1 1 f (t 1) #E/( tpn 1 f (t 1) )
13 Z p Γ X Z p Z p Λ Γ X k n p k n X n := Gal( k n /k n ) = CFT Cl(k n ) (p) k n /k k := k n k Γ := Gal(k /k) = t = Z p X = lim X n = Gal( k /k ) X Λ Lemma Z p k /k (1) k /k Cl(k n ) (p) = X /(t p n 1) (2) k /k s Y s.t., (X : Y) < Cl(k n ) (p) = X /( t pn 1 t 1 )Y
14 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 Zp 拡大の岩澤類数公式 分岐 Zp 被覆と岩澤型公式 Λ 加群の補題 Γ と X Zp 拡大の Γ と X の図 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres
15 Z p Γ X Z p Z p Λ Γ X n p M n M n X n := Gal( M n /M n ) = Hur H 1 (M n ) (p) M n M M n Γ = t := lim Gal(M n M) = Z p X := lim X n X Λ Lemma L Z p -cover (1) L H 1 (M n ) (p) = X /(t p n 1)X (2) L (X : Y) < Y H 1 (M n ) (p) = X /( t pn 1 t 1 )Y Remark.
16 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 Zp 拡大の岩澤類数公式 分岐 Zp 被覆と岩澤型公式 Λ 加群の補題 Γ と X 分岐 Zp 被覆の Γ と X の図 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres
17 Z p 1 Z p Z p Λ Γ X Λ G n,n := Gal( M N M n ), G n := lim N G n,n (t pn 1)X = [G n, G n ] G n /X = Γ n Γ n = t pn X [G n, G n ] (t pn 1)x = t pn x t pn x 1 = [ t pn, x], (x X ) (t pn 1)X compact Hasudorff (1) I n < G n < G 1 I n X = 1, G n = In X H 1 (M n ) (p) = Gal( Mn /M n ) = G n /I n [G n, G n ] = I n X /I n (t pn 1)X = X /(t pn 1)X
18 Z p 2 Z p Z p Λ Γ X (2) L = K 1... K s K i I i < G 1 Γ = Z p Z p t i I i t Y = (t 1)X, t 2 t1 1,..., t st1 1 < G 1 (X : Y) < H 1 (M n ) (p) = X /( t pn 1 t 1 )Y X /(t pn 1) = Gn ab X n t pn i < G 1 H 1 (M n ) (p) = Xn = (X /(t p n 1)X )/( t pn 1 = X /( (t pn 1)X, t pn 1 tj n,..., tpn s,..., tpn s X ) X ) = X / (t pn 1)X, t pn 2 t pn 1,..., ts pn t pn 1. = (t j t 1 1 )(t 1 t j t 1 1 t 1 1 )(t1 2 t j t 1 1 t 2 1 )...(t n 1 1 t j t 1 1 t (n 1) 1 )t1 n t X (x X tx = x t = t 1 xt 1 1 ) t pn j t pn 1 = (t j t 1 1 ) 1+t+...+tp n 1 = ( tpn 1 ) (1 t)x, t t 1 2t 1 1,..., t s t 1 1 = ( tp n 1 t 1 )Y.
19 Z p Z p Λ Γ X TLNcover QHS 3 Z p -cover Remark. (1) [K i ] = 0 in H 1 (M) iff K i (2) L = K i [K i ] = 0 τ : µ i 1 Z p -cover TLN Alexander TLN L (t) iff M n QHS 3.
20 Theorem (periodic knot, Gordon 1972) K S 3 Z-cover M S 3 r M r S 3 H 1 (M r ) = H 1 (M r+m ), for all r m 1 (t)/ 2 (t) t m 1 i (t) i-th elementary ideal (= (i + 1)-th Fitting ideal) #H 1 (M r ) < Q-HS 3 Z p -cover 1 (t)/ 2 (t) iff iff λ = µ = 0.
