1 What is CFT? 1. 3 Strange duality conjecture (G) Geometric strange duality conjecture... 5
|
|
|
- Μέλαινα Παπανικολάου
- 6 χρόνια πριν
- Προβολές:
Transcript
1 What is CFT? 1 2 Wess-Zumino-Witten model (R Representation theoretic formulation of WZW model (G Geometric formulation of WZW model (R=(G Strange duality conjecture (G Geometric strange duality conjecture (R (R local version ( p X (R P 1 N conformal blocks Coset construction GKO coset construction of unitary representations Coset construction of conformal blocks What is CFT? (conformal field theory, CFT Riemann conformal blocks, (various conformal field theories. Example. Belavin-Polyakov-Zamolodchikov (BPZ minimal models Wess-Zumino-Witten (WZW models W-algebras, ( 7 (, ( L A TEX.
2 2 2. Wess-Zumino-Witten model parafermions coset models Super Symmetry + ( conformal blocks. WZW models BPZ minimal models coset models. : (R Representation theoretic approach (G Geometric approach. 2 Wess-Zumino-Witten model Wess-Zumino-Witten (WZW model Lie G Riemann.,. X : compact Riemann surface, G : semisimple Lie group over C, g := Lie G = (Lie algebra of G. WZW model conformal blocks (R (G : (R affine Lie algebra ĝ, highest weight integrable representations, adelic formulation on X (G moduli space of principal G-bundles on X, determinant line bundle ( G = SL n (C (R, G = SL n (C (G. 2.1 (R Representation theoretic formulation of WZW model. K := C(X = (the field of rational functions on X, K p := (the completion of K at p X = C((z p, Ô p = C[[zp ]], A := K p X p, adelic ring (, Ô := p X Ôp, maximal compact,
3 2.1. (R Representation theoretic formulation of WZW model 3 Ô A K, ĝ := g C((z CC, affine Lie algebra, ĝ A := g A CC, adele of affine Lie algebras, g Ô ĝ A g K, k = 0, 1, 2,... (level, {L k,λ } λ Pk = {h.w. integrable representations of ĝ with level k}, p 1,..., p N X, λ 1,..., λ N P k, λ p := λ i if p = p i, λ p := 0 otherwise, λ = (λp p X, L k, λ := p X L k,λ p (. ( : Riemann X p 1,..., p N λ 1,..., λ N. WZW model adele : L k, λ;x := L k, λ /(g KL k, λ, coinvariant quotient space, L k, λ;x = [L k, λ ]g K, invariant subspace. Definition. L k, λ;x conformal blocks. L k;x ( λ1,...,λ N p 1,..., p N := [( N i=1 L k,λ i ] g H0 (X,O X ( p p N, invariant subspace. ( Lemma. L k, λ;x = L λ1,...,λ N k;x p 1,..., p N (. Conformal blocks adele, p 1,..., p N. Theorem (WZW model, [TUY]. (1 dim L k, = dim ( λ;x L λ1,...,λ N k;x p 1,..., p N <. ( (2 dim L λ1,...,λ N k;x p 1,..., p N Riemann (X; p1,..., p N. stable curve. ( Exists projectively flat connection + (3 (3 (X; p 1,..., p N stable curve, ordinary double point q, q q, q stable curve ( X; p 1,..., p N, q, q. : L k;x ( λ1,...,λ N p 1,..., p N = µ P k L k; X ( λ1,...,λ N, µ,µ p 1,..., p N,q,q. (2+(3 conformal blocks factorization property.
4 4 2. Wess-Zumino-Witten model 2.2 (G Geometric formulation of WZW model, G = SL n (C,. SU X (n := (moduli sp. of semistable vector bundles on X of rank n with trivial det., θ n := (determinant line bundle on SU X (n. Definition. H 0 (SU X (n, θ k n generalized theta functions. 2.3 (R=(G Theorem ((R=(G, [BL]. λ = 0 := (0 p X ( Riemann highest weight 0 H 0 (SU X (n, θ k n = L k, 0;X.. generalized theta functions conformal blocks. Remark. λ 0 parabolic structure (level structure vector bundles moduli space,. Problem. WZW models CFTs geometric. [BL]. trivialization vector bundles t = (t X, (t p p X, E : vector bundle of rank n with trivial det., T 0 := (E, t t p : Ôn p E p Op Ô p trivialization at p, t X : K n H 0 (X, E OX K trivialization at generic point. (E, t T 0, p X γ p γ p = ˆt 1 X,p t p : K n p E p Op Kp K n p / =.. ˆt X,p t X : K n H 0 (X, E OX K E p Op Kp. : K n p T 0 SL n (A = p X SL n( K p, (E, t (γ p p X. SU X (n SL n (K\SL n (A/SL n (Ô =: SL X(n. SL X (n stack well-defined. Lemma. Grassmann SL n (A/SL n (Ô determinant line bundle θ n, : L k, 0 = H 0 (SL n (A/SL n (Ô, θ k n. Lemma, generalized theta functions affine Lie algebra highrst weight integrable representations, (R=(G.
