2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W
|
|
|
- つねたけ なみこし
- 5 years ago
- Views:
Transcription
1 SGC -83
2 2 2 Belavin Polyakov Zamolodchikov (BPZ) 1984 [13] 2 BPZ BPZ Virasoro [16][18] [20], [30], [47] [1][6] [8][10], [11], [12] Affine [6],GKO [2] W [14] c = 1 CFT [8] Rational CFT [15], [56] CFT [4] [3] [7]
3 Ward Primary quasi primary primary Ward Virasoro Virasoro CFT Hilbert BPZ
4 OPE Virasoro Virasoro Descendant OPE Bootstrap Crossing Virasoro Fusion Kac monodromy OPE Kac CFT CFT Virasoro theta Poisson theta Virasoro ADE iii
5 6.5 Verlinde Boundary CFT BCFT Verlinde iv
6 1 2 d d R d x μ =(x 0,,x d 1 ) R d g μν = 0 1 (1.1) ds 2 = g μν dx μ dx ν =(dx 0 ) 2 + +(dx d 1 ) 2 (1.2) x μ x μ g μν g μν = xρ x μ x σ x ν g ρσ (1.3) x μ 1 x μ x μ = Λ μ ν x ν (1.4) d d Λ μ ρ g μν Λ μ ρ Λ ν σ = g ρσ (1.5) 2 2 (x 1 x 2 ) 2
7 2 Ward d N φ(x) =(φ a (x)) (a =1,,N) x μ x μ = Λ μ ν x ν (2.1) φ a (x) φ a (x) φ a (x )=D(Λ) a b φ b (x) (2.2)
8 3 2 d 2 2 Ward (OPE) 3.1 Primary quasi primary 2 (x 0,x 1 ) z = x 0 +ix 1, z = x 0 ix 1 ds 2 = dzd z (z, z) SL(2, C) SL(2, C) z z = az + b cz + d, z z = ā z + b c z + d (3.1) φ(z, z) (3.1) ( ) dz φ (z, z h ( ) d z h )= φ(z, z) (3.2) dz d z φ(z, z) (conformal weight) (h, h) quasi primary h = h (3.2) Δ = h + h quasi primary h h 2 z = e iθ z, z = e iθ z quasi primary φ(z, z)
9 4 Virasoro Virasoro (3.60) primary descendant primary OPE OPE 4 crossing 4.1 Virasoro φ(z, z) (h, h) primary φ(0, 0) SL(2, C) 0 φ h, h = φ(0, 0) 0 (4.1) Virasoro L n, L n (n Z) Virasoro L n, L n Virasoro L n primary φ(w, w) OPE dz [L n,φ(w, w)] = w 2πi zn+1 T (z) φ(w, w) { } dz hφ(w, w) φ(w, w) = 2πi zn+1 + (z w) 2 z w + w = h(n +1)w n φ(w, w)+w n+1 φ(w, w) (4.2) w, w 0 n>0 [L n,φ(0, 0)] = 0 (4.3) n =0 [L 0,φ(0, 0)] = hφ(0, 0) (4.4)
10 5 Virasoro primary 2 CFT 2 CFT primary monodromy OPE Dotsenko Fateev 5.1 Virasoro A primary φ i [φ i ] A = i [φ i ] φ i (h i, h i ) [φ i ] Virasoro (L n ) ( L n ) V hi, V hi [φ i ]=V hi V hi Hilbert Verma V h h descendant L n1 L nk h (n 1 n k > 0) h L n h =0(n>0), L 0 h =0 Verma V h Virasoro Verma V h h + N (N) L n h + N =0, n > 0, (5.1) L 0 h + N =(h + N) h + N (5.2) h + N L n V h+n Virasoro h + N N (null vector) (singular vector)
11 CFT L (cylinder) CFT t x φ(t, x) φ(t, x + L) =φ(t, x) (t, x) <t<, 0 x L (3.3 ) 6.1 z w w = t + ix= L log z (6.1) 2π CFT T cyl (w) z t w o L x 6.1 : z w
12 7 Boundary CFT Boundary CFT BCFT D CFT CFT C CFT primary Cardy [58], [59] [60], [61] H = {z = x + iy C ; y>0} (7.1) H R H (bulk) H primary 0 (h, h) primary φ(z, z) 2 φ(z 1, z 1 ) φ(z 2, z 2 ) 7.1 C PSL(2, C)
13 [1] C. Itzykson and J. M. Drouffe, Statistical Field Theory : Volume 2, Strong Coupling, Monte Carlo Methods, Conformal Field Theory and Random Systems, Cambridge Univ. Press, [2] P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Springer Verlag, [3] R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory, Lect. Notes Phys. 779 (2009) 1. [4] G. Mussardo, Statistical Field Theory : An Introduction to Exactly Solved Models in Statistical Physics, Oxford Univ. Press, [5] [6], [7] J. Polchinski, String Theory, Vol. 1, 2, Cambrdige Univ. Press, : [8] P. H. Ginsparg, Applied Conformal Field Theory, arxiv:hep-th/ [9] A. B. Zamolodchikov, Al. B. Zamolodchikov, Conformal Field Theory and Critical Phenomena in Two-dimensional Systems, Physics Reviews Vol. 10.4, Harwood, [10] J. Cardy, arxiv: [cond-mat.stat-mech]. [11] C. Itzykson, H. Saleur and J. B. Zuber, Conformal Invariance and Applications to Statistical Mechanics, World Scientific, [12] P. Goddard and D. I. Olive, Adv. Ser. Math. Phys. 3 (1988) 1. [13] A. A. Belavin, A. M. Polyakov and A. B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [14] P. Bouwknegt and K. Schoutens, Phys. Rept. 223 (1993) 183 [arxiv:hep-th/ ]. [15] A. Tsuchiya and Y. Kanie, Adv. Stud. Pure Math. 16 (1988) 297 [Erratum ibid. 19 (1989) 675]. [16] I1989. [17] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Perseus Books Publ., [18] [19] [20] R.V. J.W [21] A. M. Polyakov, JETP Lett. 12 (1970) 381 [Pisma Zh. Eksp. Teor. Fiz. 12 (1970) 538]. [22] L. P. Kadanoff, Phys. Rev. Lett. 23 (1969) [23] K. G. Wilson, Phys. Rev. 179 (1969) 1499.