21 Fitting ideal. Z p 1 (t)/ 2 (t) iff λ = µ = 0.
22 Fitting ideal. Z p 1 (t)/ 2 (t) iff λ = µ = 0.. Yes. Λ X j Λ/(f j ) s.t., f j f j+1 i (t) = j i f j (i + 1)-th Fitting ideal f 1 = 1 (t)/ 2 (t) iff X 0 iff λ = µ = 0. : Z/2Z Z/4Z Z/3Z = Z/2Z Z/12Z by CRT
23 Z p Z p GL 1 Z[[T ]] u = ±1 + T f (T ) p Z p = lim Z/p n Z Z p = {x + py x 0 mod p} u 1 mod p u 1 + pz p (1 + pz p ) = Z p ; 1 + p 1 non canonical Z p (1 + pz p )
24 Z p GL 1 R Z p m R R/m R = Fp ρ : π 1 (X ) GL 1 (F p ) ρ : π 1 (X ) GL 1 (R) such that ρ mod m R = ρ ρ R 1 Z p Z p = (1 + pzp ) GL 1 (F p ) R = Z p
25 Z p GL 1 Z p R s.t., R/m R = F p ρ : π 1 (X ) GL 1 (F p ) ρ ρ univ : π 1 (X ) R univ ρ ρ : π 1 (X ) GL 1 (R) φ : R univ R φ ρ univ = ρ GL 1 (F p ) R univ = Λ = Z p [[T ]] : ρ univ : π 1 (X ) Λ = Z p [[T ]]; µ 1 + T.
26 GL 1 (F p ) Z p GL 1 ρ R Defo( ρ, R) R Defo( ρ, R) 1 + m R ; ρ ρ(1), Hom(Z p [[T ]], R) 1 + m R ; φ 1 + φ(t ) R Defo( ρ, R) R Hom(Z p [[T ]], R) R 1 + m R R Defo( ρ, R) Z p [[T ]] Defo( ρ, R) Hom(Z p [[T ]], R); ρ (φ ρ : f (T ) f (ρ(1) 1)) ρ univ : H 1 (X ) GL 1 (Z p [[T ]]); µ 1 + T.
27 Z p GL 1 X = S 3 \ K ρ univ : π 1 (X ) Λ GL 1 (F p ) Alexander ( ) Λ = Z p [[Γ]] X = lim H 1 (X p n, Z p ) = H 1 (X, Λ) = H 1 (π 1 (X ), Λ) = H 1 (ρ univ ). H 1 (ρ univ ) = Λ/( K (t)).
28 Appendix: 2014 April [Morishita Takakura Terashima U.] (on arxiv) = =
29 Appendix: 2014 April [Morishita Takakura Terashima U.] (on arxiv) = = Thurston Dehn Mazur R=T
30 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 SL2 表現の変形理論の展望 Appendix: 2014 April Thank you for listening! 植木 潤 九州大学大学院数理学府 D2 On a branched Zp -cover of Q-homology 3-spheres
31 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres
32 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres
33 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres
34 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres
35 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres
36 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres
37 Zp 拡大と分岐 Zp 被覆 GL1 表現の変形理論としての岩澤理論 SL2 表現の変形理論 植木 潤 九州大学大学院数理学府 D2 SL2 表現の変形理論の展望 Appendix: 2014 April On a branched Zp -cover of Q-homology 3-spheres
SAMA- 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 information( ) 1., ([SU] ): F K k., Z p -, (cf. [Iw2], [Iw3], [Iw6]). K F F/K Z p - k /k., Weil., K., K F F p- ( 4.1).,, Z p -,., Weil..,,. Weil., F, F projectiv
( ) 1 ([SU] ): F K k Z p - (cf [Iw2] [Iw3] [Iw6]) K F F/K Z p - k /k Weil K K F F p- ( 41) Z p - Weil Weil F F projective smooth C C Jac(C)/F ( ) : 2 3 4 5 Tate Weil 6 7 Z p - 2 [Iw1] 2 21 K k k 1 k K
More informationwiles05.dvi
Andrew Wiles 1953, 20 Fermat.. Fermat 10,. 1 Wiles. 19 20., Fermat 1. (Fermat). p 3 x p + y p =1 xy 0 x, y 2., n- t n =1 ζ n Q Q(ζ n ). Q F,., F = Q( 5) 6=2 3 = (1 + 5)(1 5) 2. Kummer Q(ζ p ), p Fermat
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,,.
