X = E ij (1.3 L Eij = n x jν x ν=1 iν n n (1.4 (E ij = t ( ( x ij, x ij ( t ( t(l Eij = x ij. x ij g G U(g g m m=0 g X Y Y X [X, Y ] X, Y g g G U(g Ad

Size: px
Start display at page:

Download "X = E ij (1.3 L Eij = n x jν x ν=1 iν n n (1.4 (E ij = t ( ( x ij, x ij ( t ( t(l Eij = x ij. x ij g G U(g g m m=0 g X Y Y X [X, Y ] X, Y g g G U(g Ad"

Transcription

1 1. GL(n GL(n Lie GL(n, C Lie 1 Lie G = GL(n, R GL(n, C G G X M(n, C ϕ(x d dt ϕ(xetx t=0 = d dt ϕ(x + txx t=0 M(n, C C (i, j 1 0 E ij ( n ν, µ=1 x νµe νµ E ij = n ν=1 x νie νj (1.1 E ij = Lie n x νi x ν=1 νj (1.2 [E ij, E kl ] = δ jk E il δ li E kj g G = GL(n, C Lie gl n g M(n, C = n i, CE ij G ϕ g G π g ϕ(x = ϕ(g 1 x X M(n, C 1 t e tx L X

2 X = E ij (1.3 L Eij = n x jν x ν=1 iν n n (1.4 (E ij = t ( ( x ij, x ij ( t ( t(l Eij = x ij. x ij g G U(g g m m=0 g X Y Y X [X, Y ] X, Y g g G U(g Ad(g X M(n, C g X Ad(gX = gxg 1 2. g = M(n, C P (g g Ad(g A M(n, C A A(t = Ad(e tx A X g A U(g G g g g (2.1 X, Y = Trace XY P (g g S(g V A = g G Ad(gA. V A G S(g U(g G R U(g G U(g S(g cf. [O5] (2.2 U ϵ (g := ( k=0 k g / X Y Y X ϵ[x, Y ]; X, Y g ϵ U(g = U 1 (g S(g = U 0 (g ϵ 0 U 1 (g X ϵx U ϵ (g g ϵ 1 (2.2 ϵ U ϵ (g = U(g[ϵ] (Poincaré-Birkhoff-Witt V A C n n S n λ = (λ 1,..., λ n C S n λ n! (2.3 s j (x s j (λ (j = 1,..., n, s j (x = x i1 x ij 1 i 1 < <i j n S j (x = n i=1 xj i 2

3 n (2.4 (x i λ j (j = 1,..., n, (2.5 i=1 n (x i λ j (i = 1,..., n. λ generic (2.4 λ = (µ,..., µ, ν,..., ν n! }{{}}{{} k!(n k! k n k (2.4 (2.5 (2.6 (2.7 (x i1 µ (x in k+1 µ, (x j1 ν (x jk+1 ν (1 i 1 < < i n k+1 n, 1 j 1 < < j k+1 n, (x i µ(x i ν (i = 1,..., n µ ν (2.6 (2.7 j = 1 ( (2.3 S n [OS] cf. [OP] Heckman-Opdam [HO] (2.3 x i S n (x, λ [O2] λ = 0 S n x i 3. Generalized Verma Modules n {n 1,..., n L } n j = n n j (1 j L, n 0 = 0, (3.1 Θ = {n 1, n 2,..., n L }, ι Θ (ν = j if n j 1 < ν n j (1 ν n Θ = {n 1 < n 2 < < n L = n} n g Lie n Θ, n Θ m Θ ι Θ (i > ι Θ (j ι Θ (i < ι Θ (j ι Θ (i = ι Θ (j E ij p Θ = m Θ + n Θ m k Θ = ι CE Θ(i=ι Θ(j=k ij n = 1 j<i n CE ij, n = 1 i<j n CE ij a = n CE jj p = a + n m Θ = m 1 Θ ml Θ p Borel Θ p Θ Borel p Θ = {X g; X, Y = 0 ( Y n Θ } λ = (λ 1,..., λ L C L g affine (3.2 A Θ,λ := n λ ιθ(je jj + n Θ λ 1 I n 1 A 21 λ 2 I n 0 2 = A 31 A 32 λ 3 I n 3 ; A ij M(n i, n j; C. A L1 A L2 A L3 λ L I n L I m m M(k, l; C k l 3

4 3.1. A Θ,λ Jordan (3.3 µ C, 1 k n J ( #{i; λ i = µ n i k}, µ µ 1 µ 0 J(m, µ = M(m, C 1 µ Jordan Θ λ. f U 0 (g = S(g ϵ = 0 f ( ( Ad(gA Θ,λ = 0 Ad(gf (AΘ,λ = 0 ( g G g G Ad(gf JΘ(λ ϵ ( g G f Ad(gJΘ(λ ϵ g G f Ann G ( M ϵ Θ (λ JΘ(λ ϵ := U ϵ ( (g X λθ (X, X p Θ MΘ(λ ϵ := U ϵ (g/jθ(λ, ϵ Ann ( M ϵ Θ(λ := { D U ϵ (g; DM ϵ Θ(λ = 0 }, I ϵ Θ(λ := Ann G ( M ϵ Θ (λ := { D U ϵ (g; Ad(gD Ann ( M ϵ Θ(λ ( g G } p Θ C Lie 1 λ Θ (3.4 λ Θ (Y + L X k := k=1 L λ k Trace(X k for X k m k Θ and Y n Θ. k=1 ϵ = 1 1 M Θ (λ = MΘ 1 (λ A Θ,λ V AΘ,λ g G Ad(gJ Θ 0 (λ g G Ad(gJ ( Θ(λ = Ann G MΘ (λ = Ann ( M Θ (λ IΘ ϵ (λ M Θ(λ m Θ λ Θ Verma Verma (3.5 M(λ Θ := U(g/J(λ Θ, J ϵ (λ Θ := X p U ϵ (g ( X λ Θ (X and J(λ Θ = J 1 (λ Θ. g Θ = {1, 2,..., n} p Θ = p M ϵ (λ Θ M ϵ (λ ϵ = 0 ϵ = 1 Verma 4