14 [24] A. M. Polyakov, Sov. Phys. JETP 30 (1970) 151. [25] F. A. Bais, P. Bouwknegt, M. Surridge and K. Schoutens, Nucl. Phys. B 304 (1988) 348. [26] L. Benoit and Y. Saint-Aubin, Phys. Lett. B 215 (1988) 517. [27] M. Bauer, P. Di Francesco, C. Itzykson and J. B. Zuber, Nucl. Phys. B 362 (1991) 515. [28] V. G. Kac, Lect. Notes. Phys. 94 (1979) 441. [29] B. L. Feigin and D. B. Fuchs, Funct. Anal. and Appl. 16 (1984) 114, 17 (1983) 241. [30] [31] D. Friedan, Z. a. Qiu and S. H. Shenker, Phys. Rev. Lett. 52 (1984) [32] D. Friedan, S. H. Shenker and Z. a. Qiu, Commun. Math. Phys. 107 (1986) 535. [33] D. Friedan, Z. a. Qiu and S. H. Shenker, Conformal Invariance, Unitarity and Two Dimensional Critical Exponents in Vertex Operators in Mathematical Physics, Springer Verlag, [34] J. L. Cardy, Phys. Rev. Lett. 54 (1985) [35] P. Goddard, A. Kent and D. I. Olive, Commun. Math. Phys. 103 (1986) 105. [36] V. S. Dotsenko and V. A. Fateev, Nucl. Phys. B 240 (1984) 312. [37] V. S. Dotsenko and V. A. Fateev, Nucl. Phys. B 251 (1985) 691. [38] V. S. Dotsenko and V. A. Fateev, Phys. Lett. B 154 (1985) 291. [39] V. S. Dotsenko, Adv. Stud. Pure. Math. 16 (1988) 123. [40] B. L. Feigin and D..B. Fuchs, Representation of the Virasoro algebra, Adv. Stud. Contemp. Math. 7 (1990) 465. [41] M. Kato and S. Matsuda, Adv. Stud. Pure Math. 16 (1988) 205. [42] A. Tsuchiya and Y. Kanie, Rubl. RIMS 22 (1986) 259. [43] G. Felder, Nucl. Phys. B 317 (1989) 215 [Erratum-ibid. B 324 (1989) 548]. [44] SGC [45] SGC [46] A. Rocha-Caridi, Vacuum Vector Representations of the Virasoro Algebra, in Vertex Operators in Mathematics and Physics, MSRI Publications No.3, Springer, [47] A. R [48] V. G. Kac, Infinite-Dimensional Lie Algebra, 3rd ed., Cambridge Univ. Press, [49] [50] D. Mumford, Tata Lectures on Theta I, Birkhäuser, [51] A. Cappelli, C. Itzykson and J. B. Zuber, Nucl. Phys. B 280 (1987) 445. [52] A. Kato, Mod. Phys. Lett. A 2 (1987) 585. [53] A. Cappelli, C. Itzykson and J. B. Zuber, Commun. Math. Phys. 113 (1987) 1. [54] E. P. Verlinde, Nucl. Phys. B 300 (1988) 360. [55] G. W. Moore and N. Seiberg, Phys. Lett. B 212 (1988)
15 [56] G. W. Moore and N. Seiberg, Commun. Math. Phys. 123 (1989) 177. [57] J. L. Cardy, Nucl. Phys. B 270 (1986) 186. [58] J. L. Cardy, Nucl. Phys. B 240 (1984) 514. [59] J. L. Cardy, Nucl. Phys. B 275 (1986) 200. [60] J. L. Cardy, arxiv:hep-th/ [61] J. L. Cardy, arxiv:math-ph/ [62] N. Ishibashi, Mod. Phys. Lett. A 4 (1989) 251. [63] J. L. Cardy, Nucl. Phys. B 324 (1989) 581. [64] J. L. Cardy and D. C. Lewellen, Phys. Lett. B 259 (1991) 274. [65] R. E. Behrend, P. A. Pearce, V. B. Petkova and J. B. Zuber, Nucl. Phys. B 570 (2000) 525 [arxiv:hep-th/ ]. [66] I. Runkel, Nucl. Phys. B 549 (1999) 563 [arxiv:hep-th/ ]. 144
16 (), 38 (Ishibashi state), 138, 17, 79 (annulus), 135 (operator product expansion), 33 (boundary operator), 133 (boundary state), 137 (boundary scaling dimension), 133 Virasoro (boundary Virasoro operator), 132 (boundary changing operator), 133 (conformal weight), 26 Killing, 3, 6 (conformal tower), 56 (conformal field theory), 27 (conformal block), 64, 3 (conformal class), 56 Ward, 20, 31, 130, 10 (Coulomb Gas), 44, 43 (highest weight state), 54 (character), 103 (screening operator), 96 (period), 103, 49, 44 (degenerate representation), 66, 43 (scaling dimension), 14, 102, 17 (spin), 102, 79 (normal ordered product), 45, 2 (central charge), 37, 120, 9, 73, 72 (vertex operator), 47, 2, 1 (radial ordering), 35, 35 (torus), 102 (singular vector), 65 (special conformal transformation), 6 2 (double contraction), 47 2 (doubling trick), 131 (null field), 66