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重力方向に基づくコントローラの向き決定方法
( ) 2/Sep 09 1 ( ) ( ) 3 2 X w, Y w, Z w +X w = +Y w = +Z w = 1 X c, Y c, Z c X c, Y c, Z c X w, Y w, Z w Y c Z c X c 1: X c, Y c, Z c Kentaro Yamaguchi@bandainamcogames.co.jp 1 M M v 0, v 1, v 2 v 0 v
More informationII R n k +1 v 0,, v k k v 1 v 0,, v k v v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ k σ dimσ = k 1.3. k
II 231017 1 1.1. R n k +1 v 0,, v k k v 1 v 0,, v k v 0 1.2. v 0,, v k R n 1 a 0,, a k a 0 v 0 + a k v k v 0 v k k k v 0,, v k σ kσ dimσ = k 1.3. k σ {v 0,...,v k } {v i0,...,v il } l σ τ < τ τ σ 1.4.
More information16 B
16 B (1) 3 (2) (3) 5 ( ) 3 : 2 3 : 3 : () 3 19 ( ) 2 ax 2 + bx + c = 0 (a 0) x = b ± b 2 4ac 2a 3, 4 5 1824 5 Contents 1. 1 2. 7 3. 13 4. 18 5. 22 6. 25 7. 27 8. 31 9. 37 10. 46 11. 50 12. 56 i 1 1. 1.1..
More informationA µ : A A A µ(x, y) x y (x y) z = x (y z) A x, y, z x y = y x A x, y A e x e = e x = x A x e A e x A xy = yx = e y x x x y y = x A (1)
7 2 2.1 A µ : A A A µ(x, y) x y (x y) z = x (y z) A x, y, z x y = y x A x, y A e x e = e x = x A x e A e x A xy = yx = e y x x x y y = x 1 2.1.1 A (1) A = R x y = xy + x + y (2) A = N x y = x y (3) A =
More informationMazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R M End R M) M R ut R M) M R R G R[G] R G Sets 1 Λ Noether Λ k Λ m Λ k C Λ
Galois ) 0 1 1 2 2 4 3 10 4 12 5 14 16 0 Galois Galois Galois TaylorWiles Fermat [W][TW] Galois Galois Galois 1 Noether 2 1 Mazur [Ma1] Schlessinger [Sch] [SL] [Ma1] [Ma1] [Ma2] Galois [] 17 R m R R R
More informationk + (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 information17 Θ Hodge Θ Hodge Kummer Hodge Hodge
Teichmüller ( ) 2015 11 0 3 1 4 2 6 3 Teichmüller 8 4 Diophantus 11 5 13 6 15 7 19 8 21 9 25 10 28 11 31 12 34 13 36 14 41 15 43 16 47 1 17 Θ 50 18 55 19 57 20 Hodge 59 21 63 22 67 23 Θ Hodge 69 24 Kummer
More informationkawa (Spin-Orbit Tomography: Kawahara and Fujii 21,Kawahara and Fujii 211,Fujii & Kawahara submitted) 2 van Cittert-Zernike Appendix A V 2
Hanbury-Brown Twiss (ver. 1.) 24 2 1 1 1 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 3 3 Hanbury-Brown Twiss ( ) 4 3.1............................................
More informationZ[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,
More informationmeiji_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 informationD 24 D D D
5 Paper I.R. 2001 5 Paper HP Paper 5 3 5.1................................................... 3 5.2.................................................... 4 5.3.......................................... 6
More informationHanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence
Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................
More 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 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 informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
More informationkeisoku01.dvi
2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.
