L(A) l(a)
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- まな しのしま
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2 L(A) l(a) K Bessel Airy Schrödinger Airy Airy
3 2 Contents 1. Akira ASADA: Old Tale: Riemann Roch Theorem 2. Kenji SETO: Bessel Function and Airy Function 3. Shozo NIIZEKI: Editorial Comments
4 3 Old Tale: Riemann Roch Theorem 1 Akira Asada 2 C - K- K- K- 1.1 : l(a) = n(a) g l(k A), X X - A = i n ip i n(a) = i n i, X g l(a) = diml(a), L(A) A X l(k A) S : χ(s) = 1 KdS, 2π χ(s) K l(a) n(a) g + 1 l(k A) K K(x) ([1]) ([16]) 1 asada-a@poporo.ne.jp 2 Professor emeritus, Sinsyu University S
5 4 A ( L(A) A L(K A) A X D(A) l(a) d(a) = n(a) g + 1, d(a) = dimd(a) l(a) d(a) χ(s) = dimh 2 (S, R) dimh 1 (S, R)+ dimh (S, R) l(a) A (l(a) d(a)) (n(a) g + 1) ([3]) X ξ Ω(ξ) χ(ξ) = n ( 1) p dimh p (X, Ω(ξ)) p= ([2]) K- Ω(ξ) Ω(ξ) Ω(ξ) ([9],[1]) K- K- X F H G H = F + G K- f : X Y Y Y K- K- Ω(ξ) χ(ξ) = dim(ker( + )) codim(ker( + )), + : n p= H (X, Ω 2p (ξ)) n p= H (X, Ω 2p+1 (ξ)) M D indexd = dim(kerd) dim(cokerd) = dim(kerd) dim(kerd )
6 5 ([4]) ([5],[6],[7]) indd = tre tdd tre td D ([5],[7]) ([11],[12]) l(a) χ(a) χ(a) 1.2 f(z) f(p i ) =, i = 1,..., n, p i f N i - q j, j = 1,..., m M j - f (f) (f) = i N i p i j M j q j, (fg) = (f) + (g) X (f) = (g) f = cg, c C (f) X - X X - r(z) = C P (z) Q(z), P (z) = k i=1 (z c i) n i, Q(z) = l j=1 (z c j) m j, degp = n, degq = m S 2 = C { } (r(z)) = i n i c i j m j c j + (m n)p A = k n k p k k n k = (R(z)) = A C - - ( ) 2 = 4 3 g 2 g 3 = w = 4z 2 g 2 z g 3 A ζ (z) = (z), σ (z) σ(z) = ζ(z) σ A σ(z c i )
7 6 X 1 X (f) ( ϑ- 1.3 L(A) l(a) A = k n k p k ; n k Z, B = k m k p k X - n k m k A B L(A) = {f (f) + A } {}, l(a) = diml(a) g X L(A) f fg L(A + (g)) L(A) = L(A + (g)) l(a) = l(a + (g)) l(a) l(n p) = l(n 1) p + 1 X p n A = k n k p k n(a) = k n k n((f)) = - - l() = l((f)) = 1, l(a) =, n(a) < n(a) = A = (f) X ϕ (ϕ) dz 2 p dz 1 z 2 z = ±1 t 2 = 1 ± z 1 2 dz 1 z 4 dimh 1 (X, R) = 2g X g g f(z) = 1 z 2(g+1) z k 1 z 2(g+1)) dz, k g 1 D(A) = {ϕ A + (ϕ) } {}, l(a) ([1]) d(a) = dimd(a) l(a) d(a) = n(a) g + 1, D(A) = L(A K) K canonical divisor l(k) = g, n(k) = 2g n(a) > 2g l(a) = n(a) g + 1 g m > 2g m
8 7 1.4, p : F X p 1 (x) = F x p f x F x U(f x ) U(x) F x α : F G F x G x α F x x f f = g, f U(x) = g U(x), U(x) x ; f x f x x X F x f x U(f x ; f, U(x)) = {f y ; y U(x)} p s(x) = x s : U F U F U U U = {U i } X s i,...,i p : U i... U ip : F {s i,...,i p } C p (U, F), δ : C p (U, F) C p+1 (U.F) p+1 δs i,...,i p+1 = ( 1) j s i,...,i j 1,i j+1,...,i p+1 j= X F- H p (X, F) (X H (X, F) X F dimx = m H q (X, F) =, q > m, χ(f) = p ( 1) p dim(h p (X, F)) δ ;- ϕ p dϕ p 1 U dϕ p 2 UV,..., δϕp k U i,...,u ik F i G j H H (X, F) H (X, G) H (X, H) H 1 (X, F), ϕp 1 U ϕp 1 V = χ(g) = χ(f) + χ(h) H (X, F) dimh (X, F) G, H G F H χ(f)