17 (null vector), 65 (background charge), 94 (bulk), 126 (bulk operator), 133, 81, 7 (cross ratio), 25, 81 (partition function), 18, 102 (minimal model), 78 S, 119 (modular parameter), 103, 2, 124, 54, 2 Feynman (Feynman propagator), 19 Fuchs, 71 fusion (fusion algebra), 74 fusion (fusion rule), 74 Green, 31 Ising, 79 Jacobi 3, 106 Jacobi theta, 107 Kac, 82 Kac, 69 monodromy, 85 Noether, 16 OPE, 33 Pochhammer, 98 Poisson, 113 primary, 28 ADE, 120 bootstrap, 59 BPZ, 43 Cardy, 124 Cardy, 139 Cardy, 139 CFT, 27 crossing, 64 Dedekind eta, 109 descendant, 54 descendant, 56 dilatation, 17 quasi primary, 20 Schwarzian, 38 secondary, 56 Shapovarov, 81 theta, 110 Verlinde, 125 Verma (Verma module), 55 Virasoro, 39 Wick, 19 Yang Lee,
18 J. SGC ISBN TEL.(03) FAX.(03) sk@saiensu.co.jp C TEL.(03) ()
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
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](/thumbs/94/118614691.jpg "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]
『共形場理論』
T (z) SL(2, C) T (z) SU(2) S 1 /Z 2 SU(2) (ŜU(2) k ŜU(2) 1)/ŜU(2) k+1 ŜU(2)/Û(1) G H N =1 N =1 N =1 N =1 N =2 N =2 N =2 N =2 ĉ>1 N =2 N =2 N =4 N =4 1 2 2 z=x 1 +ix 2 z f(z) f(z) 1 1 4 4 N =4 1 = = 1.3
(τ τ ) τ, σ ( ) w = τ iσ, w = τ + iσ (w ) w, w ( ) τ, σ τ = (w + w), σ = i (w w) w, w w = τ w τ + σ w σ = τ + i σ w = τ w τ + σ w σ = τ i σ g ab w, w
S = 4π dτ dσ gg ij i X µ j X ν η µν η µν g ij g ij = g ij = ( 0 0 ) τ, σ (+, +) τ τ = iτ ds ds = dτ + dσ ds = dτ + dσ δ ij ( ) a =, a = τ b = σ g ij δ ab g g ( +, +,... ) S = 4π S = 4π ( i) = i 4π dτ dσ
R R P N (C) 7 C Riemann R K ( ) C R C K 8 (R ) R C K 9 Riemann /C /C Riemann 10 C k 11 k C/k 12 Riemann k Riemann C/k k(c)/k R k F q Riemann 15
(Gen KUROKI) 1 1 : Riemann Spec Z 2? 3 : 4 2 Riemann Riemann Riemann 1 C 5 Riemann Riemann R compact R K C ( C(x) ) K C(R) Riemann R 6 (E-mail address: kuroki@math.tohoku.ac.jp) 1 1 ( 5 ) 2 ( Q ) Spec
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](/thumbs/94/118766307.jpg "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
φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)
φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x
,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising
![,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising](/thumbs/94/118770363.jpg ",, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising")
,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising Model 1 Ising 1 Ising Model N Ising (σ i = ±1) (Free
Introduction SFT Tachyon condensation in SFT SFT ( ) at 1 / 38
( ) 2011 5 14 at 1 / 38 Introduction? = String Field Theory = SFT 2 / 38 String Field : ϕ(x, t) x ϕ x / ( ) X ( σ) (string field): Φ[X(σ), t] X(σ) Φ (Φ X(σ) ) X(σ) & / 3 / 38 SFT with Lorentz & Gauge Invariance
統計力学模型とＳＬＥ 中央大学理工学部 香取眞理
1 Loewner 0. 1. 1.1 (loop erased RW : LERW) (self avoiding walk : SAW) (critical percolation model) Ising (critical Ising model) 1.2 Markov (conformal cov./inv.) Markov (domain Markov property) 1.3 LERW
ʪ¼Á¤Î¥È¥Ý¥í¥¸¥«¥ë¸½¾Ý (2016ǯ¥Î¡¼¥Ù¥ë¾Þ¤Ë´ØÏ¢¤·¤Æ)
(2016 ) Dept. of Phys., Kyushu Univ. 2017 8 10 1 / 59 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER 2 / 59 ( ) ( ) (Dirac, t Hooft-Polyakov)
Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib
![Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib](/thumbs/78/77523767.jpg "Donaldson Seiberg-Witten [GNY] f U U C 1 f(z)dz = Res f(a) 2πi C a U U α = f(z)dz dα = 0 U f U U P 1 α 0 a P 1 Res a α = 0. P 1 Donaldson Seib")
( ) Donaldson Seiberg-Witten Witten Göttsche [GNY] L. Göttsche, H. Nakajima and K. Yoshioka, Donaldson = Seiberg-Witten from Mochizuki s formula and instanton counting, Publ. of RIMS, to appear Donaldson