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 information2001 Miller-Rabin Rabin-Solovay-Strassen self-contained RSA RSA RSA ( ) Shor RSA RSA 1 Solovay-Strassen Miller-Rabin [3, pp
200 Miller-Rabin 2002 3 Rabin-Solovay-Strassen self-contained RSA RSA RSA ( ) Shor 996 2 RSA RSA Solovay-Strassen Miller-Rabin [3, pp. 8 84] Rabin-Solovay-Strassen 2 Miller-Rabin 3 4 Miller-Rabin 5 Miller-Rabin
More information( ) ( ) 1729 (, 2016:17) = = (1) 1 1
1729 1 2016 10 28 1 1729 1111 1111 1729 (1887 1920) (1877 1947) 1729 (, 2016:17) 12 3 1728 9 3 729 1729 = 12 3 + 1 3 = 10 3 + 9 3 (1) 1 1 2 1729 1729 19 13 7 = 1729 = 12 3 + 1 3 = 10 3 + 9 3 13 7 = 91
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 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 information非可換Lubin-Tate理論の一般化に向けて
Lubin-Tate 2012 9 18 ( ) Lubin-Tate 2012 9 18 1 / 27 ( ) Lubin-Tate 2012 9 18 2 / 27 Lubin-Tate p 1 1 ( ) Lubin-Tate 2012 9 18 2 / 27 Lubin-Tate p 1 1 Lubin-Tate GL n n 1 Lubin-Tate ( ) Lubin-Tate 2012
More information1 Slodowy 005 3 9 3 1 ADE ADE (0) ADE (1) SL(, C), R 3 (), (3) ADE, II 1 (4) SL(, Z)- (5) F 4, B, M McKay E 6, E 7, E 8 4 A n 5 5 5... 5 5 5 D n 5 5 5... 5 5 5 E 6 5 5 5 5 5 5 E 7 5 5 5 5 5 5 5 E 8 5
More information2 2 MATHEMATICS.PDF 200-2-0 3 2 (p n ), ( ) 7 3 4 6 5 20 6 GL 2 (Z) SL 2 (Z) 27 7 29 8 SL 2 (Z) 35 9 2 40 0 2 46 48 2 2 5 3 2 2 58 4 2 6 5 2 65 6 2 67 7 2 69 2 , a 0 + a + a 2 +... b b 2 b 3 () + b n a
More informationx 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a +
1 1 22 1 x 3 (mod ) 2 2.1 ( )., b, m Z b m b (mod m) b m 2.2 (Z/mZ). = {x x (mod m)} Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} + b = + b, b = b Z/mZ 1 1 Z Q R Z/Z 2.3 ( ). m {x 0, x 1,..., x m 1 } modm 2.4
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 information006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
More information数学概論I
{a n } M >0 s.t. a n 5 M for n =1, 2,... lim n a n = α ε =1 N s.t. a n α < 1 for n > N. n > N a n 5 a n α + α < 1+ α. M := max{ a 1,..., a N, 1+ α } a n 5 M ( n) 1 α α 1+ α t a 1 a N+1 a N+2 a 2 1 a n
More information( 9 1 ) 1 2 1.1................................... 2 1.2................................................. 3 1.3............................................... 4 1.4...........................................
More informationRIMS Kôkyûroku Bessatsu B32 (2012), (Iwasawa invariants of rea abeian number fieds with prime power conductors) By (Keiichi Komatsu), (Takashi
Tite 素数巾導手実アーベル体の岩澤不変量 (Agebraic Number Theory and Reated Topics 2010) Author(s) 小松, 啓一 ; 福田, 隆 ; 森澤, 貴之 Citation 数理解析研究所講究録別冊 = RIMS Kokyuroku Bessa (2012), B32: 105-124 Issue Date 2012-07 URL http://hd.hande.net/2433/196246
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
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 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 informationセアラの暗号
1 Cayley-Purser 1 Sarah Flannery 16 1 [1] [1] [1]314 www.cayley-purser.ie http://cryptome.org/flannery-cp.htm [2] Cryptography: An Investigation of a New Algorithm vs. the RSA(1999 RSA 1999 9 11 2 (17
More informationI A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
More informationexpander graph [IZ89] Nii (NII) Lec. 11 October 22, / 16
Lecture 11: PSRGs via Random Walks on Graphs October 22, 2013 Nii (NII) Lec. 11 October 22, 2013 1 / 16 expander graph [IZ89] Nii (NII) Lec. 11 October 22, 2013 2 / 16 Expander Graphs Expander Graph (
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 information6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2
1 6 6.1 (??) (P = ρ rad /3) ρ rad T 4 d(ρv ) + PdV = 0 (6.1) dρ rad ρ rad + 4 da a = 0 (6.2) dt T + da a = 0 T 1 a (6.3) ( ) n ρ m = n (m + 12 ) m v2 = n (m + 32 ) T, P = nt (6.4) (6.1) d [(nm + 32 ] )a
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 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 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 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 informationMilnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P
Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, PC ( 4 5 )., 5, Milnor Milnor., ( 6 )., (I) Z modulo
More informationArmstrong culture Web
2004 5 10 M.A. Armstrong, Groups and Symmetry, Springer-Verlag, NewYork, 1988 (2000) (1989) (2001) (2002) 1 Armstrong culture Web 1 3 1.1................................. 3 1.2.................................