9 8 L(A) x L(A) x ; A U(x) A U(x) L(A U(x)) L(A V (x)), V (x) U(x) L(A) x U(x) L(A U(x)) Ω(A) = x L(A) x L(A) Ω(A) Ω(A + (p)) C x χ(ω(a + (p))) = l(a) + 1 H (X, Ω(A)) = L(A), χ(ω()) = 1 g l(a) dim ( H 1 (X, Ω(A)) ) = n(a) + 1 g 1.5 F p- C p H p (X, F) =, p 1 C t C C 1 Φ p p- Φ p C p Φ p+1 H p (X, C t ) = H p (X, C) H p (X, C) = H (X, Φ p )/dh (X, C p 1 ) C t (C) X C χ(c t ) X X d δ = dδ + δd H p (X, C) = ker( H (X.C p )) p- X z k = x k + iy k, dz = dx + idy, d z = dx idy, df = k ( f x k dx k + f y k dy k ) = f + f, f = f dz k, z k k f z =1 2 ( f x + 1 f i y ), f = k f z k d z k, f z = 1 2 ( f y 1 f i y ) X (p, q) X i,j g ijdz i d z j ij dz i d z j (p, q)- Π p,q Π p,q = Π p,q
10 9 X H r (X, C) = p+q=r H p,q (X, C), H p,q (X, C) = H q,p (X, C) ) f = f 2 =, ϕ = ϕ ψ ϕ = i 1,...,i p f i1,...,i p dz i1 dz ip f i1,...,i p ϕ p- p- Ω p H q (X, Ω p ) = H p,q (X, C) 1.6 M F F G U = U} X g UV : U V G, g UV g V W g W U = e, e {g UV } {h U }, h U G g UV = h U g UV h 1 V {g UV } {g UV } g UV X X {U} {f U }, f U U f U f 1 V, {g UV } = {f U f 1 V } f U X D D + (f), f X X ξ = {g UV } X (GL(m, C)- X C ω (C H 1 (X, C ω ) C d H1 (X, C d ) H 1 (X, C d ) = H 2 (X, Z) 1 K- ϕ U g UV ϕ V = ϕ U ξ ξ (p, q)- C p,q (ξ), Ω(ξ), p- Ω p (ξ) ξ (g UV (g UV ϕ V ) = g UV ϕv
11 1 ξ Ω(ξ) C,1 (ξ) H p (X, Ω(ξ)) -closed (, p)- -exact (, p)- ( X = + X 2 = H p (X, Ω(ξ)) ξ p- X H (X, Ω(A)) = L(A), H 1 (X, Ω(A)) = D(A) X {g UV } {detg UV } K H p (X, Ω(ξ)) = H n p (X, Ω(K ξ )) d(a) = l(k A) 1.7 χ(ω(a)) = n(a) + 1 g n(a) A 1 g X (2g = dimh 1 (X, R). 1 g X ) l(a) χ(ω(ξ)) ξ Ch(ξ) = Π i (1 + γ i ) X Ch(X) = Π i (1 + α i ) χ(ω(ξ)) = ch(ξ)td(x), ch(ξ) = X i e γ i, td(x) = ΠQ(α i ), x Q(x) = ([3]) 1 e x ξ χ(ω) ([2]) {g UV } { 1 2πi (log(g V W ) log(g UW ) + log(g UV ))} D D c(d) ( A X A = (f) A Z C d C d n(a) A X 1 g
12 11 γ i ξ F q (= GL(q, C)/ (n, C) = U(q)/T q )- X ξ ξ ξ T q - γ i p : X ξ X p : H p (X, Z) H p (X ξ, Z) F q γ 1,..., γ q H (X, Z) ([3]) D 1, D 2 c(d 1 ) c(d 2 ) = c(d 1 D 2 ) D 1 D 2 D 1 D 2 [1]; χ(d) = 1 2 D (D K) (K2 + c 2 ) c 2 [17] - l(a) + δ δ l(a) + δ = χ(a) X GL(n, C)- ( Z Ω C ω, C ω C ρ Φ,1 ρ(g) = 1 g 1 g 2πi H 1 (X, R)/ι (H 1 (X, Z)) (ι ι : Z R 1.8 K- ξ Ω(ξ) ( dimh p (X, Ω(ξ)) χ(ω(ξ)) ( - p χ(f), F F G H χ(g) = χ(f)+χ(h) G (F + H) K(X) χ K(X) Z f : X Y F f 1 (p) F f! (F) Y f! (F) = n ( 1)n H n (X, F) ch(f! (F)td(Y ) = f (ch(ftd(x)) Y ([9],[1])