Anderson ( ) Anderson / 14
Anderson 2008 12 ( ) Anderson 2008 12 1 / 14 Anderson ( ) Anderson 2008 12 2 / 14 Anderson P.W.Anderson 1958 ( ) Anderson 2008 12 3 / 14 Anderson tight binding Anderson tight binding Z d u (x) = V i u
* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *
* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *1 2004 1 1 ( ) ( ) 1.1 140 MeV 1.2 ( ) ( ) 1.3 2.6 10 8 s 7.6 10 17 s? Λ 2.5 10 10 s 6 10 24 s 1.4 ( m
2016 ǯ¥Î¡¼¥Ù¥ëʪÍý³Ø¾Þ²òÀ⥻¥ß¥Ê¡¼ Kosterlitz-Thouless ž°Ü¤È Haldane ͽÁÛ
2016 Kosterlitz-Thouless Haldane Dept. of Phys., Kyushu Univ. 2016 11 29 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER ( ) ( ) (Dirac,
反D中間子と核子のエキゾチックな 束縛状態と散乱状態の解析
.... D 1 in collaboration with 1, 2, 1 RCNP 1, KEK 2 . Exotic hadron qqq q q Θ + Λ(1405) etc. uudd s? KN quasi-bound state? . D(B)-N bound state { { D D0 ( cu) B = D ( cd), B = + ( bu) B 0 ( bd) D(B)-N
YITP50.dvi
1 70 80 90 50 2 3 3 84 first revolution 4 94 second revolution 5 6 2 1: 1 3 consistent 1-loop Feynman 1-loop Feynman loop loop loop Feynman 2 3 2: 1-loop Feynman loop 3 cycle 4 = 3: 4: 4 cycle loop Feynman
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,
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
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
1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr ψ σ + (r)ψ +
![1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr ψ σ + (r)ψ +](/thumbs/93/112757023.jpg "1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr ψ σ + (r)ψ +")
1 1.1 21 11 22 10 33 cm 10 29 cm 60 6 8 10 12 cm 1cm 1 1.2 2 1 1 1: Sheldon L. Glashow (Ouroboros) [1] 1 v(r) u(r, r ) ( e 2 / r r ) H 2 [2] H = ( dr ψ σ + (r) 1 2 ) σ 2m r 2 + v(r) µ ψ σ (r) + 1 dr dr
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](/thumbs/91/105692760.jpg "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...................................
橡超弦理論はブラックホールの謎を解けるか？
1999 3 (Can String Theory Solve the Puzzles of Black Holes?) 305-0801 1-1 makoto.natsuume@kek.jp D-brane 1 Schwarzschild 60 80 2 [1] 1 1 1 2 2 [2] 25 2.2 2 2.1 [7,8] Schwarzschild 2GM/c 2 Schwarzschild
, 0 = U 1 (g) U 0 (g) U 1 (g)..., U(g) = p U p (g) U p (g)u q (g) U p+q (g), [U p (g), U q (g)] U p+q 1 (g). U(g) PBW,. Associated graded algebra gr U
![, 0 = U 1 (g) U 0 (g) U 1 (g)..., U(g) = p U p (g) U p (g)u q (g) U p+q (g), [U p (g), U q (g)] U p+q 1 (g). U(g) PBW,. Associated graded algebra gr U](/thumbs/92/109646782.jpg ", 0 = U 1 (g) U 0 (g) U 1 (g)..., U(g) = p U p (g) U p (g)u q (g) U p+q (g), [U p (g), U q (g)] U p+q 1 (g). U(g) PBW,. Associated graded algebra gr U")
W ( ) 1. ( )W Kac-Moody Virasoro,,,,, 4, Langlands.,, W., W, W ([A1, A2, A3, A7]). Premet[Pre] W ( )W, Kostant[Kos]. W Slodowy, primitive ideal. Premet Losev[Los2]. primitive ideal. W. ( )W Losev. Kac-Moody
(a) (b) (c) (d) 1: (a) (b) (c) (d) (a) (b) (c) 2: (a) (b) (c) 1(b) [1 10] 1 degree k n(k) walk path 4
![(a) (b) (c) (d) 1: (a) (b) (c) (d) (a) (b) (c) 2: (a) (b) (c) 1(b) [1 10] 1 degree k n(k) walk path 4](/thumbs/91/105665060.jpg "(a) (b) (c) (d) 1: (a) (b) (c) (d) (a) (b) (c) 2: (a) (b) (c) 1(b) [1 10] 1 degree k n(k) walk path 4")
1 vertex edge 1(a) 1(b) 1(c) 1(d) 2 (a) (b) (c) (d) 1: (a) (b) (c) (d) 1 2 6 1 2 6 1 2 6 3 5 3 5 3 5 4 4 (a) (b) (c) 2: (a) (b) (c) 1(b) [1 10] 1 degree k n(k) walk path 4 1: Zachary [11] [12] [13] World-Wide
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
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
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)
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................................