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 information, = = 7 6 = 42, =
http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1 1 2016.9.26, http://www.ss.u-tokai.ac.jp/~mahoro/2016autumn/alg_intro/ 1.1 1 214 132 = 28258 2 + 1 + 4 1 + 3 + 2 = 7 6 = 42, 4 + 2 = 6 2 + 8
More informationO x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0
9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )
More information20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33
More informationtokei01.dvi
2. :,,,. :.... Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 3. (probability),, 1. : : n, α A, A a/n. :, p, p Apr. - Jul., 26FY Dept. of Mechanical Engineering, Saga Univ., JAPAN
More informationAkito 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
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 information( ) (, ) ( )
( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................
More information1. Γ, R 2,, M R. M R. M M Map(M, M) 3, Aut R (M). ρ : Γ Aut R (M) Γ. M R n, R, R ρ : Γ Aut R (M) GL n (R) := {g M n (R) det(g) R } 4. ρ Γ R R M.,,.,,
I ( ) (i) l, l, l (ii) (Q p ) l, l, l (iii) Artin (iv). (i),(ii). (iii) 1. (iv),.. [9]. [4] L-,.. Contents 1. 2 2. 4 2.1. 4 2.2. l 5 2.3. l 9 2.4. l 10 2.5. 12 2.6. Artin 13 3. 15 3.1. l, l, l 15 3.2.
More information1
1 Borel1956 Groupes linéaire algébriques, Ann. of Math. 64 (1956), 20 82. Chevalley1956/58 Sur la classification des groupes de Lie algébriques, Sém. Chevalley 1956/58, E.N.S., Paris. Tits1959 Sur la classification
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 informationprime number theorem
For Tutor MeBio ζ Eite by kamei MeBio 7.8.3 : Bernoulli Bernoulli 4 Bernoulli....................................................................................... 4 Bernoulli............................................................................
More information数学Ⅱ演習(足助・09夏)
II I 9/4/4 9/4/2 z C z z z z, z 2 z, w C zw z w 3 z, w C z + w z + w 4 t R t C t t t t t z z z 2 z C re z z + z z z, im z 2 2 3 z C e z + z + 2 z2 + 3! z3 + z!, I 4 x R e x cos x + sin x 2 z, w C e z+w
More information1 1.1 R (ring) R1 R4 R1 R (commutative [abelian] group) R2 a, b, c R (ab)c = a(bc) (associative law) R3 a, b, c R a(b + c) = ab + ac, (a + b)c = ac +
ALGEBRA II Hiroshi SUZUKI Department of Mathematics International Christian University 2004 1 1 1 2 2 1 3 3 1 4 4 1 5 5 1 6 6 1 7 7 1 7.1....................... 7 1 7.2........................... 7 4 8
More informationIII 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)
More informationSiegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo
Siegel Hecke 1 Siege Hecke L L Fourier Dirichlet Hecke Euler L Euler Fourier Hecke [Fr] Andrianov [An2] Hecke Satake L van der Geer ([vg]) L [Na1] [Yo] 2 Hecke ( ) 0 1n J n =, Γ = Γ n = Sp(n, Z) = {γ GL(2n,
More informationAI n Z f n : Z Z f n (k) = nk ( k Z) f n n 1.9 R R f : R R f 1 1 {a R f(a) = 0 R = {0 R 1.10 R R f : R R f 1 : R R 1.11 Z Z id Z 1.12 Q Q id
1 1.1 1.1 R R (1) R = 1 2 Z = 2 n Z (2) R 1.2 R C Z R 1.3 Z 2 = {(a, b) a Z, b Z Z 2 a, b, c, d Z (a, b) + (c, d) = (a + c, b + d), (a, b)(c, d) = (ac, bd) (1) Z 2 (2) Z 2? (3) Z 2 1.4 C Q[ 1] = {a + bi
More information,.,. 2, R 2, ( )., I R. c : I R 2, : (1) c C -, (2) t I, c (t) (0, 0). c(i). c (t)., c(t) = (x(t), y(t)) c (t) = (x (t), y (t)) : (1)
( ) 1., : ;, ;, ; =. ( ).,.,,,., 2.,.,,.,.,,., y = f(x), f ( ).,,.,.,., U R m, F : U R n, M, f : M R p M, p,, R m,,, R m. 2009 A tamaru math.sci.hiroshima-u.ac.jp 1 ,.,. 2, R 2, ( ).,. 2.1 2.1. I R. c