13 12 K- K- + : p H (X, Ω 2p (ξ)) p H (X, Ω 2p+1 (ξ)) χ(ω(ξ)) = dim(ker( + )) dim(coker( + )) E 1, E 2 M D : H (M, E 1 ) H (M, E 2 ) D = k m C k (x) k x k 1 1 xk n n, k = k k n ξ 1,..., ξ σ(d) = k =m C k (x)ξ k1 1 ξk n n p : SM M p E 1 p E 2 D σ(d) K (BM/SM), BM, ch(σ(d)) ch(d) D kerd, cokerd indd = dim(kerd) dim(cokerd) dim(cokerd) = dim(kerd ) indd = dim(kerd) dim(kerd ) indd = ϕ(ch(d))td(m) M ϕ : Hc (T M) H (M) M n n d + δ : H (X, C 2p ) H (X, C 2p+1 ), p= p= C p p- K- K- ch(d) K-
14 13 K- K- λ DD ϕ = λϕ (D D)D = λd ϕ D D DD indd = tre td D tre tdd ([4],[6]) Q, F indq = tr( 1) F ([11],[12]) ([11]) 1.9 l(a) χ(a) K - l(a) l(a) A l(k + A) l(k + A) = a(x) + a(a) 1, a(x) X ([2]) l(a) χ(a) H p (X, Ω(ξ)) p 1 ( η- ([8]) ([15]) X ΩX X K 1 (X) ([16]; K (X) ) ([13]) ([14] )
15 14 [1], [2] Kodaira,K.: Some results in the transcendental theory of algebraic variety, Ann. Math. 59(1954), [3] Hirzebruch, F.: Topological Method in Algebraic Geometry, Springer, 1978 [4] Gel fand,i.m.: On elliptic differential equations, Russian Mat. Surveys,15(196), (Uspekhi Mat. Nauka,15(196), ); Collected Works I, [5] Atiyah,M.F. Bott,R. Patodi, V.K.: On the heat equation and the index theorem, Invent.Math.19(1973), , 28(1975), [6] Booss-Bavnbek, B. Bleecker,D.: Topology and Analysis. The Atiyah-SInger index formula and gaugetheoretic phyisics, Springer, [7] Berline,N. Getzler,E. Vergne,M.: Heat Kernels and Dirac Operators, Springer, 24. [8] Merlose,R.B.: The Atiyah-Patodi-Singer Index Theorem, Peters [9] Borel,A. Serre,J.P.: Le théorèm de Riemann-Roch, Bull. Soc. Math. Fr.86(1958), [1] Berthlot,P. et al(ed).: Théories des intersections et théorème de Riemann-Roch, SGA6. Lect. Notes in Math. 225, Springer [11] Fujikawa,K.: Path-Integrals and Quantum Anomalies, Clarendon Press, 24. [12] Alvarez-Gaume,L.: Supersymmetry and the Atiyah-Singer index theorem, Commun. Math. Phys. 9(1983), [13] Higson, N.: The local index formula in noncommutative geometry. Contemporary developments in K-thoery, Karoubi, M. (ed.), ICTP Lect. Notes 15(23), [14] Perrot,D.: A Riemann-Roch Theorem for one-dimensional complex groupoids, Commun. Math. Phys.218(21), [15] Alvarez, O. Killingback,T.P. Mangano, K. Windey,P.: String theory and loop space index theorem, Commun. Math. Phys. 111(1987), 1-1. [16] Asada,A.: Characteristic classes of loop group bundles and generalized string classes. Colloq. SOc. Math. János Bolyai 56(1992), [17] Zariski,O.: Algebraic Surfaces, Springer 1995.