5 5 3 Strange duality conjecture,. 3.1 (G Geometric strange duality conjecture. SU X (n := (moduli sp. of semistable vector bundles on X of rank n with trivial det., U X (n, d := (moduli sp. of semistable vector bundles on X of rank n and of degree d, UX (n := U X(n, n(g 1 (g := (genus of X, τ r,l : SU X (r U (l U (rl, (E, F E OX F, Θ n := (divisor { E U X (n H0 (X, E 0 } U (n line bundle, θ n := (divisor { E SU X (n H 0 (X, E OX L 0 } SU (n line bundle. L J g 1 (X = Pic g 1 (X. Lemma. τ r,l (Θ rl = θ l r Θ r l. Lemma. dim H 0 (U X (n, Θ n = 1. : C = H 0 (UX(rl, Θ rl H 0 (SU X (r UX(l, τ r,l (Θ rl = H 0 (SU X (r, θr l H 0 (UX(l, Θ r l. : ν r,l : H 0 (SU X (r, θr l H 0 (UX(l, Θ r l. Conjecture ([B],[DT]. ν r,l. Beauville [B] strange duality. strange duality conjecture. Remark (. Beauville-Narasimhan-Ramanan [BNR] proved the case of l = 1. Verlinde formula = dim H 0 (SU X (r, θr l = dim H 0 (UX (l, Θ r l. Donagi-Tu [DT] generalized the conjecture to an arbitrary degree. 3.2 (R WZW model? : local version P 1 N conformal blocks.
6 6 3. Strange duality conjecture (R local version ( p X Affine Lie algebra ĝ level k highest weight integrable representation Lĝk λ. λ ĝ highest weight g-part., sl n (C sl(n. r level l Young diagrams : Y r,l = { Y = (y i r i=1 Z r y 1 y r, y 1 y r l }. Young diagram Y = (y i r i=1 Y r,l Y : Y = y 1 + y y r Young diagram Y Y r,l Maya diagram M = M(Y : { 1 (j yi M = M(Y = (m ij, m ij = 0 (j > y i. Maya diagram M = (m ij t M = ( t m ij : t m j,νr+i = m i,νl+j (ν Z, i = 1,..., r, j = 1,..., l. t M(Y Y l,r Y, t Y. Young diagrams Y r,l : Y r,l = { Y = (ȳ i r i=1 Y r,l ȳ r = 0 }. Young diagram Y Y r,l Young diagram Y Yr, l : Y = (ȳ i r i=1, ȳ i = y i y r. Y r,l ŝl(r level l highest weight integrable representation highest weight sl(r-part. Y Y r,l ŝl(r level l highest weight integrable representation Lŝl(r,l Y : F = p Z Lŝl(rl 1 Λ p Lĝl(1 rl p = Y =p Lŝl(rl Y Y Y r,l. Lŝl(l r t Y Lĝl(1 rl Y F Fermion Fock space. : Lŝl(rl 1 = Λ Lŝl(r l p Y Lŝl(lr. t Y (R P 1 N conformal blocks Affine Lie algebra ĝ WZW model conformal blocks Lĝ, k, λ;x.