5 4. Harish-Chandra n n U 0 (g G Z 0 (g Z ϵ (g = {D U ϵ (g; Ad(gD = D ( g G} ϵ = 1 Z(g g (Ad(gE ij = t g 1 E t g E = (E ij (4.1 Z k := Trace E k (k = 1, 2,... Z k Z ϵ (g cf. Gelfand [Ge] ϵ = 0 k Harish-Chandra (4.2 γ : U ϵ (g D Γ(D U ϵ ( (a = S(a D Γ(D nu ϵ (g + U ϵ (gn U ϵ (g = U ϵ ( n U ϵ (a U ϵ (n U ϵ (a a U ϵ (a = S(a D = D + D, ϵρ := ϵ n n+1 (j 2 E jj Γ(D = γ(d Harish-Chandra (4.3 Γ : Z ϵ (g S(a Sn. Z ϵ (g Z 1,..., Z n Γ(Z k 7 [Go1] [Ca1] 100 γ ( det(e, t = n i=1( Eii t + ϵ(n i (4.4 det(e, t := det (E ij t + ϵ(n iδ ij Z ϵ (g ( t C. (4.5 det (A ij = sgn(σa σ(11 A σ(nn. σ S n Capelli E ij E ij tδ ij (4.4 ( (4.6 det (x ij det = det (E ij iδ ij. x ij 4.1. i (4.4 t k k (4.7 Z ϵ (g = C[Z 1,..., Z n ] = C[ 1,..., n ]. ii (4.1 Lie cf. 7 (4.4 o n cf. [HU] [Wa] iii λ = (λ 1,..., λ n ϵ = 0 (4.8 I ϵ λ := { D U ϵ (g; γ(ad(gd, λ = 0 ( g G } 5

6 G B(G Verma M(λ g L(λ I λ = Ann ( L(λ V {1,...,n},λ {I λ ; λ C n } g [Du] w S n w.λ = w(λ + ϵρ ϵρ ϵ = 0 generic λ I w.λ = I λ (Z 1,..., Z n λ = ρ λ ( iv Verma M ϵ (λ Ann G M ϵ (λ D γ(d, λ D Z ϵ (g gl n k γ( k, λ k = 1,..., n 5. Generalized Capelli Elements rank (5.1 A {k,n} (µ, ν = ( µik νi n k rank ( A {k,n} (µ, ν µ n k, rank ( A {k,n} (µ, ν ν k ϵ = 0 ( E ij µ n k + 1 V A{k,n} (µ,ν ( (5.2 D{i ϵ (t := det 1,...,i m}{j 1,...,j m} E ip,i q + ( ϵ(m q t δ ip i q 1 p m U ϵ (g 1 q m, Capelli { DIJ ϵ ; #I = #J = m, I, J {1,..., n} } G G ϵ [x ij, µν ] = ϵδ iµ δ jν Capelli [O4] (5.3 = ( n det x νik νjl + ϵ(m lδ ik j l 1 k m ν=1 1 l m det (x νp i q 1 p m det ( νp i q 1 q m 1 ν 1 < <ν m n 1 p m 1 q m ϵ (5.4 D ϵ IJ(µ, D ϵ I J (ν + kϵ (#I = #J = n k + 1, #I = #J = k + 1 IΘ ϵ (λ cf. (2.6 ϵ = 0 µ ν IΘ ϵ (λ (5.5 µ ν / {ϵ, 2ϵ,..., (n 1ϵ} IΘ ϵ (λ µ = ν 6

7 ([O5]. MΘ ϵ (λ m = 1,..., n d ϵ m(x := d ϵ m(x; Θ, λ = L ( (n x j λj ϵn +m n j 1, (6.1 d m = d m (Θ := deg x d ϵ m(x; Θ, λ = L max{n j + m n, 0}, e ϵ m(x := e ϵ m(x; Θ, λ = d ϵ m(x/d ϵ m 1(x, q ϵ (x := q ϵ (x; Θ, λ = L ( x λj ϵn j 1. d ϵ m(x MΘ ϵ (λ m, {eϵ m(x; 1 m n} MΘ ϵ (λ, q ϵ (x MΘ ϵ (λ dϵ n(x MΘ ϵ (λ { (6.2 z (l z ( z ϵ (z ϵ(l 1 if l > 0, := 1 if l ϵ = A Θ,λ U ϵ (g d 0 m(x xi n A Θ,λ m 6.3 ([O5]. d ϵ m(x = k m ν=1 (x λ m,ν N m,ν ν ν λ m,ν λ m,ν (6.3 V ϵ Θ(λ := n k m m=1 ν=1 N m,ν 1 j=0 #I=#J=m ( d j C dx IJ(x j Dϵ x=λm,ν IΘ ϵ (λ = U ϵ (gvθ ϵ(λ dϵ n(x = 0 regular ϵ = 0 Jordan L (6.4 IΘ(λ ϵ = U ϵ (gdij(λ ϵ k + ϵn k 1. k=1 #I=#J=n+1 n k 6.4. i I ϵ Θ (λ Iϵ Θ (λ d ϵ m(x; Θ, λ d ϵ m(x; Θ, λ (m = 1,..., n ϵ = 0 A Θ,λ g G Ad(gA Θ,λ ϵ = 0, λ = 0 d m (Θ d m (Θ m = 1,..., n ii ϵ = 0, λ = [Ta1] [We] (6.3 ϵ = 0, λ = 0, Θ = {n} [Ko] GL(n normal variety [KP] 7. (5.1 A {k,n} (µ, ν (x µ(x ν (E µ(e ν V A{k,n} (µ,ν µ ν n 2 Trace E kµ (n kν I{k,n} 0 (µ, ν cf. (2.7. µ = ν k (x µ(x ν ϵk Lie 7.1 ([O6]. Lie g π : g M(N, C End(C N g M(N, C M(N, C X, Y = Trace XY 2 g g 2 7