16 15 Bessel Airy Bessel Function and Airy Function 3 Kenji Seto Bessel Airy Airy Bessel Bessel Airy ±1/3 Bessel Airy George Biddell Airy( ) Airy Airy (1838) Airy 2 1 ±1/3 Bessel σ y = 2MPa, ρ = 8kg/m 3 g = 1m/s 2 h[m] ρgh h = σ y /ρg = 25m (buckling by gravity) 3 4 seto@pony.ocn.ne.jp
17 h x Young 2 ρ, E, A, I x θ(x), V (x) 2 V (x) = x sin θ(x )dx (2.2.1) θ x U 1, U 2 U 1 = EI 2A θ 2 x, U 2 = ρg x cos θ(x )dx (2.2.2) 5 U U = h ( U1 + U 2 ) Adx = EI 2 h θ 2 x dx + ρga h [ x cos θ(x )dx ] dx (2.2.3) 2 x x U = EI 2 h θ 2 x dx + ρga h (h x) cos θ(x)dx (2.2.4) U θ Euler-Lagrange EIθ xx + ρga(h x) sin θ = (2.2.5) θ sin θ = θ EIθ xx + ρga(h x)θ = (2.2.6) h x µ x/h x (2.2.7) µ = ρgah 3 EI (2.2.8) (2.2.6) θ xx + µ 2 (1 x)θ = (2.2.9) x θ z = 2 3 µ(1 x)3/2, θ(x) = z 1/3 f(z) (2.2.1) d 2 f dz df ( z dz ) 9z 2 f = (2.2.11) 5 θ x x U 1
18 17 ±1/3 Bessel C, D f(z) = CJ 1/3 (z) + DJ 1/3 (z) (2.2.12) θ(x) = [ ( 2µ ( 2µ 1 x CJ 1/3 3 (1 x)3/2) + DJ 1/3 3 (1 x)3/2)] (2.2.13) (2µ/3) 1/3 C, D (2.2.9) ±1/3 Bessel Airy Airy 4 Bessel (2.2.13) x d dz J ν(z) = ν z J ν(z) J ν+1 (z) = J ν 1 (z) ν z J ν(z) (2.2.14) θ x (x) = Cµ(1 x)j 2/3 ( 2µ 3 (1 x)3/2) Dµ(1 x)j 2/3 ( 2µ 3 (1 x)3/2) (2.2.15) x = x = 1 θ() =, θ x (1) = (2.2.16) θ x (1) = (2.2.15) 1 x = 1 2 D = θ() = (2.2.13) ( 2µ ) J 1/3 = (2.2.17) 3 2µ/3 1/3 Bessel 2µ i /3, (i = 1, 2, 3, ) 6 2µ i 3 = 1.866, 4.987, 8.166, , , 17.7 (2.2.18) θ(x) µ i θ(x, µ i ) θ(x, µ i ) = C 1 xj 1/3 ( 2µi 3 (1 x)3/2) (2.2.19) µ (2.2.8) ρ E ρ = 8kg/m 3, E = 2GPa (2.2.2) A I r A = πr 2, I = π 4 r4 (2.2.21)
19 18 A/I A I = 4 r 2 (2.2.22) r r d d d r A = π[r 2 (r d) 2 ] = 2πrd, I = π 4 [r4 (r d) 4 ] = πr 3 d (2.2.23) A/I A I = 2 r 2 (2.2.24) d (2.2.2) (2.2.22) (2.2.24) g = 9.8m/s 2 (2.2.8) µ µ 1 h r 1 r, h m r =1m h 8m 1m 25m 1 4 x, t θ θ(x, t) cos(ωt) ω ω i, (i = 1, 2, 3, ) µ µ ω i µ = µ 1 ω 1 = µ ω 1 µ = µ i, (i = 2, 3, ) ω i = µ ω i ω cos(ωt) cosh( ω t) ±1/3 Bessel