sakigake1.dvi
(Zin ARAI) arai@cris.hokudai.ac.jp http://www.cris.hokudai.ac.jp/arai/ 1 dynamical systems ( mechanics ) dynamical systems 3 G X Ψ:G X X, (g, x) Ψ(g, x) =:Ψ g (x) Ψ id (x) =x, Ψ gh (x) =Ψ h (Ψ g (x)) (
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
Kaluza-Klein(KK) SO(11) KK 1 2 1
Maskawa Institute, Kyoto Sangyo University Naoki Yamatsu 2016 4 12 ( ) @ Kaluza-Klein(KK) SO(11) KK 1 2 1 1. 2. 3. 4. 2 1. 標準理論 物質場 ( フェルミオン ) スカラー ゲージ場 クォーク ヒッグス u d s b ν c レプトン ν t ν e μ τ e μ τ e h
Seiberg Witten 1994 N = 2 SU(2) Yang-Mills 1 1 3 2 5 2.1..................... 5 2.2.............. 8 2.3................................. 9 3 N = 2 Yang-Mills 11 3.1............................... 11 3.2
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
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
Gelfand 3 L 2 () ix M : ϕ(x) ixϕ(x) M : σ(m) = i (λ M) λ (L 2 () ) ( 0 ) L 2 () ϕ, ψ L 2 () ((λ M) ϕ, ψ) ((λ M) ϕ, ψ) = λ ix ϕ(x)ψ(x)dx. λ /(λ ix) ϕ,
A spectral theory of linear operators on Gelfand triplets MI (Institute of Mathematics for Industry, Kyushu University) (Hayato CHIBA) chiba@imi.kyushu-u.ac.jp Dec 2, 20 du dt = Tu. (.) u X T X X T 0 X
, 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
main.dvi
SGC - 48 208X Y Z Z 2006 1930 β Z 2006! 1 2 3 Z 1930 SGC -12, 2001 5 6 http://www.saiensu.co.jp/support.htm http://www.shinshu-u.ac.jp/ haru/ xy.z :-P 3 4 2006 3 ii 1 1 1.1... 1 1.2 1930... 1 1.3 1930...
. 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](/thumbs/91/105488587.jpg ". 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
163 KdV KP Lax pair L, B L L L 1/2 W 1 LW = ( / x W t 1, t 2, t 3, ψ t n ψ/ t n = B nψ (KdV B n = L n/2 KP B n = L n KdV KP Lax W Lax τ KP L ψ τ τ Cha
63 KdV KP Lax pair L, B L L L / W LW / x W t, t, t 3, ψ t n / B nψ KdV B n L n/ KP B n L n KdV KP Lax W Lax τ KP L ψ τ τ Chapter 7 An Introduction to the Sato Theory Masayui OIKAWA, Faculty of Engneering,
& 3 3 ' ' (., (Pixel), (Light Intensity) (Random Variable). (Joint Probability). V., V = {,,, V }. i x i x = (x, x,, x V ) T. x i i (State Variable),
.... Deeping and Expansion of Large-Scale Random Fields and Probabilistic Image Processing Kazuyuki Tanaka The mathematical frameworks of probabilistic image processing are formulated by means of Markov
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,
Tohoku Mathematical Publications No.3
Tohoku Mathematical Publications Editors: Shigetoshi Bando Ryoshi Hotta Satoru Igari Masanori Ishida Junji Kato Katsuei Kenmotsu Kyûya Masuda Yasuo Morita Tetsuo Nakamura Seiki Nishikawa Tadao Oda Norio
·«¤ê¤³¤ß·²¤È¥ß¥ì¥Ë¥¢¥àÌäÂê
.. 1 10-11 Nov., 2016 1 email:keiichi.r.ito@gmail.com, ito@kurims.kyoto-u.ac.jp ( ) 10-11 Nov., 2016 1 / 45 Clay Institute.1 Construction of 4D YM Field Theory (Jaffe, Witten) Jaffe, Balaban (1980).2 Solution
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,
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,
26 3 ( 29 5 ) D 3 1 (KEK) 1 Minkowski (conformal field theory, CFT) Minkowski Minkowski Hamilton Hermite Euclid
26 3 ( 29 5 ) D 3 (KEK) http://research.kek.jp/people/hamada/ Minkowski (conformal field theory, CFT) Minkowski Minkowski Hamilton Hermite Euclid Euclid Minkowski () Euclid Minkowski Euclid D 4 (, 206)
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
[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)
![[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)](/thumbs/47/23763429.jpg "[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)")
. D............................................... : E = κ ............................................ 3.................................................