7 7 Theorem ([NT]. : C = Lŝl(rl 1, λ;p 1 µ Y(λ Lŝl(r l, µ;p 1 Lŝl(lr, t µ;p 1. λ = (λ p p X, λ pi = Λ mi (i = 1,..., N, λ p 0, Y(λ = { (µ p p X µ pi Y r,l, µ pi = m i (i = 1,..., N, µ p 0 }, t µ = ( t µ p p X. Lŝl(r l µ;p 1 : Lŝl(r l µ;p 1 = Lŝl(l r, t µ;p 1. Lŝl(lr, t µ;p 1 Remark. P 1 N p 1,..., p N [NT]. conformal blocks factorization property Knizhnik- Zamolodchikov (KZ equation braid monodromy. Nakanishi-Tsuchiya [NT] Riemann, Beauville- Laszlo [BL] (R=(G, strange duality conjecture. Nakanishi-Tsuchiya [NT] Riemann conformal blocks factorization property., strange duality KZ equation braid monodromy,. 4 Coset construction, Goddard-Kent-Olive [GKO] Virasoro algebra coset construction. 4.1 GKO coset construction of unitary representations. g = sl 2 (C, ĝ = sl 2 (C C[t, t 1 ] CC, affine Lie algebra, k = 1, 2, 3,... (level, C, P k = {0, 1, 2,..., k} λ, L k,λ = (level k, highest weight λ highest weight integrable representation of ĝ, Vir = C[t, t 1 ] d dt CC, the Virasoro algebra, c := 1 6 (k + 2(k + 3 (central charge, C, h λ,µ = ((k + 3(λ + 1 (k + 2(µ , 4(k + 2(k + 3 V c,h = (cenral charge c, h.w. h h.w. irreducible representation of Vir.
8 8 4. Coset construction Theorem ([GKO]. λ P k, ε P 1, : L k,λ L 1,ε = L k+1,µ V c,hλ,µ µ P k+1, λ+µ+ε 0 mod 2 Remark. [GKO] V c,hλ,µ Vir unitary representation. central charge c 1 Vir unitary representations ([FQS], [L]. 4.2 Coset construction of conformal blocks Vir WZW model conformal blocks. WZW model : V c, h;x = V c;x ( h1,...,h N p 1,...,p N. [GKO] : L k, λ;x L 1, ε;x L k+1, µ;x V c,. h λ, µ ;X : µ:λ p +µ p +ε p 0 mod 2 φ : L k+1, λ;x L k, ε;x V c, h λ, µ ;X L 1, ε;x. Conjecture. φ. Remark. k = 1, 2,..., c = 1 6/((k + 2(k + 3 R c R c = { h λ,µ λ P k, µ P k+1 }. h k λ,k+1 µ = h λ,µ R p (k + 1(k + 2/2. : P k P k+1 = Rc P 1, (λ, µ (h, ε. λ + µ + ε 0 (mod 2, h = h λ,µ. Remark. X = P 1 3 conformal blocks φ. WZW model Virasoro algebra CFT factorization properties : (1 φ. (2 P 1 3 conformal blocks. Example. k = 1, c = 1/2 R c = {0, 1/2, 1/16}. L 1,0, L 1,1, V 1/2,0, V 1/2,1/2, V 1/2,1/16 Fermion. P 1 3 conformal blocks k = 1. [I].
9 9 [B] Beauville, Arnaud: Vector bundles on curves and generalized theta functions: recent results and open problems. Current topics in complex algebraic geometry (Berkeley, CA, 1992/93, 17 33, Math. Sci. Res. Inst. Publ., 28, Cambridge Univ. Press, Cambridge, [BL] Beauville, Arnaud and Laszlo, Yves: Conformal blocks and generalized theta functions. Comm. Math. Phys. 164 (1994, no. 2, [BNR] Beauville, Arnaud, Narasimhan, M. S., and Ramanan, S.: Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398 (1989, [DT] Donagi, Ron and Tu, Loring W.: Theta functions for SL(n versus GL(n. Math. Res. Lett. 1 (1994, no. 3, [FQS] Friedan, Daniel, Qiu, Zongan, and Shenker, Stephen: Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett. 52 (1984, no. 18, [GKO] Goddard, P., and Kent, A., and Olive, D.: Virasoro algebras and coset space models. Phys. Lett. B 152 (1985, no. 1-2, [I] [L] Ikeda, Takeshi: Coset constructions of conformal blocks. Dissertation, Tohoku University, Sendai, Tohoku Mathematical Publications, 3. Tohoku University, Mathematical Institute, Sendai, ii+55 pp. Langlands, Robert P.: On unitary representations of the Virasoro algebra. Infinitedimensional Lie algebras and their applications (Montreal, PQ, 1986, , World Sci. Publ., Teaneck, NJ, [NT] Nakanishi, Tomoki, and Tsuchiya, Akihiro: Level-rank duality of WZW models in conformal field theory. Comm. Math. Phys. 144 (1992, no. 2, [TUY] Tsuchiya, Akihiro, Ueno, Kenji, and Yamada, Yasuhiko: Conformal field theory on universal family of stable curves with gauge symmetries. Integrable systems in quantum field theory and statistical mechanics, , Adv. Stud. Pure Math., 19, Academic Press, Boston, MA, 1989.