8 ( M(N, C g π F π = π (E ij 1 i N M(N, g 1 j N Z ϵ (g[x] q(x q(f π = 0 F π π q π (x g V q(fv = 0 C q(x 1 (π, V q π,v (x 7.2. i g = gl n π (7.1 q π (x = det (x E ij ϵ(n iδ ij (7.2 (7.3 q π (E = 0 (Cayley-Hamilton, L q π,m ϵ Θ (λ = (x λ j ϵn j 1. 1 i n 1 j n Z ϵ (g[x], ii q π,m ϵ Θ (λ(e N 2 L ϵ Θ (λ Lϵ Θ (λ G λ generic MΘ ϵ (λ regular IΘ ϵ (λ Lϵ Θ (λ ( k λ Θ γ( k k = 1,..., L i V g V ii g GL(n Lie π V = MΘ 0 (λ q π,m 0 Θ (λ(x A Θ,λ iii O(n Lie o n π F π =. ( Eij E ji 2 1 i n 1 j n iv Trace F k π Z ϵ (g γ(trace F k π [Go1] q π (x [Go2] q π,m ϵ (λ(x Cayley- Hamilton cf. [OO] v 6 [I1] [Um] [Sg]. vi g π q π,m ϵ Θ (λ [O6] [OO] 8. Grassmann Poisson Penrose U ϵ (g 2 Verma Verma Gap 8.1 ([O5], [OO]. λ generic regular (8.1 J ϵ Θ(λ = I ϵ Θ(λ + J ϵ (λ Θ (GAP. G GL(n SL(n GL(n, C GL(n, R, U(p, q SU (n Lie G P P Θ G/P Θ P Θ 1 λ (8.2 B(G/P Θ ; λ := { f B(G; f(gp = λ(p 1 f(g ( p P Θ } P Θ Lie 2 p Θ λ p Θ 1 (8.3 Ann ( B(G/P Θ ; λ := { D U(g; L D f = 0 ( f B(G/P ; λ } = I Θ (λ. B(G/P Θ ; λ G G G/P Θ I Θ (λ 8

9 Poisson G Lie K G (8.4 P λθ P Θ,λ : B(G/P Θ ; λ ( B(G/P ; λ Θ A(G/K; Mλ. f (P λ f(g = f(gkdk Poisson M λ λ Riemann G Z(g Ann ( M(λ Θ A(G/K; M λ G/P Θ Riemann G/K cf. [Sa] P Θ Poisson A(G/K; M λ Helgason [H1] G = SL(2, R, λ = 0 Poisson G [K ] generic λ λ = 0 Shilov Stein Hua [BV] [La] [KM], [Sh1] [Sh2] λ = 0 [Jn] [K ] P Θ λ 8 [K ] I Θ (λ (8.1 B(G/P ; λ Θ I Θ (λ B(G/P Θ ; λ λ generic λ = U(g P Θ,λ P Θ,λ I Θ (λ U(g 8.2. i Hua 7 Shilov tube 2 3 SU(m, n Shilov m = n 2 m n 3 G K m n 2 [BV] [OSh] ii Lie iii [BOS] iv Helgason Poisson [O1] Penrose G C Lie G G C P C G C V G O λ G C /P C 9 K

10 T P en : H V (O λ S G C Penrose 6 7 S Riemann G G = U(n, n Θ = {k, 2n} GL(n, C Grassmann Gr k (C n V S [Se] 6 (5.3 k + 1 n = 2 k = 1 Penrose F Grassmann Gr k (F n n F F n k F = R Gr k (R n := {k R n } ( Grassmann x 11 x 1k M o (n, k; R := X =.. M(n, k; R; rank X = k x n1 x nk = M o (n, k; R/GL(k, R. G = GL(n, R Gr k (R n Gr k (R n = GL(n, R/P k,n (= O(n/O(k O(n k { ( } g1 0 P k,n := p = ; g y g 1 GL(k, R, g 2 GL(n k, R, y M(n k, k, R 2 B(G/P k,n ; λ := { f B(G; f(xp = f(x det g 1 λ 1 det g 2 λ 2 }, p P k,n ( ( = B O(n/O(k O(n k = { f B ( M o (n, k; R ; f(xg 1 = f(x det g 1 λ1, g 1 GL(k, R } (x t x 1 X Gr 1 (F n P n 1 (F F = R Θ = {k, n} F = C Θ = {k, n} {k, n} Ann ( B(Gr k (F n ; λ 6 k + 1 n k Trace Z(g 1 9. Radon B(G/P Θ ; λ B(G/P Θ ; λ G Radon G Grassmann Radon 0 < k < l < n R k l : B ( Gr k (R n φ (R k l φ(x = φ(xydy B ( Gr l (R n GL(n, R O(l/O(k O(l k (9.1 R k l : B ( G/P k,n ; (l, 0 B ( G/P l,n ; (k, 0 G k + l < n dim Gr k (R n < dim Gr l (R n I Θ (λ 10

11 9.1 ([O4]. 0 < k < l < n k + l < n R k l M 0 (n, l; R G { Φ ( ( (x ij 1 i l B M 0 (n, l; R ; 1 j n Φ(xg = det g k Φ(x for g GL(l, R, ( det Φ(x = 0 (Capelli x iµ j ν 1 µ k+1 1 ν k+1 } for 1 i 1 < < i k+1 n, 1 j 1 < < j k+1 l. M 0 (n, l; R = {A M(n, l; R; rank A = l} 9.2. i C 9.1 [Hi] ii [Ka] [Ka], [GR] 9.3 ([O4]. G Lie P Θ Q j j = 1, 2 G/P Θ G λ µ j P Θ Q j 1 φ j G (9.2 φ 1 (q 1 xp = µ 1 (q 1 λ(pφ 1 (x (q 1 Q 1, p P Θ, φ 2 (q 2 xp = µ 2 (q 2 λ(pφ 2 (x (q 2 Q 2, p P Θ, λ = λ 2ρ PΘ (9.3 Φ φ1,φ 2 (x := K φ 1 (xkφ 2 (kdk ( = K φ(kφ 2 (x 1 kdk 9.4. i Φ φ1,φ 2 (x Q 1 Lie Q 2 Lie 6 7 I Θ (λ (9.2 φ 1, φ 2 (9.3 ii Q 1 = Q 2 = K µ j Φ φ1,φ 2 (x P Θ Lauricella F D [Kr] P = P k,n Q 2 = P l,n λ = (l, 0 µ 2 = ( k, 0 φ 2 R k l H GL(n, R H C V (H C GL(k, C, V C k l = 1 (H C, V (9.2 φ 1 ( k = 1 ( GL(1, C, C n ( H = GL(1 GL(1, C n -Gelfand [Ao], [GG] Φ(α, x = t t2 l =1 ν=1 n l αj t ν x jν ω ± 11 (α 1 + α α n = l.