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.,
[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin Clifford Spin 10 A 12 B 17 1 Cliffo
![[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin Clifford Spin 10 A 12 B 17 1 Cliffo](/thumbs/101/149329436.jpg "[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin Clifford Spin 10 A 12 B 17 1 Cliffo")
[1] convention Minkovski i Polchinski [2] 1 Clifford Spin 1 2 Euclid Clifford 2 3 Euclid Spin 6 4 Euclid Pin + 8 5 Clifford Spin 10 A 12 B 17 1 Clifford Spin D Euclid Clifford Γ µ, µ = 1,, D {Γ µ, Γ ν
(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](/thumbs/97/133596579.jpg "(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)
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](/thumbs/91/106734137.jpg "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
1: (Emmy Noether; ) (Feynman) [3] [4] {C i } A {C i } (A A )C i = 0 [5] 2
![1: (Emmy Noether; ) (Feynman) [3] [4] {C i } A {C i } (A A )C i = 0 [5] 2](/thumbs/91/106312522.jpg "1: (Emmy Noether; ) (Feynman) [3] [4] {C i } A {C i } (A A )C i = 0 [5] 2")
2003 1 1 (Emmy Noether 1) [1] [2] [ (Paul Gordan Clebsch-Gordan ] 1915 habilitation habilitation außerordentlicher Professor Außerordentlich(=extraordinary) 1 1: (Emmy Noether; 1882-1935) (Feynman) [3]
1 2 LDA Local Density Approximation 2 LDA 1 LDA LDA N N N H = N [ 2 j + V ion (r j ) ] + 1 e 2 2 r j r k j j k (3) V ion V ion (r) = I Z I e 2 r
![1 2 LDA Local Density Approximation 2 LDA 1 LDA LDA N N N H = N [ 2 j + V ion (r j ) ] + 1 e 2 2 r j r k j j k (3) V ion V ion (r) = I Z I e 2 r](/thumbs/91/104998910.jpg "1 2 LDA Local Density Approximation 2 LDA 1 LDA LDA N N N H = N [ 2 j + V ion (r j ) ] + 1 e 2 2 r j r k j j k (3) V ion V ion (r) = I Z I e 2 r")
11 March 2005 1 [ { } ] 3 1/3 2 + V ion (r) + V H (r) 3α 4π ρ σ(r) ϕ iσ (r) = ε iσ ϕ iσ (r) (1) KS Kohn-Sham [ 2 + V ion (r) + V H (r) + V σ xc(r) ] ϕ iσ (r) = ε iσ ϕ iσ (r) (2) 1 2 1 2 2 1 1 2 LDA Local
ver.1 / c /(13)
1 -- 11 1 c 2010 1/(13) 1 -- 11 -- 1 1--1 1--1--1 2009 3 t R x R n 1 ẋ = f(t, x) f = ( f 1,, f n ) f x(t) = ϕ(x 0, t) x(0) = x 0 n f f t 1--1--2 2009 3 q = (q 1,..., q m ), p = (p 1,..., p m ) x = (q,
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
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
July 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i
July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac
Venkatram and Wyngaard, Lectures on Air Pollution Modeling, m km 6.2 Stull, An Introduction to Boundary Layer Meteorology,
65 6 6.1 No.4 1982 1 1981 J. C. Kaimal 1993 1994 Turbulence and Diffusion in the Atmosphere : Lectures in Environmental Sciences, by A. K. Blackadar, Springer, 1998 An Introduction to Boundary Layer Meteorology,
3 1 7 1.1........................ 7 1.2 Calabi-Yau Gepner.......... 9 2 11 2.1 WZW.............................. 11 2.1.1 WZW Lie......... 11 2.1.2............... 13 2.1.3 SU(2)................. 15 2.2...........................
untitled
2 : n =1, 2,, 10000 0.5125 0.51 0.5075 0.505 0.5025 0.5 0.4975 0.495 0 2000 4000 6000 8000 10000 2 weak law of large numbers 1. X 1,X 2,,X n 2. µ = E(X i ),i=1, 2,,n 3. σi 2 = V (X i ) σ 2,i=1, 2,,n ɛ>0
1 : ( ) ( ) ( ) ( ) ( ) etc (SCA)
START: 17th Symp. Auto. Decentr. Sys., Jan. 28, 2005 Symplectic cellular automata as a test-bed for research on the emergence of natural systems 1 : ( ) ( ) ( ) ( ) ( ) etc (SCA) 2 SCA 2.0 CA ( ) E.g.