Higher spin gauge theories and their CFT duals
Higher spin gauge theories and their CFT duals E-mail: hikida@phys-h.keio.ac.jp 2 AdS Vasiliev AdS/CFT 4 Vasiliev 3 O(N) 3 Vasiliev 2 W N 1 AdS/CFT g µν Vasiliev AdS [1] AdS/CFT anti-de Sitter (AdS) (CFT)
g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King
Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i
( ) 1.1. (2 ),,.,.,.,,,,,.,,,,.,,., K, K.
( ),.,,, 1, [17]. 1. 1.1. (2 ),,.,.,.,,,,,.,,,,.,,., K, K. 1.2. Σ g g. M g, Σ g. g 1 Σ g,, Σ g Σ g. Σ g, M g,, Σ g.. g = 1, M 1 M 1, SL(2, Z). Q. g = 2, 2000 M 2 (Korkmaz [20], Bigelow Budney [5])., Bigelow
On the Galois Group of Linear Difference-Differential Equations
On the Galois Group of Linear Difference-Differential Equations Ruyong Feng KLMM, Chinese Academy of Sciences, China Ruyong Feng (KLMM, CAS) Galois Group 1 / 19 Contents 1 Basic Notations and Concepts
Table 1. morphism U P 1 dominant (MMP) 2. dim = 3 (MMP) 3. (cf. [Ii77], [Miy01]) (Table 1) 3.
![Table 1. morphism U P 1 dominant (MMP) 2. dim = 3 (MMP) 3. (cf. [Ii77], [Miy01]) (Table 1) 3.](/thumbs/92/108955011.jpg "Table 1. morphism U P 1 dominant (MMP) 2. dim = 3 (MMP) 3. (cf. [Ii77], [Miy01]) (Table 1) 3.")
338-8570 255 e-mail: tkishimo@rimath.saitama-u.ac.jp 1 C T κ(t ) 1 [Projective] κ = κ =0 κ =1 κ =2 κ =3 dim = 1 P 1 elliptic others dim = 2 P 2 or ruled elliptic surface general type dim = 3 uniruled bir.
J. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5
Vol. 37 ( 2017 ) No. 5 J. of Math. (PRC) 1,2, 1, 1 (1., 225002) (2., 225009) :. I +AT +, T + = T + (I +AT + ) 1, T +. Banach Hilbert Moore-Penrose.. : ; ; Moore-Penrose ; ; MR(2010) : 47L05; 46A32 : O177.2
Discriminantal arrangement
Discriminantal arrangement YAMAGATA So C k n arrangement C n discriminantal arrangement 1989 Manin-Schectman Braid arrangement Discriminantal arrangement Gr(3, n) S.Sawada S.Settepanella 1 A arrangement
IUTeich. [Pano] (2) IUTeich
![IUTeich. [Pano] (2) IUTeich](/thumbs/76/73790805.jpg "IUTeich. [Pano] (2) IUTeich")
2014 12 2012 8 IUTeich 2013 12 1 (1) 2014 IUTeich 2 2014 02 20 2 2 2014 05 24 2 2 IUTeich [Pano] 2 10 20 5 40 50 2005 7 2011 3 2 3 1 3 4 2 IUTeich IUTeich (2) 2012 10 IUTeich 2014 3 1 4 5 IUTeich IUTeich
Wishart α-determinant, α-hafnian
Wishart α-determinant, α-hafnian (, JST CREST) (, JST CREST), Wishart,. ( )Wishart,. determinant Hafnian analogue., ( )Wishart,. 1 Introduction, Wishart. p ν M = (µ 1,..., µ ν ) = (µ ij ) i=1,...,p p p
N. P. Mozhey Belarusian State University of Informatics and Radioelectronics NORMAL CONNECTIONS ON SYMMETRIC MANIFOLDS
Òðóäû ÁÃÒÓ 07 ñåðèÿ ñ. 9 54.765.... -. -. -. -. -. : -. N. P. Mozhey Belarusian State University of Inforatics and Radioelectronics NORMAL CONNECTIONS ON SYMMETRIC MANIFOLDS In this article we present
The form factor program - a review and new results - the nested SU(N)-off-shell Bethe ansatz - 1/N expansion
The form factor program - a review and new results - the nested SU(N-off-shell Bethe ansatz - 1/N expansion H. Babujian, A. Foerster, and M. Karowski Yerevan-Tbilisi, October 2007 Babujian, Foerster, Karowski