12 l Φ x ij = α j Φ for 1 i n H, x ij n ν=1 x νi Φ = kδ ij Φ for 1 i, j l GL(l, R, x νj 2 Φ x i1 j 1 x i2 j 2 = 2 Φ x i2 j 1 x i1 j 2 for 1 i 1 < i 2 n, 1 j 1 < j 2 l Capelli i n = 4, l = 2 Gauss ii 9.5 G [Ta2] iii Penrose [Se] References [Ao] K. Aomoto, On the structure of integrals of power products of linear functions, Sc. Papers Coll. Gen. Education, Univ. of Tokyo 27(1977, [BOS] S. Ben Saïd, T. Oshima and N. Shimeno, Fatou s theorems and Hardy-type spaces for eigenfunctions of the invariant differential operators on symmetric spaces, Intern. Math. Research Notice 16(2003, [BV] N. Berline and M. Vergne, Equations du Hua et noyau de Poisson, Lect. Note. in Math. 880(1981, [Ca1] A. Capelli, Üeber die Zurückführung der Cayley schen Operation Ω auf gewöhnliche Polar- Operationen, Math. Ann. 29(1887, [Ca2], Sur les opérations dans la théorie des formes algébriques, Math. Ann. 37(1890, [DP] C. DeConcini and C. Procesi, Symmetric functions, conjugacy classes and the flag variety, Invent. Math. 64(1981, [Du] M. Duflo, Sur la classification des idéaux primitifs dans l algèbre enveloppante d une algèbre de Lie semi-simple, Ann. of Math. 105(1977, [Ge] I. M. Gelfand, Center of the infinitesimal group ring, Mat. Sb., Nov. Ser. 26(68(1950, ; English transl. in Collected Papers, Vol. II, pp [GG] I. M. Gelfand and S. I. Gelfand, Generalized hypergeometric equations, Soviet Math. Dokl 33(1986, [Go1] M. D. Gould, A trace formula for semi-simple Lie algebras, Ann. Inst. Henri Poincaré, Sect. A 32(1980, [Go2], Characteristic identities for semi-simple Lie algebras, J. Austral. Math. Soc. Ser. B 26(1985, [GR] E. L. Grinberg and B. Lubin, Radon inversion and Grassmannians via Gårding-Gindikin fractional integrals, to appear in Ann. of Math. [HO] G. J. Heckman and E. M. Opdam, Root system and hypergeometric functions. I, Comp. Math. 64(1987, [H1] S. Helgason, A duality for symmetric spaces with applications to group representations II, Advances in Math., 22(1976, [H2], The Radon transform (2nd ed., Birkhauser, Boston, [Hi] T. Higuchi, Generalized Capelli Operator Grassmann Radon,, 2004 [HU] R. Howe and T. Umeda, The Capelli identity, the double commutation theorem, and multiplicity free actions, Math. Ann. 290(1991, [I1] M. Itoh, Explicit Newton s formula for gl n, J. Alg. 208(1998, [I2], The Capelli elements for the orthogonal Lie algebras, J. Lie Theory 10(2000, [IU] M. Itoh and T. Umeda, On the central elements in the universal enveloping algebra of the orthogonal Lie algebras, Compositio Math. 127(2001, [Jn] K. D. Johnson, Generalized Hua operators and parabolic subgroups, Ann. of Math. 120(1984, [Ka] T. Kakehi, Integral geometry on Grassmannian manifolds and calculus of invariant differential operators, J. Funct. Anal., 168(1999,

13 [K ] M. Kashiwara, A. Kowata, K. Minemura, K. Okamoto, T. Oshima and M. Tanaka, Eigenfunctions of invariant differential operators on a symmetric space, Ann. of Math. 107(1978, [Kr] A. Korányi, Hua-type integrals, hypergeometric functions and symmetric polynomials, International symposium in memory of Hua Loo Keng, vol. II, Analysis, Science Press, Beijing and Springer-Verlag, Berlin, 1991, pp [KM] A. Korányi and P. Malliavin, Poisson formula and compound diffusion associated to an overdetermined elliptic system on the Siegel half plane of rank two, Acta Math. 134(1975, [Ko] B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85(1963, [KP] H. Kraft and C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math. 53(1979, [La] M. Lassale, Les équations de Hua d un domaine borneé symétrique de type tube, Invent. Math. 77(1984, [Od1] H. Oda, Annihilator operators of the degenerate principal series for simple Lie groups of type (B and (D,, 2000, 1183(2001, [OO] H. Oda and T. Oshima, Minimal polynomials and annihilators of generalized Verma modules of the scalar type, UTMS , preprint, 2004, [OP] M. A. Olshanetsky and A. M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rep. 94(1983, [O1] T. Oshima,, 281(1976, [O2], Asymptotic behavior of spherical functions on semisimple symmetric spaces, Advanced Studies in Pure Math. 14 (1988, [O3], Capelli identities, degenerate series and hypergeometric functions, Proceedings of a symposium on Representation Theory at Okinawa, 1995, [O4], Generalized Capelli identities and boundary value problems for GL(n, Structure of Solutions of Differential Equations, World Scientific, 1996, [O5], A quantization of conjugacy classes of matrices, UTMS , preprint, 2000, to appear in Adv. in Math. [O6], Annihilators of generalized Verma modules of the scalar type for classical Lie algebras, UTMS , preprint, [OS] T. Oshima and H. Sekiguchi, Commuting families of differential operators invariant under the action of a Weyl group, J. Math. Sci. Univ. Tokyo 2(1995, [OSh] T. Oshima and N. Shimeno, Boundary value problems on Riemannian symmetric spaces of the noncompact type, in preparation. [Sg] H. Sakaguchi, U(g,, [Sa] I. Satake, On realizations and compactifications of symmetric spaces, Ann. of Math. 71(1960, [Se] H. Sekiguchi, The Penrose transform for certain non-compact homogeneous manifolds of U(n, n, J. Math. Sci. Univ. Tokyo 3(1996, [Sh1] N. Shimeno, Boundary value problems for the Shilov boundary of a bounded symmetric domain of tube type, J. of Funct. Anal. 140(1996, [Sh2], Boundary value problems for various boundaries of Hermitian symmetric spaces, J. of Funct. Anal. 170(2000, [Ta1] T. Tanisaki, Defining ideals of the closure of conjugacy classes and representation of the Weyl groups, Tohoku Math. J. 34(1982, [Ta2], Hypergeometric systems and Radon transforms for Hermitian symmetric spaces, Adv. Studies in Pure Math. 26(2000, [Um] T. Umeda, Newton s formula for gl n, Proc. Amer. Math. Soc. 126 (1998, [Wa] A. Wachi, Lie, 1348(2003, [We] J. Weyman, The equations of conjugacy classes of nilpotent matrices, Invent. Math. 98(1989,

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. R-space ( ) Version 1.1 (2012/02/29) i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. ii 1 Lie 1 1.1 Killing................................

More information

1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, webpage,.,,.