2 5W1H = a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] (: ) 2
![2 5W1H = a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] (: ) 2](/thumbs/91/105692443.jpg "2 5W1H = a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] (: ) 2")
1 vs. 90 mescoscopic physics 1 2 5W1H = a) [ ]= (= ) : b) [ ] : c) [ ] = (to characterize the observed system) : d) [ ] (: ) 2 (: ) [1]: 1. Newton =[( ) vs. ] (a) =0 x v ( p = mv) [ a), b), c)] (b) = :
重力と宇宙 新しい時空の量子論
Summer Institute at Fujiyoshida, 2009/08/06 KEK/ Conformal Field Theory on R x S^3 from Quantized Gravity, arxiv:0811.1647[hep-th]. Renormalizable 4D Quantum Gravity as A Perturbed Theory from CFT, arxiv:0907.3969[hep-th].
OTO研究会スライド
OTO @, 2017 5 26 2 CFT OTO ) PTEP 2016 (2016) no.11, 113B06 (arxiv1602.06542[hep-th]) Collaboration with P. Caputa(YITP) and A.Veliz-Osorio(Queen Mary U.) i i +1 H = X i i z z i+1 + h X i i x i z = 1
D v D F v/d F v D F η v D (3.2) (a) F=0 (b) v=const. D F v Newtonian fluid σ ė σ = ηė (2.2) ė kl σ ij = D ijkl ė kl D ijkl (2.14) ė ij (3.3) µ η visco
post glacial rebound 3.1 Viscosity and Newtonian fluid f i = kx i σ ij e kl ideal fluid (1.9) irreversible process e ij u k strain rate tensor (3.1) v i u i / t e ij v F 23 D v D F v/d F v D F η v D (3.2)
1. 1.1....................... 1.2............................ 1.3.................... 1.4.................. 2. 2.1.................... 2.2..................... 2.3.................... 3. 3.1.....................
TOP URL 1
TOP URL http://amonphys.web.fc2.com/ 1 6 3 6.1................................ 3 6.2.............................. 4 6.3................................ 5 6.4.......................... 6 6.5......................
1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,
![1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,](/thumbs/91/105754557.jpg "1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition) A = {x; P (x)} P (x) x x a A a A Remark. (i) {2, 0, 0,")
2005 4 1 1 2 2 6 3 8 4 11 5 14 6 18 7 20 8 22 9 24 10 26 11 27 http://matcmadison.edu/alehnen/weblogic/logset.htm 1 1 = = = (set) (element) a A a A a A a A a A {2, 5, (0, 1)}, [ 1, 1] = {x; 1 x 1}. (proposition)
Euler, Yang-Mills Clebsch variable Helicity ( Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity (i) Yang-Mills 3 A T (T A) Poisson Ha
Euler, Yang-ills Clebsch variable Helicity Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity i) Yang-ills 3 A T T A) Poisson Hamilton ii) Clebsch parametrization iii) Y- Y-iv) Euler,v)
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
Outline I. Introduction: II. Pr 2 Ir 2 O 7 Like-charge attraction III.
Masafumi Udagawa Dept. of Physics, Gakushuin University Mar. 8, 16 @ in Gakushuin University Reference M. U., L. D. C. Jaubert, C. Castelnovo and R. Moessner, arxiv:1603.02872 Outline I. Introduction:
TOP 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.............................
untitled
1 Constructive field theory and renormalization group Hiroshi WATANABE 1 2001 7,. 1 2 2 Euclidean field theory 3 2.1...................................... 4 2.2..................................... 6 3
( ) (, ) ( )
( ) (, ) ( ) 1 2 2 2 2.1......................... 2 2.2.............................. 3 2.3............................... 4 2.4.............................. 5 2.5.............................. 6 2.6..........................
( ) ) AGD 2) 7) 1
( 9 5 6 ) ) AGD ) 7) S. ψ (r, t) ψ(r, t) (r, t) Ĥ ψ(r, t) = e iĥt/ħ ψ(r, )e iĥt/ħ ˆn(r, t) = ψ (r, t)ψ(r, t) () : ψ(r, t)ψ (r, t) ψ (r, t)ψ(r, t) = δ(r r ) () ψ(r, t)ψ(r, t) ψ(r, t)ψ(r, t) = (3) ψ (r,
120 9 I I 1 I 2 I 1 I 2 ( a) ( b) ( c ) I I 2 I 1 I ( d) ( e) ( f ) 9.1: Ampère (c) (d) (e) S I 1 I 2 B ds = µ 0 ( I 1 I 2 ) I 1 I 2 B ds =0. I 1 I 2
9 E B 9.1 9.1.1 Ampère Ampère Ampère s law B S µ 0 B ds = µ 0 j ds (9.1) S rot B = µ 0 j (9.2) S Ampère Biot-Savart oulomb Gauss Ampère rot B 0 Ampère µ 0 9.1 (a) (b) I B ds = µ 0 I. I 1 I 2 B ds = µ 0
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
BPZ (Survey) ( ) Virasoro [FeEhl] Belavin- Polyakov-Zamolodchikov (BPZ) 1 Virasoro : $\text{ }$ $\mathrm{v}i\mathrm{r}$ 11viras $or