Diderot (Paris VII) les caractères des groupes de Lie résolubles
Βιογραφικο Σημειωμα Μ. Ανουσης Προσωπικά στοιχεία Εκπαίδευση Μιχάλης Ανούσης Πανεπιστήμιο Αιγαίου 83200 Καρλόβασι Σάμος Τηλ.: (3022730) 82127 Email: mano@aegean.gr 1980 Πτυχίο από το Τμήμα Μαθηματικών
11 Drinfeld. k( ) = A/( ) A K. [Hat1, Hat2] k M > 0. Γ 1 (M) = γ SL 2 (Z) f : H C. ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d Γ 1 (M))
![11 Drinfeld. k( ) = A/( ) A K. [Hat1, Hat2] k M > 0. Γ 1 (M) = γ SL 2 (Z) f : H C. ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d Γ 1 (M))](/thumbs/91/105488540.jpg "11 Drinfeld. k( ) = A/( ) A K. [Hat1, Hat2] k M > 0. Γ 1 (M) = γ SL 2 (Z) f : H C. ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d Γ 1 (M))")
Drinfeld Drinfeld 29 8 8 11 Drinfeld [Hat3] 1 p q > 1 p A = F q [t] A \ F q d > 0 K A ( ) k( ) = A/( ) A K Laurent F q ((1/t)) 1/t C Drinfeld Drinfeld p p p [Hat1, Hat2] 1.1 p 1.1.1 k M > 0 { Γ 1 (M) =
Space-Time Symmetries
Chapter Space-Time Symmetries In classical fiel theory any continuous symmetry of the action generates a conserve current by Noether's proceure. If the Lagrangian is not invariant but only shifts by a
ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ Ä Œμ Ìμ. ±É- É Ê ± μ Ê É Ò Ê É É, ±É- É Ê, μ Ö
ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ 2017.. 48.. 5.. 740Ä744 ˆ Œˆ ƒ Š Œ ˆ Œˆ ˆŸ ˆ ˆ ˆŸ ˆˆ ƒ ˆ Šˆ ˆ.. Œμ Ìμ ±É- É Ê ± μ Ê É Ò Ê É É, ±É- É Ê, μ Ö ±μ³ ² ± ÒÌ ³μ ʲÖÌ Ð É Ò³ ² ³ Š² ËËμ Î É μ - ³ μ É Ò Ë ³ μ Ò ³ Ò Å ²μ ÉÉ. Ì
Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics
Sachdev-Ye-Kitaev Model as Liouville Quantum Mechanics Dmitry Bagrets Nucl. Phys. B 9, 9 (06) arxiv: 607.00694 Alexander Altland Univ. zu Köln Alex Kamenev Univ. of Minnesota PCS IBS Workshop, Daejeon,
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
( ) Kähler X ( ),. Floer -Oh- - [6]. X Fano *, X ( = (C ) N ) W : X C ( ) (X,W). X = P, W (y) =y + Q/y. Q P. Φ:X R N, Δ=Φ(X). u Int Δ, Lagrange L(u) =
![( ) Kähler X ( ),. Floer -Oh- - [6]. X Fano *, X ( = (C ) N ) W : X C ( ) (X,W). X = P, W (y) =y + Q/y. Q P. Φ:X R N, Δ=Φ(X). u Int Δ, Lagrange L(u) =](/thumbs/87/96954144.jpg "( ) Kähler X ( ),. Floer -Oh- - [6]. X Fano *, X ( = (C ) N ) W : X C ( ) (X,W). X = P, W (y) =y + Q/y. Q P. Φ:X R N, Δ=Φ(X). u Int Δ, Lagrange L(u) =")
Floer Cohomologes of Non-torus Fbers of the Gelfand-Cetln System (X, ω) 2N. X N Φ=(ϕ,...,ϕ N ):X R N, Posson, Φ. Φ, Arnold-Louvlle Largange. Φ (u) = T N, ω Φ (u) =0.. Gelfand-Cetln, Gullemn-Sternberg [9]
Single-value extension property for anti-diagonal operator matrices and their square
1 215 1 Journal of East China Normal University Natural Science No. 1 Jan. 215 : 1-56412151-95-8,, 71119 :, Hilbert. : ; ; : O177.2 : A DOI: 1.3969/j.issn.1-5641.215.1.11 Single-value extension property
Reminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, RIMS
Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, 2017 @ RIMS Contents Introduction Generalized Karcher equation Ando-Hiai inequalities Problem Introduction PP: The set of all positive definite operators
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Heisenberg Uniqueness pairs