20 ρ T l x t, x V (x, t) ρ(x) ρ(x)v tt = T V xx (2.3.1) x 1 ( ρ(x) = ρ 1 x ) l (2.3.2) x = ρ = ρ x = l c l c = T ρ (2.3.3) τ = l/c (2.3.4) l, τ x, t, V x/l x, t/τ t, V/l V (2.3.5) (1 x)v tt = V xx (2.3.6) V (x, t) x, t ω V (x, t) = X(x) sin(ωt) (2.3.7) X xx + ω 2 (1 x)x = (2.3.8) ω µ, X θ (2.2.9) x = x = 1 X() =, X x (1) = (2.3.9) (2.2.9) (2.2.19) µ ω, θ X (2.2.17) (2.2.18) (2.2.19) ω ( 2ω ) J 1/3 = (2.3.1) 3 ω i, (i = 1, 2, 3, ) 6 2ω i 3 = 1.866, 4.987, 8.166, , , 17.7 (2.3.11) X(x) ω i X(x, ω i ) X(x, ω i ) = 1 xj 1/3 ( 2ωi 3 (1 x)3/2) (2.3.12) (2.2.19) C 1
21 ω, ω (2.3.12) ω i ω ω X(x, ω), X(x, ω ) (2.3.8) X xx (x, ω) + ω 2 (1 x)x(x, ω) =, X xx (x, ω ) + ω 2 (1 x)x(x, ω ) = (2.3.13) 1 X(x, ω ) 2 X(x, ω) d [ X(x, ω )X x (x, ω) X(x, ω)x x (x, ω ) ] + (ω 2 ω 2 )(1 x)x(x, ω)x(x, ω ) = (2.3.14) dx 1 (1 x)x(x, ω)x(x, ω )dx = 1 [ X(x, ω)xx ω 2 ω 2 (x, ω ) X(x, ω )X x (x, ω) ] 1 (2.3.15) ω, ω ω i, ω j (2.3.9) ω = ω i X x (1, ω) = ω ω i N i = 1 ω 2 ω i 2 X(, ω)x x(, ω i ) (2.3.16) = 1 [ ω X(, ω)] 2ω ω=ωi X x (, ω i ) = 1 [ ( 2ωi )] 2 J 2/3 (2.3.17) i (1 x)x(x, ω i )X(x, ω j )dx = N i 2 δ i,j (2.3.18) N i = 1 ( 2ωi ) J 2/3 3 3 (2.3.19) X(x, ω i )/N i 2 6 2
22 t = v (x) V (x, ) =, V t (x, ) = v (x) (2.3.2) V (x, t) X(x, ω i ) K i V (x, t) = K i X(x, ω i ) sin(ω i t) (2.3.21) i=1 t t = ω i K i X(x, ω i ) = v (x) (2.3.22) i=1 (1 x)x(x, ω j ) x (2.3.18) K j = 1 ω j N 2 j 1 (2.3.21) V (x, t) = i=1 (1 x)v (x)x(x, ω j )dx (2.3.23) 1 [ 1 ω i Ni 2 (1 x )v (x )X(x, ω i )dx ] X(x, ω i ) sin(ω i t) (2.3.24) 2.4 Schrödinger 2 Hermite 1 α WKB 1 Schrödinger Schrödinger 1 Schrödinger [ ħ2 d 2 ] 2m dx 2 + V (x) ψ = Eψ (2.4.1) V (x) V (x) = V () + V ()x (2.4.2) x x + V () E V () x (2.4.3)
23 22 [ ħ2 d 2 ] 2m dx 2 + V ()x ψ = (2.4.4) x = ± ψ = V () ħ 2 /(2mV ()) 3 l l 3 l x (2.4.4) ħ 2 2mV () (2.4.5) x/l x (2.4.6) ψ xx xψ = (2.4.7) x 2 x 1 (2.2.9) x > (2.2.1) z = 2 3 x3/2, ψ = z 1/3 g(z) (2.4.8) d 2 g dz dg ( z dz ) 9z 2 g = (2.4.9) ±1/3 Bessel (modified Bessel function) Bessel ( z ±1/3 I ±1/3 (z) e πi/6 J ±1/3 (iz) = 2) (z/2) 2n n! Γ (± n + 1) (2.4.1) ±1/3 z 2 n= K 1/3 (z) π 2 I 1/3 (z) I 1/3 (z) sin(π/3) (2.4.11) K 1/3 (z) z (2.4.9) g(z) = /3 π K 1/3(z) (2.4.12) (2.4.8) ψ(x) = Ai(x) (2.4.13) Ai(x) x > Airy Ai(x) 1 x ( 2 π 3 K 1/3 3 x3/2), x > (2.4.14) x (entire function)