1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, webpage,.,,. 1 1, 2016 D B. 1.1,.,,. (1). (2). (3) Milnor., (1) (2)., (3). 1.2,.,, ( )..,.,,. 1.3, 2015. webpage,.,,. 2 1 (1),, ( ). (2),,. (3),.,, : Hashinaga, T., Tamaru, H.: Three-dimensional solvsolitons and the

More information

1

1 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 information

Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu

Gauss Fuchs rigid rigid rigid Nicholas Katz Rigid local systems [6] Fuchs Katz Crawley- Boevey[1] [7] Katz rigid rigid Katz middle convolu rigidity 2014.9.1-2014.9.2 Fuchs 1 Introduction y + p(x)y + q(x)y = 0, y 2 p(x), q(x) p(x) q(x) Fuchs 19 Fuchs 83 Gauss Fuchs rigid rigid rigid 7 1970 1996 Nicholas Katz Rigid local systems [6] Fuchs Katz

More information

CAPELLI (T\^o $\mathrm{r}\mathrm{u}$ UMEDA) MATHEMATICS, KYOTO UNIVERSITY DEPARTMENT $\mathrm{o}\mathrm{p}$ $0$:, Cape i,.,.,,,,.,,,.

CAPELLI (T\^o $\mathrm{r}\mathrm{u}$ UMEDA) MATHEMATICS, KYOTO UNIVERSITY DEPARTMENT $\mathrm{o}\mathrm{p}$ $0$:, Cape i,.,.,,,,.,,,. 1508 2006 1-11 1 CAPELLI (T\^o $\mathrm{r}\mathrm{u}$ UMEDA) MATHEMATICS KYOTO UNIVERSITY DEPARTMENT $\mathrm{o}\mathrm{p}$ $0$: Cape i Capelli 1991 ( ) (1994 ; 1998 ) 100 Capelli Capelli Capelli ( ) (

More information

compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1

compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) a, Σ a {0} a 3 1 014 5 4 compact compact Hermann compact Hermite ( - ) Hermann Hermann ( ) compact Hermite Lagrange compact Hermite ( ) 1 1.1. a, Σ a {0} a 3 1 (1) a = span(σ). () α, β Σ s α β := β α,β α α Σ. (3) α, β

More information

Macdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona

Macdonald, ,,, 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 information

untitled

untitled Lie L ( Introduction L Rankin-Selberg, Hecke L (,,, Rankin, Selberg L (GL( GL( L, L. Rankin-Selberg, Fourier, (=Fourier (= Basic identity.,,.,, L.,,,,., ( Lie G (=G, G.., 5, Sp(, R,. L., GL(n, R Whittaker

More information

A11 (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

A11 (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

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, 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, 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 information

Mazur [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

Mazur [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 information

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c

2.1 H f 3, SL(2, Z) Γ k (1) f H (2) γ Γ f k γ = f (3) f Γ \H cusp γ SL(2, Z) f k γ Fourier f k γ = a γ (n)e 2πinz/N n=0 (3) γ SL(2, Z) a γ (0) = 0 f c GL 2 1 Lie SL(2, R) GL(2, A) Gelbart [Ge] 1 3 [Ge] Jacquet-Langlands [JL] Bump [Bu] Borel([Bo]) ([Ko]) ([Mo]) [Mo] 2 2.1 H = {z C Im(z) > 0} Γ SL(2, Z) Γ N N Γ (N) = {γ SL(2, Z) γ = 1 2 mod N} g SL(2,

More information

2016 Course Description of Undergraduate Seminars (2015 12 16 ) 2016 12 16 ( ) 13:00 15:00 12 16 ( ) 1 21 ( ) 1 13 ( ) 17:00 1 14 ( ) 12:00 1 21 ( ) 15:00 1 27 ( ) 13:00 14:00 2 1 ( ) 17:00 2 3 ( ) 12

More information

0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t

0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo t e-mail: koyama@math.keio.ac.jp 0. I II I II (1) linear type: GL( ), Sp( ), O( ), (2) loop type: loop current Kac-Moody affine, hyperbolic (3) diffeo type: diffeo universal Teichmuller modular I. I-. Weyl

More information

第5章 偏微分方程式の境界値問題

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 information

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

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 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 information

等質空間の幾何学入門

等質空間の幾何学入門 2006/12/04 08 tamaru@math.sci.hiroshima-u.ac.jp i, 2006/12/04 08. 2006, 4.,,.,,.,.,.,,.,,,.,.,,.,,,.,. ii 1 1 1.1 :................................... 1 1.2........................................ 2 1.3......................................

More information

all.dvi

all.dvi 29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan

More information

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi

1 M = (M, g) m Riemann N = (N, h) n Riemann M N C f : M N f df : T M T N M T M f N T N M f 1 T N T M f 1 T N C X, Y Γ(T M) M C T M f 1 T N M Levi-Civi 1 Surveys in Geometry 1980 2 6, 7 Harmonic Map Plateau Eells-Sampson [5] Siu [19, 20] Kähler 6 Reports on Global Analysis [15] Sacks- Uhlenbeck [18] Siu-Yau [21] Frankel Siu Yau Frankel [13] 1 Surveys

More information

takei.dvi

takei.dvi 0 Newton Leibniz ( ) α1 ( ) αn (1) a α1,...,α n (x) u(x) = f(x) x 1 x n α 1 + +α n m 1957 Hans Lewy Lewy 1970 1 1.1 Example 1.1. (2) d 2 u dx 2 Q(x)u = f(x), u(0) = a, 1 du (0) = b. dx Q(x), f(x) x = 0

More information

main.dvi

main.dvi SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1

More information

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x

1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x . P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +

More information

Feynman 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, [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

, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n

, CH n. CH n, CP n,,,., CH n,,. RH n ( Cartan )., CH n., RH n CH n,,., RH n, CH n., RH n ( ), CH n ( 1.1 (v), (vi) )., RH n,, CH n,., CH n,. 1.2, CH n ( ), Jürgen Berndt,.,. 1, CH n.,,. 1.1 ([6]). CH n (n 2), : (i) CH k (k = 0,..., n 1) tube. (ii) RH n tube. (iii). (iv) ruled minimal, equidistant. (v) normally homogeneous submanifold F k tube. (vi) normally

More information

Λ (Kyo Nishiyama) 1 p q r ( determinantal variety) n n r Kostant ( Rallis, Steinberg ) D 1980 Borho-Brylinski Vogan Springer theta theta theta