Heisenberg Uniqueness pairs Philippe Jaming Bordeaux Fourier Workshop 2013, Renyi Institute Joint work with K. Kellay Heisenberg Uniqueness Pairs µ : finite measure on R 2 µ(x, y) = R 2 e i(sx+ty) dµ(s,
Global nonlinear stability of steady solutions of the 3-D incompressible Euler equations with helical symmetry and with no swirl
Around Vortices: from Cont. to Quantum Mech. Global nonlinear stability of steady solutions of the 3-D incompressible Euler equations with helical symmetry and with no swirl Maicon José Benvenutti (UNICAMP)
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
1. 3. ([12], Matsumura[13], Kikuchi[10] ) [12], [13], [10] ( [12], [13], [10]
![1. 3. ([12], Matsumura[13], Kikuchi[10] ) [12], [13], [10] ( [12], [13], [10]](/thumbs/103/159380891.jpg "1. 3. ([12], Matsumura[13], Kikuchi[10] ) [12], [13], [10] ( [12], [13], [10]")
3. 3 2 2) [2] ) ) Newton[4] Colton-Kress[2] ) ) OK) [5] [] ) [2] Matsumura[3] Kikuchi[] ) [2] [3] [] 2 ) 3 2 P P )+ P + ) V + + P H + ) [2] [3] [] P V P ) ) V H ) P V ) ) ) 2 C) 25473) 2 3 Dermenian-Guillot[3]
: 1. 10:20 12:40. 12:50 13:50 14:00 14:50 15:00 16:30 Selberg ( ) 18:45 20:00 20:15 21:45 Selberg ( ) 7:00 9:00
: 2010 9 6 ( ) 9 10 : 1. 9/6( ) 10:20 12:40 GL(2) Hecke ( ) 12:50 13:50 14:00 14:50 15:00 16:30 Selberg ( ) 16:45 18:15 GL(2) I ( ) 18:45 20:00 20:15 21:45 Selberg ( ) 9/7( ) 7:00 9:00 9:15 10:30 GL(2)
The q-commutators of braided groups
206 ( ) Journal of East China Normal University (Natural Science) No. Jan. 206 : 000-564(206)0-0009-0 q- (, 20024) : R-, [] ABCD U q(g).,, q-. : R- ; ; q- ; ; FRT- : O52.2 : A DOI: 0.3969/j.issn.000-564.206.0.002
6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Jordan Form of a Square Matrix
Jordan Form of a Square Matrix Josh Engwer Texas Tech University josh.engwer@ttu.edu June 3 KEY CONCEPTS & DEFINITIONS: R Set of all real numbers C Set of all complex numbers = {a + bi : a b R and i =
J. of Math. (PRC) 6 n (nt ) + n V = 0, (1.1) n t + div. div(n T ) = n τ (T L(x) T ), (1.2) n)xx (nt ) x + nv x = J 0, (1.4) n. 6 n
Vol. 35 ( 215 ) No. 5 J. of Math. (PRC) a, b, a ( a. ; b., 4515) :., [3]. : ; ; MR(21) : 35Q4 : O175. : A : 255-7797(215)5-15-7 1 [1] : [ ( ) ] ε 2 n n t + div 6 n (nt ) + n V =, (1.1) n div(n T ) = n
Lecture 13 - Root Space Decomposition II
Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).
THE BIGRADED RUMIN COMPLEX. 1. Introduction
THE BIGRADED RUMIN COMPLEX JEFFREY S. CASE Abstract. We give a detailed description of the bigraded Rumin complex in dimensions three and five, including formulae for all of its constituent operators.
([28] Bao-Feng Feng (UTP-TX), ( ), [20], [16], [24]. 1 ([3], [17]) p t = 1 2 κ2 T + κ s N -259-
![([28] Bao-Feng Feng (UTP-TX), ( ), [20], [16], [24]. 1 ([3], [17]) p t = 1 2 κ2 T + κ s N -259-](/thumbs/89/98133812.jpg "([28] Bao-Feng Feng (UTP-TX), ( ), [20], [16], [24]. 1 ([3], [17]) p t = 1 2 κ2 T + κ s N -259-")
5,..,. [8]..,,.,.., Bao-Feng Feng UTP-TX,, UTP-TX,,. [0], [6], [4].. ps ps, t. t ps, 0 = ps. s 970 [0] []. [3], [7] p t = κ T + κ s N -59- , κs, t κ t + 3 κ κ s + κ sss = 0. T s, t, Ns, t., - mkdv. mkdv.