24 23 (2.4.1) (2.4.11) 2 Ai(x) = 1 3 2/3 Γ ( 2 3 ) 1 3 1/3 Γ ( 1 + (2.4.15) 3 )x I 1/3 I 1/3 x Bessel I ν (z) Ai(x) e 2 3 x3/2 2 πx 1/4 (2.4.16) x > (2.4.2) (2.4.3) E E < V (x) x > x < (2.4.8) z = 2 3 ( x)3/2, ψ = z 1/3 g(z) (2.4.17) d 2 g dz dg ( z dz ) 9z 2 g = (2.4.18) ±1/3 Bessel J ±1/3 (z) C, D (2.4.17) ψ g(z) = CJ 1/3 (z) + DJ 1/3 (z) (2.4.19) ψ(x) = (2/3) 1/3 [ ( 2 ( 2 x CJ 1/3 3 ( x)3/2) + DJ 1/3 3 ( x)3/2)] (2.4.2) ψ x = Bessel ( z ν J ν (z) = 2) ( 1) n (z/2) 2n n!γ (ν + n + 1) n= (2.4.2) x = (2.4.21) ( 2 xj 1/3 3 ( x)3/2) 1 = 3 1/3 Γ ( 2 3 ) +, ( 2 xj1/3 3 ( x)3/2) 1 = 3 2/3 Γ ( 1 + (2.4.22) 3 )x (2.4.2) x > Ai(x) (2.4.15) 1 C = D = 1 3 ( 2 3 ) 1/3 (2.4.23) (2.4.2) ψ(x) = Ai(x) (2.4.24) x < Airy x [ ( 2 ( 2 Ai(x) J 1/3 3 3 ( x)3/2) + J 1/3 3 ( x)3/2)], x < (2.4.25) (2.4.14) x > Airy x < Airy 2 Taylor 1
25 24 (2.4.14) Airy K 1/3 (z) (2.4.11), I ±1/3 (z) (2.4.1) Ai(x) = 1 3 2/3 n= x 3n n! Γ ( n)32n 1 3 4/3 n= x 3n+1 n! Γ ( n)32n (2.4.26) (2.4.25) Airy (2.4.21) Airy x (2.4.7) ψ(x) = C n x 3n + D n x 3n+1 (2.4.27) n= x C n, D n n= 1 C = 3 2/3 Γ ( 2 3 ), D 1 ( 1 ) = 3 4/3 Γ ( 4 3 ) = 3 1/3 Γ ( 1 3 ) x Bessel J ν (z) (2.4.28) Ai(x) sin ( 2 3 ( x)3/ π) π( x) 1/4 (2.4.29) x Airy x 2 Airy ; Bi(x) 3 Airy Ai(x) (2.4.26) x 5 x 14 (2.4.29) 14 x 5 (2.4.26) 5 x 25 (2.4.16) 3 Airy Ai(x) Airy Airy Bessel Airy Ai(x) = 1 e 1 3 t3 xt dt (2.4.3) 2πi t π/3 +π/3 C C Me (π/3)i ( Me (π/3)i Me (π/3)i ) = lim, or = lim M Me (π/3)i M (2.4.31)
26 25 e 1 3 t3 e 1 3 t 3 x z Airy (2.4.7) ( d 2 ) dx 2 x Ai(x) = 1 2πi C (t 2 x)e 1 3 t3 xt dt = lim M 1 2πi e 1 3 t 3 xt Me(π/3)i Me (π/3)i = (2.4.32) x (2.4.26) e xt Taylor Ai(x) = 1 e 1 t3( ( xt) k ) 3 dt = 1 ( 1) k ( 2πi k! 2πi k! C k= k= C ) e 1 3 t3 t k dt x k (2.4.33) (2.4.31) t e (π/3)i t t e (π/3)i t = 1 π ( 1) k sin ( 1 k! 3 (k + 1)π)( k= ) e 1 3 t3 t k dt x k (2.4.34) 1 3 t3 = ξ gamma e 1 3 t3 t k dt = (k 2) k! gamma 3 (2.4.34) e ξ ξ 1 3 (k 2) dξ = (k 2) Γ ( 1 3 (k + 1)) (2.4.35) k! = Γ (k + 1) = 3k+1 2π 3 Γ ( 1 3 (k + 1)) Γ ( 1 3 (k + 2)) Γ ( 1 3 (k + 3)) (2.4.36) Ai(x) = 2 3 k= ( 1) k (2k+5) sin ( 1 3 (k + 1)π) Γ ( 1 3 (k + 2)) Γ ( 1 3 (k + 3)) xk (2.4.37) n k = 3n k = 3n + 1 k = 3n k = 3n + 2 sin 2 k = 3n 1 3 2/3 n= x 3n n! Γ ( n)32n (2.4.26) 1 k = 3n /3 n= x 3n+1 n! Γ ( n)32n (2.4.26) 2 (2.4.26) Airy Ai(x) (2.4.3) t π/3 π/3 t 3 π/2 π/6 π/6 π/2 π/2 π/2 (2.4.3) t it Ai(x) = 1 2π e i( 1 3 t3 +xt) dt = 1 2π e i( 13 t3 +xt) dt = 1 π ( t 3 ) cos 3 + xt dt (2.4.38) t ( t 3 ) cos 3 + xt dt = 1 ( t 3 ) t 2 + x sin 3 + xt 2t ( t 3 ) + (t 2 + x) 2 sin 3 + xt dt (2.4.39)