![BPZ (Survey) ( ) Virasoro [FeEhl] Belavin- Polyakov-Zamolodchikov (BPZ) 1 Virasoro : $\text{ }$ $\mathrm{v}i\mathrm{r}$ 11viras $or](/thumbs/96/128166122.jpg "BPZ (Survey) ( ) Virasoro [FeEhl] Belavin- Polyakov-Zamolodchikov (BPZ) 1 Virasoro : $\text{ }$ $\mathrm{v}i\mathrm{r}$ 11viras $or")
1218 2001 15-25 15 BPZ (Survey) ( ) Virasoro [FeEhl] Belavin- Polyakov-Zamolodchikov (BPZ) 1 Virasoro $\text{ }$ $\mathrm{v}i\mathrm{r}$ 11viras $oro$ $\mathbb{c}$-vector space Vir $=\oplus \mathbb{c}l_{n}n\in
1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1
![1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1](/thumbs/94/121802164.jpg "1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1")
1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2
untitled
17 5 13 1 2 1.1... 2 1.2... 2 1.3... 3 2 3 2.1... 3 2.2... 5 3 6 3.1... 6 3.2... 7 3.3 t... 7 3.4 BC a... 9 3.5... 10 4 11 1 1 θ n ˆθ. ˆθ, ˆθ, ˆθ.,, ˆθ.,.,,,. 1.1 ˆθ σ 2 = E(ˆθ E ˆθ) 2 b = E(ˆθ θ). Y 1,,Y
スケーリング理論とはなにか? - --尺度を変えて見えること--
? URL: http://maildbs.c.u-tokyo.ac.jp/ fukushima mailto:hukusima@phys.c.u-tokyo.ac.jp DEX-SMI @ 2006 12 17 ( ) What is scaling theory? DEX-SMI 1 / 40 Outline Outline 1 2 3 4 ( ) What is scaling theory?
Bose-Einstein Hawking Hawking Hawking Hawking nk Hawking Bose-Einstein Hawking 1 Bekenstein[1] Hawking 1974 [2,
![Bose-Einstein Hawking Hawking Hawking Hawking nk Hawking Bose-Einstein Hawking 1 Bekenstein[1] Hawking 1974 [2,](/thumbs/96/127226526.jpg "Bose-Einstein Hawking Hawking Hawking Hawking nk Hawking Bose-Einstein Hawking 1 Bekenstein[1] Hawking 1974 [2,")
Bose-Einstein Hawking E-mail: moinai@yukawa.kyoto-u.ac.jp Hawking Hawking Hawking nk Hawking Bose-Einstein Hawking 1 Bekenstein[1] Hawking 1974 [2, 3] Hawking Hawking 6nK Hawking Hawking 3K Hawking Hawking
Hanbury-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............................................
SUSY 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
1980年代半ば，米国中西部のモデル 理論，そして未来－モデル理論賛歌
2016 9 27 RIMS 1 2 3 1983 9-1989 6 University of Illinois at Chicago (UIC) John T Baldwin 1983 9-1989 6 University of Illinois at Chicago (UIC) John T Baldwin Y N Moschovakis, Descriptive Set Theory North
Broadhurst-Kreimer Brown ( D3) 1 Broadhurst-Kreimer Zagier Gangl- -Zagi
Broadhurst-Kreimer Brown ( D3) 1 Broadhurst-Kreimer 2 2 - -Zagier 5 2.1............................. 5 2.2........................... 8 3 Gangl- -Zagier 11 3.1.................................. 11 3.2
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](/thumbs/94/118365979.jpg "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
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.
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
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
QMI13a.dvi
I (2013 (MAEDA, Atsutaka) 25 10 15 [ I I [] ( ) 0. (a) (b) Plank Compton de Broglie Bohr 1. (a) Einstein- de Broglie (b) (c) 1 (d) 2. Schrödinger (a) Schrödinger (b) Schrödinger (c) (d) 3. (a) (b) (c)
/ Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiat
/ Christopher Essex Radiation and the Violation of Bilinearity in the Thermodynamics of Irreversible Processes, Planet.Space Sci.32 (1984) 1035 Radiation and the Continuing Failure of the Bilinear Formalism,
kawa (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............................................
Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x
7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)
1/2 ( ) 1 * 1 2/3 *2 up charm top -1/3 down strange bottom 6 (ν e, ν µ, ν τ ) -1 (e) (µ) (τ) 6 ( 2 ) 6 6 I II III u d ν e e c s ν µ µ t b ν τ τ (2a) (
August 26, 2005 1 1 1.1...................................... 1 1.2......................... 4 1.3....................... 5 1.4.............. 7 1.5.................... 8 1.6 GIM..........................
d (i) (ii) 1 Georges[2] Maier [3] [1] ω = 0 1
![d (i) (ii) 1 Georges[2] Maier [3] [1] ω = 0 1](/thumbs/94/121850661.jpg "d (i) (ii) 1 Georges[2] Maier [3] [1] ω = 0 1")
16 5 19 10 d (i) (ii) 1 Georges[2] Maier [3] 2 10 1 [1] ω = 0 1 [4, 5] Dynamical Mean-Field Theory (DMFT) [2] DMFT I CPA [10] CPA CPA Σ(z) z CPA Σ(z) Σ(z) Σ(z) z - CPA Σ(z) DMFT Σ(z) CPA [6] 3 1960 [7]
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
1 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