arxiv: v1 [math.ra] 19 Dec 2017
![arxiv: v1 [math.ra] 19 Dec 2017](/thumbs/73/68347977.jpg "arxiv: v1 [math.ra] 19 Dec 2017")
TWO-DIMENSIONAL LEFT RIGHT UNITAL ALGEBRAS OVER ALGEBRAICALLY CLOSED FIELDS AND R HAHMED UBEKBAEV IRAKHIMOV 3 arxiv:7673v [mathra] 9 Dec 7 Department of Math Faculty of Science UPM Selangor Malaysia &
Fractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
ADE. 1 Introduction. (Ryo Fujita) Lie. U q (Lg) U(Lg) Dynkin. Dynkin. Dynkin. 4 A n (n Z 1 ), B n (n Z 2 ), C n (n Z 2 ), D n (n Z 4 )
ADE (Ryo Fujita) 1 Introduction Lie g U(g) q 2 q q Hopf Drinfeld- U q (g) C S 1 g U(g) q U q (g) U(Lg) q U q (Lg) Lg := g C[t ±1 ] Lie U q (Lg) 2 R ADE Lie g U q (Lg) ADE Dynkin Dynkin Q Dynkin Q Hernandez-Leclerc
Lecture 10 - Representation Theory III: Theory of Weights
Lecture 10 - Representation Theory III: Theory of Weights February 18, 2012 1 Terminology One assumes a base = {α i } i has been chosen. Then a weight Λ with non-negative integral Dynkin coefficients Λ
Vol. 37 ( 2017 ) No. 3. J. of Math. (PRC) : A : (2017) k=1. ,, f. f + u = f φ, x 1. x n : ( ).
Vol. 37 ( 2017 ) No. 3 J. of Math. (PRC) R N - R N - 1, 2 (1., 100029) (2., 430072) : R N., R N, R N -. : ; ; R N ; MR(2010) : 58K40 : O192 : A : 0255-7797(2017)03-0467-07 1. [6], Mather f : (R n, 0) R
EE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
L p approach to free boundary problems of the Navier-Stokes equation
L p approach to free boundary problems of the Navier-Stokes equation e-mail address: yshibata@waseda.jp 28 4 1 e-mail address: ssshimi@ipc.shizuoka.ac.jp Ω R n (n 2) v Ω. Ω,,,, perturbed infinite layer,
ECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Symmetric Stress-Energy Tensor
Chapter 3 Symmetric Stress-Energy ensor We noticed that Noether s conserved currents are arbitrary up to the addition of a divergence-less field. Exploiting this freedom the canonical stress-energy tensor
Parametrized Surfaces
Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some
Abstract Storage Devices
Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
PHASE TRANSITIONS IN QED THROUGH THE SCHWINGER DYSON FORMALISM
PHASE TRANSITIONS IN THROUGH THE SCHWINGER DYSON FORMALISM Spyridon Argyropoulos University of Athens Physics Department Division of Nuclear Physics and Elementary Particles Supervisor: C.N. Ktorides Athens
The ε-pseudospectrum of a Matrix
The ε-pseudospectrum of a Matrix Feb 16, 2015 () The ε-pseudospectrum of a Matrix Feb 16, 2015 1 / 18 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of 2 2 5 Problems
n+1 v2 2 1 + x 3 1 + x 3 u2 1 + u2 2 1 ) + 1 (u 1, u 2 ) = 1 v2 1 ) (v 1, v 2 ) =
Κεφάλαιο 2 Λείες πολλαπλότητες Σύνοψη Παρουσιάζουμε τον ορισμό μιας λείας (διαφορικής) πολλαπλότητας και αναλύουμε δύο βασικά παραδείγματα, τη μοναδιαία σφαίρα και τον προβολικό χώρο. Στη συνέχεια, μελετάμε
Partial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
arxiv: v1 [math-ph] 4 Jun 2016
![arxiv: v1 [math-ph] 4 Jun 2016](/thumbs/74/70333136.jpg "arxiv: v1 [math-ph] 4 Jun 2016")
On commuting ordinary differential operators with polynomial coefficients corresponding to spectral curves of genus two Valentina N. Davletshina, Andrey E. Mironov arxiv:1606.0136v1 [math-ph] Jun 2016
SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Wavelet based matrix compression for boundary integral equations on complex geometries