More informationZ: 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 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 information研究集会 結び目の数学 X 報告集 254 Alexander The profinite completions of knot groups determine the Alexander polynomials ( ) X preprint[uek17b] [Uek18] Ẑ[
研究集会 結び目の数学 X 報告集 254 Alexander The profinite completions of knot groups determine the Alexander polynomials ( ) 30 1 28 X preprint[uek17b] [Uek18] Ẑ[[tẐ]] Alexander 2 J, K J K Alexander J (t) K (t) S
More information2 Riemann Im(s) > 0 ζ(s) s R(s) = 2 Riemann [Riemann]) ζ(s) ζ(2) = π2 6 *3 Kummer s = 2n, n N ζ( 2) = 2 2, ζ( 4) =.3 2 3, ζ( 6) = ζ( 8)
(Florian Sprung) p 2 p * 9 3 p ζ Mazur Wiles 4 5 6 2 3 5 2006 http://www.icm2006.org/video/ eighth session [ ] Coates [Coates] 2 735 Euler n n 2 = p p 2 p 2 = π2 6 859 Riemann ζ(s) = n n s = p p s s ζ(s)
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 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 informationp.2/76
kino@info.kanagawa-u.ac.jp p.1/76 p.2/76 ( ) (2001). (2006). (2002). p.3/76 N n, n {1, 2,...N} 0 K k, k {1, 2,...,K} M M, m {1, 2,...,M} p.4/76 R =(r ij ), r ij = i j ( ): k s r(k, s) r(k, 1),r(k, 2),...,r(k,
More information2 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
More information(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fou
(Bessel) (Legendre).. (Hankel). (Laplace) V = (x, y, z) n (r, θ, ϕ) r n f n (θ, ϕ). f n (θ, ϕ) n f n (θ, ϕ) z = cos θ z θ ϕ n ν. P ν (z), Q ν (z) (Fourier) (Fourier Bessel).. V ρ(x, y, z) V = 4πGρ G :.
More informationB
B YES NO 5 7 6 1 4 3 2 BB BB BB AA AA BB 510J B B A 510J B A A A A A A 510J B A 510J B A A A A A 510J M = σ Z Z = M σ AAA π T T = a ZP ZP = a AAA π B M + M 2 +T 2 M T Me = = 1 + 1 + 2 2 M σ Te = M 2 +T
More informationMacintosh_HD:Users:toshi:myDocuments:classes:過去の非常勤:東工大非常勤2007(情報):Markov_chain:note.dvi
1 2 2 3 2.1................................ 3 2.1.1................................... 3 2.1.2............................................. 4 2.1.3................................................. 5 2.1.4........................................
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 information) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8)
4 4 ) a + b = i + 6 b c = 6i j ) a = 0 b = c = 0 ) â = i + j 0 ˆb = 4) a b = b c = j + ) cos α = cos β = 6) a ˆb = b ĉ = 0 7) a b = 6i j b c = i + 6j + 8) a b a b = 6i j 4 b c b c 9) a b = 4 a b) c = 7
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 informationTOP URL 1
TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................
More informationSUSY DWs
@ 2013 1 25 Supersymmetric Domain Walls Eric A. Bergshoeff, Axel Kleinschmidt, and Fabio Riccioni Phys. Rev. D86 (2012) 085043 (arxiv:1206.5697) ( ) Contents 1 2 SUSY Domain Walls Wess-Zumino Embedding
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 information‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í
Markov 2009 10 2 Markov 2009 10 2 1 / 25 1 (GA) 2 GA 3 4 Markov 2009 10 2 2 / 25 (GA) (GA) L ( 1) I := {0, 1} L f : I (0, ) M( 2) S := I M GA (GA) f (i) i I Markov 2009 10 2 3 / 25 (GA) ρ(i, j), i, j I
More informationI , : ~/math/functional-analysis/functional-analysis-1.tex
I 1 2004 8 16, 2017 4 30 1 : ~/math/functional-analysis/functional-analysis-1.tex 1 3 1.1................................... 3 1.2................................... 3 1.3.....................................