Λ (Kyo Nishiyama) 1 p q r ( determinantal variety) n n r Kostant ( Rallis, Steinberg ) D 1980 Borho-Brylinski Vogan Springer theta theta theta Λ (Kyo Nishiyama) 1 p q r ( determinantal variety) n n r 1960 70 Kostant ( Rallis, Steinberg ) D 1980 Borho-Brylinski Vogan Springer theta theta theta theta ( ) Λ ( ) August 9, 2000 Theta lifting of representations

More information

1 4 1 ( ) ( ) ( ) ( ) () 1 4 2

1 4 1 ( ) ( ) ( ) ( ) () 1 4 2 7 1995, 2017 7 21 1 2 2 3 3 4 4 6 (1).................................... 6 (2)..................................... 6 (3) t................. 9 5 11 (1)......................................... 11 (2)

More information

24 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. 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 information

数学Ⅱ演習(足助・09夏)

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 information

Step 2 O(3) Sym 0 (R 3 ), : a + := λ 1 λ 2 λ 3 a λ 1 λ 2 λ 3. a +. X a +, O(3).X. O(3).X = O(3)/O(3) X, O(3) X. 1.7 Step 3 O(3) Sym 0 (R 3 ),

Step 2 O(3) Sym 0 (R 3 ), : a + := λ 1 λ 2 λ 3 a λ 1 λ 2 λ 3. a +. X a +, O(3).X. O(3).X = O(3)/O(3) X, O(3) X. 1.7 Step 3 O(3) Sym 0 (R 3 ), 1 1 1.1,,. 1.1 1.2 O(2) R 2 O(2).p, {0} r > 0. O(3) R 3 O(3).p, {0} r > 0.,, O(n) ( SO(n), O(n) ): Sym 0 (R n ) := {X M(n, R) t X = X, tr(x) = 0}. 1.3 O(n) Sym 0 (R n ) : g.x := gxg 1 (g O(n), X Sym 0

More information

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p.

tomocci ,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. tomocci 18 7 5...,. :,,,, Lie,,,, Einstein, Newton. 1 M n C. s, M p. M f, p d ds f = dxµ p ds µ f p, X p = X µ µ p = dxµ ds µ p. µ, X µ.,. p,. T M p. M F (M), X(F (M)).. T M p e i = e µ i µ. a a = a i

More information

' , 24 :,,,,, ( ) Cech Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing 66 1

' , 24 :,,,,, ( ) Cech Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing 66 1 1998 1998 7 20 26, 44. 400,,., (KEK), ( ) ( )..,.,,,. 1998 1 '98 7 23, 24 :,,,,, ( ) 1 3 2 Cech 6 3 13 4 Index theorem 22 5 Stability 44 6 compact 49 7 Donaldson 58 8 Symplectic structure 63 9 Wall crossing

More information

R C Gunning, Lectures on Riemann Surfaces, Princeton Math Notes, Princeton Univ Press 1966,, (4),,, Gunning, Schwarz Schwarz Schwarz, {z; x}, [z; x] =

R C Gunning, Lectures on Riemann Surfaces, Princeton Math Notes, Princeton Univ Press 1966,, (4),,, Gunning, Schwarz Schwarz Schwarz, {z; x}, [z; x] = Schwarz 1, x z = z(x) {z; x} {z; x} = z z 1 2 z z, = d/dx (1) a 0, b {az; x} = {z; x}, {z + b; x} = {z; x} {1/z; x} = {z; x} (2) ad bc 0 a, b, c, d 2 { az + b cz + d ; x } = {z; x} (3) z(x) = (ax + b)/(cx

More information

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K

II 2 3.,, A(B + C) = AB + AC, (A + B)C = AC + BC. 4. m m A, m m B,, m m B, AB = BA, A,, I. 5. m m A, m n B, AB = B, A I E, 4 4 I, J, K II. () 7 F 7 = { 0,, 2, 3, 4, 5, 6 }., F 7 a, b F 7, a b, F 7,. (a) a, b,,. (b) 7., 4 5 = 20 = 2 7 + 6, 4 5 = 6 F 7., F 7,., 0 a F 7, ab = F 7 b F 7. (2) 7, 6 F 6 = { 0,, 2, 3, 4, 5 },,., F 6., 0 0 a F

More information

2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i

2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i 1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x,

More information

K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X

K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X 2 E 8 1, E 8, [6], II II, E 8, 2, E 8,,, 2 [14],, X/C, f : X P 1 2 3, f, (O), f X NS(X), (O) T ( 1), NS(X), T [15] : MWG(f) NS(X)/T, MWL(f) 0 (T ) NS(X), MWL(f) MWL(f) 0, : {f λ : X λ P 1 } λ Λ NS(X λ

More information

Armstrong culture Web

Armstrong 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 information

1 Affine Lie 1.1 Affine Lie g Lie, 2h A B = tr g ad A ad B A, B g Killig form., h g daul Coxeter number., g = sl n C h = n., g long root 2 2., ρ half

1 Affine Lie 1.1 Affine Lie g Lie, 2h A B = tr g ad A ad B A, B g Killig form., h g daul Coxeter number., g = sl n C h = n., g long root 2 2., ρ half Wess-Zumino-Witten 1999 3 18 Wess-Zumino-Witten., Knizhnik-Zamolodchikov-Bernard,,. 1 Affine Lie 2 1.1 Affine Lie.............................. 2 1.2..................................... 3 2 WZW 4 3 Knizhnik-Zamolodchikov-Bernard

More information

December 28, 2018

December 28, 2018 e-mail : kigami@i.kyoto-u.ac.jp December 28, 28 Contents 2............................. 3.2......................... 7.3..................... 9.4................ 4.5............. 2.6.... 22 2 36 2..........................