27 26 (2.4.7) ( d 2 dx 2 x )Ai(x) = 1 π (t 2 + x) cos ( t xt ) dt = 1 π sin ( t 3 ) 3 + xt (2.4.4) t = t (2.4.7) (2.4.38) x 1 (2.4.15) (2.4.38) Airy Ai(x) (2.4.33) (2.4.38) x e ixt t x Airy Airy (2.4.6) (2.4.3) x (2.4.3) x ξ = V () E lv () (2.4.41) ξ x ψ = Ai(x + ξ) Kronecker-δ Dirac-δ (2.4.38) 2 ξ, ξ Airy Ai(x + ξ)ai(x + ξ )dx = δ(ξ ξ ) (2.4.42) (2.4.38) Ai(x + ξ)ai(x + ξ )dx = 1 4π 2 e i[ 1 3 (t3 +t 3 )+(x+ξ)t+(x+ξ )t ] dtdt dx (2.4.43) x e i(t+t )x dx = lim M e i(t+t )x i(t + t ) M M sin ( (t + t )M ) = 2π lim M π(t + t = 2πδ(t + t ) (2.4.44) ) (2.4.43) Ai(x + ξ)ai(x + ξ )dx = 1 e i(ξ ξ )t dt (2.4.45) 2π (2.4.44) 1 e i(ξ ξ )t sin ( (ξ ξ )M ) dt = lim 2π M π(ξ ξ = δ(ξ ξ ) (2.4.46) )
28 27 (2.4.42) Ai(x + ξ)ai(x + ξ)dξ = δ(x x ) (2.4.47) (, ) f(x) g(ξ) = f(x) = Ai(ξ + x)f(x)dx (2.4.48) Ai(x + ξ)g(ξ)dξ (2.4.49) x + ξ (, ) Fourier g(ξ) = 1 2π [, ) Fourier-Bessel g(ξ) = e iξx f(x)dx, f(x) = 1 2π J ν (ξx)f(x)xdx, f(x) = e ixξ g(ξ)dξ (2.4.5) J ν (xξ)g(ξ)ξdξ, ν > 1 (2.4.51) xξ x + ξ Bessel Airy ( ) p.181, p Airy (2.4.7) Airy (2.4.3) x t 4 ψ xxxx + αψ xx xψ =, ψ(x) e 1 5 t5 +α 1 3 t3 xt dt D t π/5 π/5 Airy WKB 3 3 (213 3 ) Bessel Airy Ai(x) D
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n=1 1 n 2 = π = π f(z) f(z) 2 f(z) = u(z) + iv(z) *1 f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x
n= n 2 = π2 6 3 2 28 + 4 + 9 + = π2 6 2 f(z) f(z) 2 f(z) = u(z) + iv(z) * f (z) u(x, y), v(x, y) f(z) f (z) = f/ x u x = v y, u y = v x f x = i f y * u, v 3 3. 3 f(t) = u(t) + v(t) [, b] f(t)dt = u(t)dt
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
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
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x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin
2 2.1 F (t) 2.1.1 mẍ + kx = F (t). m ẍ + ω 2 x = F (t)/m ω = k/m. 1 : (ẋ, x) x = A sin ωt, ẋ = Aω cos ωt 1 2-1 x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ
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Junji OHTSUBO 2012 FFT FFT SN sin cos x v ψ(x,t) = f (x vt) (1.1) t=0 (1.1) ψ(x,t) = A 0 cos{k(x vt) + φ} = A 0 cos(kx ωt + φ) (1.2) A 0 v=ω/k φ ω k 1.3 (1.2) (1.2) (1.2) (1.1) 1.1 c c = a + ib, a = Re[c],
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f f x, y, u, v, r, θ r > = x + iy, f = u + iv C γ D f f D f f, Rm,. = x + iy = re iθ = r cos θ + i sin θ = x iy = re iθ = r cos θ i sin θ x = + = Re, y = = Im i r = = = x + y θ = arg = arctan y x e i =
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grad φ(p ) φ P grad φ(p ) p P p φ P p l t φ l t = 0 g (0) g (0) (31) grad φ(p ) p grad φ φ (P, φ(p )) xy (x, y) = (ξ(t), η(t)) ( )
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25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 3.1.2 σ τ 2 2 ux, t) = ux, t) 3.1) 2 x2 ux, t) σ τ 2 u/ 2 m p E E = p2 3.2) E ν ω E = hν = hω. 3.3) k p k = p h. 3.4) 26 3 hω = E = p2 = h2 k 2 ψkx ωt) ψ 3.5) h
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25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 σ τ x u u x t ux, t) u 3.1 t x P ux, t) Q θ P Q Δx x + Δx Q P ux + Δx, t) Q θ P u+δu x u x σ τ P x) Q x+δx) P Q x 3.1: θ P θ Q P Q equation of motion P τ Q τ σδx
量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
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,
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
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
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
http://www.ike-dyn.ritsumei.ac.jp/ hyoo/wave.html 1 1, 5 3 1.1 1..................................... 3 1.2 5.1................................... 4 1.3.......................... 5 1.4 5.2, 5.3....................
ma22-9 u ( v w) = u v w sin θê = v w sin θ u cos φ = = 2.3 ( a b) ( c d) = ( a c)( b d) ( a d)( b c) ( a b) ( c d) = (a 2 b 3 a 3 b 2 )(c 2 d 3 c 3 d
A 2. x F (t) =f sin ωt x(0) = ẋ(0) = 0 ω θ sin θ θ 3! θ3 v = f mω cos ωt x = f mω (t sin ωt) ω t 0 = f ( cos ωt) mω x ma2-2 t ω x f (t mω ω (ωt ) 6 (ωt)3 = f 6m ωt3 2.2 u ( v w) = v ( w u) = w ( u v) ma22-9