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 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 information[ ] 0.1 lim x 0 e 3x 1 x IC ( 11) ( s114901) 0.2 (1) y = e 2x (x 2 + 1) (2) y = x/(x 2 + 1) 0.3 dx (1) 1 4x 2 (2) e x sin 2xdx (3) sin 2 xdx ( 11) ( s
[ ]. lim e 3 IC ) s49). y = e + ) ) y = / + ).3 d 4 ) e sin d 3) sin d ) s49) s493).4 z = y z z y s494).5 + y = 4 =.6 s495) dy = 3e ) d dy d = y s496).7 lim ) lim e s49).8 y = e sin ) y = sin e 3) y =
More information1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1)
1 α X (path) α I = [0, 1] X α(0) = α(1) = p α p (base point) loop α(1) = β(0) X α, β α β : I X (α β)(s) = ( )α β { α(2s) (0 s 1 2 ) β(2s 1) ( 1 2 s 1) X α α 1 : I X α 1 (s) = α(1 s) ( )α 1 1.1 X p X Ω(p)
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 informationIII 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F
III 1 (X, d) d U d X (X, d). 1. (X, d).. (i) d(x, y) d(z, y) d(x, z) (ii) d(x, y) d(z, w) d(x, z) + d(y, w) 2. (X, d). F X.. (1), X F, (2) F 1, F 2 F F 1 F 2 F, (3) F λ F λ F λ F. 3., A λ λ A λ. B λ λ
More information°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 informationBasic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.
Basic Mathematics 16 4 16 3-4 (10:40-12:10) 0 1 1 2 2 2 3 (mapping) 5 4 ε-δ (ε-δ Logic) 6 5 (Potency) 9 6 (Equivalence Relation and Order) 13 7 Zorn (Axiom of Choice, Zorn s Lemma) 14 8 (Set and Topology)
More informationChern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q
Chern-Simons E-mail: fuji@th.phys.nagoya-u.ac.jp Jones 3 Chern-Simons - Chern-Simons - Jones J(K; q) []Jones q J (K + ; q) qj (K ; q) = (q /2 q /2 )J (K 0 ; q), () J( ; q) =. (2) K Figure : K +, K, K 0
More informationρ /( ρ) + ( q, v ) : ( q, v ), L < q < q < q < L 0 0 ( t) ( q ( t), v ( t)) dq ( t) v ( t) lmr + 0 Φ( r) dt lmr + 0 Φ ( r) dv ( t) Φ ( q ( t) q ( t)) + Φ ( q+ ( t) q ( t)) dt ( ) < 0 ( q (0), v (0)) (
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 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 informationλ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T
0 2 8 8 6 3 0 0 Young Young [F] 0.. Young λ n λ n λ = (λ,, λ l ) λ λ 2 λ l λ = ( m, 2 m 2, ) λ = n, l(λ) = l {λ n n 0} P λ = (λ, ), µ = (µ, ) n λ µ k k k λ i µ i λ µ λ = µ k i= i= i < k λ i = µ i λ k >
More information4.6: 3 sin 5 sin θ θ t θ 2t θ 4t : sin ωt ω sin θ θ ωt sin ωt 1 ω ω [rad/sec] 1 [sec] ω[rad] [rad/sec] 5.3 ω [rad/sec] 5.7: 2t 4t sin 2t sin 4t
1 1.1 sin 2π [rad] 3 ft 3 sin 2t π 4 3.1 2 1.1: sin θ 2.2 sin θ ft t t [sec] t sin 2t π 4 [rad] sin 3.1 3 sin θ θ t θ 2t π 4 3.2 3.1 3.4 3.4: 2.2: sin θ θ θ [rad] 2.3 0 [rad] 4 sin θ sin 2t π 4 sin 1 1
More information