More information

Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p

Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara Sp(2, R) p Siegel modular forms of middle parahoric subgroups and Ihara lift ( Tomoyoshi Ibukiyama Osaka University 1. Introduction [8] Ihara 80 1963 Sp(2, R) p L holomorphic discrete series Eichler Brandt Eichler

More information

all.dvi

all.dvi 72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G

More information

平成 15 年度 ( 第 25 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 15 月年 78 日開催月 4 日 ) X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (

平成 15 年度 ( 第 25 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 15 月年 78 日開催月 4 日 ) X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = ( 1 1.1 X 2 = 1 ( ) f 1 (X 1,..., X n ) = 0,..., f r (X 1,..., X n ) = 0 X = (X 1,..., X n ) ( ) X 1,..., X n f 1,..., f r A T X + XA XBR 1 B T X + C T QC = O X 1.2 X 1,..., X n X i X j X j X i = 0, P i

More information

20 9 19 1 3 11 1 3 111 3 112 1 4 12 6 121 6 122 7 13 7 131 8 132 10 133 10 134 12 14 13 141 13 142 13 143 15 144 16 145 17 15 19 151 1 19 152 20 2 21 21 21 211 21 212 1 23 213 1 23 214 25 215 31 22 33

More information

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

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

xia2.dvi

xia2.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 information

Chern-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   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 information

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

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

More information

2019 1 5 0 3 1 4 1.1.................... 4 1.1.1......................... 4 1.1.2........................ 5 1.1.3................... 5 1.1.4........................ 6 1.1.5......................... 6 1.2..........................

More information

n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................

More information

2000年度『数学展望 I』講義録

2000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 information

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

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

More information

( ),.,,., C A (2008, ). 1,, (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,, (M, g) p M, s p : M M p, : (1) p s p, (

( ),.,,., C A (2008, ). 1,, (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,, (M, g) p M, s p : M M p, : (1) p s p, ( ( ),.,,., C A (2008, ). 1,,. 1.1. (M, g) (Riemannian symmetric space), : p M, s p : M M :.,.,.,, (, ).,,. 1.2. (M, g) p M, s p : M M p, : (1) p s p, (2) s 2 p = id ( id ), (3) s p ( )., p ( s p (p) = p),,

More information

E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1

E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 E1 (4/12)., ( )., 3,4 ( ). ( ) Allen Hatcher, Vector bundle and K-theory ( HP ) 1 (4/12) 1 1.. 2. F R C H P n F E n := {((x 0,..., x n ), [v 0 : : v n ]) F n+1 P n F n x i v i = 0 }. i=0 E n P n F P n

More information

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i 1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [

More information

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5. A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c

More information

Twist knot orbifold Chern-Simons

Twist knot orbifold Chern-Simons Twist knot orbifold Chern-Simons 1 3 M π F : F (M) M ω = {ω ij }, Ω = {Ω ij }, cs := 1 4π 2 (ω 12 ω 13 ω 23 + ω 12 Ω 12 + ω 13 Ω 13 + ω 23 Ω 23 ) M Chern-Simons., S. Chern J. Simons, F (M) Pontrjagin 2.,

More information

[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

2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := 0

2 Three-wave Painlevé VI 21 -Wilson three-wave Painlevé VI Gauss -Wilson [KK3] n 3 3 t = t 1 t 2 t 3 -Wilson W z; t := I + W 1 z + W 2 z 2 + z; t := 0 1473 : de nouvelles perspectives 2006 2 pp 102 119 VI q 1 Tetsuya Kikuchi Sabro Kakei Drinfel d-sokolov Painlevé [KK1] [KK2] [KK3] [KIK] [ ] [ ] [KK3] three-wave equation Painlevé VI q q Drinfel d-sokolov

More information

ADM-Hamiltonian Cheeger-Gromov 3. Penrose

ADM-Hamiltonian Cheeger-Gromov 3. Penrose ADM-Hamiltonian 1. 2. Cheeger-Gromov 3. Penrose 0. ADM-Hamiltonian (M 4, h) Einstein-Hilbert M 4 R h hdx L h = R h h δl h = 0 (Ric h ) αβ 1 2 R hg αβ = 0 (Σ 3, g ij ) (M 4, h ij ) g ij, k ij Σ π ij = g(k

More information

note1.dvi

note1.dvi (1) 1996 11 7 1 (1) 1. 1 dx dy d x τ xx x x, stress x + dx x τ xx x+dx dyd x x τ xx x dyd y τ xx x τ xx x+dx d dx y x dy 1. dx dy d x τ xy x τ x ρdxdyd x dx dy d ρdxdyd u x t = τ xx x+dx dyd τ xx x dyd

More information

1.1 foliation M foliation M 0 t Σ t M M = t R Σ t (12) Σ t t Σ t x i Σ t A(t, x i ) Σ t n µ Σ t+ t B(t + t, x i ) AB () tα tαn µ Σ t+ t C(t + t,

1.1 foliation M foliation M 0 t Σ t M M = t R Σ t (12) Σ t t Σ t x i Σ t A(t, x i ) Σ t n µ Σ t+ t B(t + t, x i ) AB () tα tαn µ Σ t+ t C(t + t, 1 Gourgoulhon BSSN BSSN ϕ = 1 6 ( D i β i αk) (1) γ ij = 2αĀij 2 3 D k β k γ ij (2) K = e 4ϕ ( Di Di α + 2 D i ϕ D i α ) + α ] [4π(E + S) + ĀijĀij + K2 3 (3) Ā ij = 2 3Āij D k β k 2αĀikĀk j + αāijk +e

More information

λ 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

λ 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 information

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

More information

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

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc2.com/ 1 30 3 30.1.............. 3 30.2........................... 4 30.3...................... 5 30.4........................ 6 30.5.................................. 8 30.6...............................

More information

( ) (, ) ( )

( ) (, ) ( ) ( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................

More information

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d

S I. 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 information

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann

More information

2010 ( )

2010 ( ) 2010 (2010 1 8 2010 1 13 ( 1 29 ( 17:00 2 3 ( e-mail (1 3 (2 (3 (1 (4 2010 1 2 3 4 5 6 7 8 9 10 11 Hesselholt, Lars 12 13 i 1 ( 2 3 Cohen-Macaulay Auslander-Reiten [1] [2] 5 [1], :,, 2002 [2] I Assem,

More information

コホモロジー的AGT対応とK群類似

コホモロジー的AGT対応とK群類似 AGT K ( ) Encounter with Mathematics October 29, 2016 AGT L. F. Alday, D. Gaiotto, Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010), arxiv:0906.3219.

More information

January 27, 2015

January 27, 2015 e-mail : kigami@i.kyoto-u.ac.jp January 27, 205 Contents 2........................ 2.2....................... 3.3....................... 6.4......................... 2 6 2........................... 6

More information

. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2

. Mac Lane [ML98]. 1 2 (strict monoidal category) S 1 R 3 A S 1 [0, 1] C 2 C End C (1) C 4 1 U q (sl 2 ) Drinfeld double. 6 2 2014 6 30. 2014 3 1 6 (Hopf algebra) (group) Andruskiewitsch-Santos [AFS09] 1980 Drinfeld (quantum group) Lie Lie (ribbon Hopf algebra) (ribbon category) Turaev [Tur94] Kassel [Kas95] (PD) x12005i@math.nagoya-u.ac.jp

More information

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

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes ) ( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)

More information

( ) 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- ( 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 information

newmain.dvi

newmain.dvi 数論 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published

More information

S K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1.

S K(S) = T K(T ) T S K n (1.1) n {}}{ n K n (1.1) 0 K 0 0 K Q p K Z/pZ L K (1) L K L K (2) K L L K [L : K] 1.1. () 1.1.. 1. 1.1. (1) L K (i) 0 K 1 K (ii) x, y K x + y K, x y K (iii) x, y K xy K (iv) x K \ {0} x 1 K K L L K ( 0 L 1 L ) L K L/K (2) K M L M K L 1.1. C C 1.2. R K = {a + b 3 i a, b Q} Q( 2, 3) = Q( 2

More information

linearal1.dvi

linearal1.dvi 19 4 30 I 1 1 11 1 12 2 13 3 131 3 132 4 133 5 134 6 14 7 2 9 21 9 211 9 212 10 213 13 214 14 22 15 221 15 222 16 223 17 224 20 3 21 31 21 32 21 33 22 34 23 341 23 342 24 343 27 344 29 35 31 351 31 352

More information

Bruhat

Bruhat SGC - 77 Bruhat ([22]) 3 3.11 2010 4 ii 1 1 1.1... 1 1.2... 5 1.3... 8 1.4 1... 11 1.5 2... 14 2 18 2.1... 18 2.2... 25 2.3... 30 3 36 3.1... 36 3.2... 42 3.3... 49 3.3.1... 49 3.3.2... 50 3.3.3... 52

More information

3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S (CMC 1), 1 ( [AA]). 3 H 3 CMC 1 Bryant ([B, UY1]).

3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S (CMC 1), 1 ( [AA]). 3 H 3 CMC 1 Bryant ([B, UY1]). 3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S 3 1 1 (CMC 1), 1 ( [AA]) 3 H 3 CMC 1 Bryant ([B, UY1]) H 3 CMC 1, Bryant ([CHR, RUY1, RUY2, UY1, UY2, UY3,

More information

(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like

(2) Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [2], [13]) Poincaré e m Poincaré e m Kähler-like 2 Kähler-like () 10 9 30 1 Fisher α (α) α Fisher α ( α) 0 Levi Civita (1) ( 1) e m (e) (m) ([1], [], [13]) Poincaré e m Poincaré e m Kähler-like Kähler-like Kähler M g M X, Y, Z (.1) Xg(Y, Z) = g( X Y, Z) + g(y, XZ)

More information

Hilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2

Hilbert, von Neuman [1, p.86] kt 2 1 [1, 2] 2 2 hara@math.kyushu-u.ac.jp 1 1 1.1............................................... 2 1.2............................................. 3 2 3 3 5 3.1............................................. 6 3.2...................................

More information

D-brane K 1, 2 ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane

D-brane K 1, 2   ( ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane K 1, 2 E-mail: sugimoto@yukawa.kyoto-u.ac.jp (2004 12 16 ) 1 K D-brane K K D-brane Witten [1] D-brane K K K K D-brane D-brane K RR BPS D-brane D-brane RR D-brane K D-brane K D-brane K K [2, 3]

More information

TOP URL 1

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

More information

[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 +

[2, 3, 4, 5] * C s (a m k (symmetry operation E m[ 1(a ] σ m σ (symmetry element E σ {E, σ} C s 32 ( ( =, 2 =, (3 0 1 v = x 1 1 + 2016 12 16 1 1 2 2 2.1 C s................. 2 2.2 C 3v................ 9 3 11 3.1.............. 11 3.2 32............... 12 3.3.............. 13 4 14 4.1........... 14 4.2................ 15 4.3................

More information

all.dvi

all.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 information

( )

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

More information

x V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R

x V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x

More information

,2,4

,2,4 2005 12 2006 1,2,4 iii 1 Hilbert 14 1 1.............................................. 1 2............................................... 2 3............................................... 3 4.............................................

More information

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

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

More information

( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h

( 3) b 1 b : b b f : a b 1 b f = f (2.7) g : b c g 1 b = g (2.8) 1 b b (identity arrow) id b f a b g f 1 b b c g (2.9) 3 C C C a, b a b Hom C (a, b) h 2011 9 5 1 Lie 1 2 2.1 (category) (object) a, b, c, a b (arrow, morphism) f : a b (2.1) f a b (2.2) ( 1) f : a b g : b c (composite) g f : a c ( 2) f f a b g f g c g h (2.3) a b c d (2.4) h (g f) = (h

More information

構造と連続体の力学基礎

構造と連続体の力学基礎 II 37 Wabash Avenue Bridge, Illinois 州 Winnipeg にある歩道橋 Esplanade Riel 橋6 6 斜張橋である必要は多分無いと思われる すぐ横に道路用桁橋有り しかも塔基部のレストランは 8 年には営業していなかった 9 9. 9.. () 97 [3] [5] k 9. m w(t) f (t) = f (t) + mg k w(t) Newton

More information

基礎数学I

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

More information

25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University

LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University LINEAR ALGEBRA I Hiroshi SUZUKI Department of Mathematics International Christian University 2002 2 2 2 2 22 2 3 3 3 3 3 4 4 5 5 6 6 7 7 8 8 9 Cramer 9 0 0 E-mail:hsuzuki@icuacjp 0 3x + y + 2z 4 x + y

More information

I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( )

I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( ) I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( ) 1 (Introduction) 1 1.1... 1 1.2 Riemann Lebesgue... 2 2 (Measurable sets and Measures) 4 2.1 σ-... 4 2.2 Borel... 5 2.3...

More information

Design of highly accurate formulas for numerical integration in weighted Hardy spaces with the aid of potential theory 1 Ken ichiro Tanaka 1 Ω R m F I = F (t) dt (1.1) Ω m m 1 m = 1 1 Newton-Cotes Gauss

More information

1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2

1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2 filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin

More information

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t

(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t 6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]

More information

untitled

untitled 0. =. =. (999). 3(983). (980). (985). (966). 3. := :=. A A. A A. := := 4 5 A B A B A B. A = B A B A B B A. A B A B, A B, B. AP { A, P } = { : A, P } = { A P }. A = {0, }, A, {0, }, {0}, {}, A {0}, {}.

More information