1 Wavelet based matrix compression for boundary integral equations on complex geometries Ulf Kähler Chemnitz University of Technology Workshop on Fast Boundary Element Methods in Industrial Applications
Βιογραφικό Σημείωμα. Γεωργίου Κ. Ελευθεράκη ΓΕΝΙΚΑ ΣΤΟΙΧΕΙΑ EKΠΑΙΔΕΥΣΗ
Βιογραφικό Σημείωμα Γεωργίου Κ. Ελευθεράκη ΓΕΝΙΚΑ ΣΤΟΙΧΕΙΑ Ημερομηνία Γέννησης: 23 Δεκεμβρίου 1962. Οικογενειακή Κατάσταση: Έγγαμος με δύο παιδιά. EKΠΑΙΔΕΥΣΗ 1991: Πτυχίο Οικονομικού Τμήματος Πανεπιστημίου
Markov chains model reduction
Markov chains model reduction C. Landim Seminar on Stochastic Processes 216 Department of Mathematics University of Maryland, College Park, MD C. Landim Markov chains model reduction March 17, 216 1 /
New bounds for spherical two-distance sets and equiangular lines
New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a
Congruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Minimal Surfaces PDE as a Monge Ampère Type Equation
Minimal Surfaces PDE as a Monge Ampère Type Equation Dmitri Tseluiko Abstract In the recent Bîlă s paper [1] it was determined the symmetry group of the minimal surfaces PDE (using classical methods).
Affine Weyl Groups. Gabriele Nebe. Summerschool GRK 1632, September Lehrstuhl D für Mathematik
Affine Weyl Groups Gabriele Nebe Lehrstuhl D für Mathematik Summerschool GRK 1632, September 2015 Crystallographic root systems. Definition A crystallographic root system Φ is a finite set of non zero
Aspects of the BMS/CFT correspondence
DAMTP, Cambridge. February 17, 2010 Aspects of the BMS/CFT correspondence Glenn Barnich Physique théorique et mathématique Université Libre de Bruxelles & International Solvay Institutes Overview Classical
The Properties of Fuzzy Relations
International Mathematical Forum, 5, 2010, no. 8, 373-381 The Properties of Fuzzy Relations Yong Chan Kim Department of Mathematics, Gangneung-Wonju National University Gangneung, Gangwondo 210-702, Korea
ADVANCES IN MECHANICS Jan. 25, Newton ( ) ,., Newton. , Euler, d Alembert. Lagrange,, , Hamilton ( )
39 1 Vol. 39 No. 1 2009 1 25 ADVANCES IN MECHANICS Jan. 25, 2009 *, 100081. 5 3. Noether, Lie,, Lagrange,,.,,, 1 1.1 1687 Newton (1642 1727), 3,., Newton. 1743 d Alembert (1717 1783), Newton, d Alembert.
The Spiral of Theodorus, Numerical Analysis, and Special Functions
Theo p. / The Spiral of Theodorus, Numerical Analysis, and Special Functions Walter Gautschi wxg@cs.purdue.edu Purdue University Theo p. 2/ Theodorus of ca. 46 399 B.C. Theo p. 3/ spiral of Theodorus 6
Lecture 16 - Weyl s Character Formula I: The Weyl Function and the Kostant Partition Function
Lecture 16 - Weyl s Character Formula I: The Weyl Function and the Kostant Partition Function March 22, 2013 References: A. Knapp, Lie Groups Beyond an Introduction. Ch V Fulton-Harris, Representation
Laplace Expansion. Peter McCullagh. WHOA-PSI, St Louis August, Department of Statistics University of Chicago
Laplace Expansion Peter McCullagh Department of Statistics University of Chicago WHOA-PSI, St Louis August, 2017 Outline Laplace approximation in 1D Laplace expansion in 1D Laplace expansion in R p Formal
CFT approach to the q-painlevé VI equation
Journal of Integrable Systems 07 7 doi: 0.093/integr/xyx009 CFT approach to the q-painlevé VI equation M. Jimbo Department of Mathematics Rikkyo University Toshima-ku Tokyo 7-850 Japan H. Nagoya School
Uniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
ACTA MATHEMATICAE APPLICATAE SINICA Nov., ( µ ) ( (
35 Þ 6 Ð Å Vol. 35 No. 6 2012 11 ACTA MATHEMATICAE APPLICATAE SINICA Nov., 2012 È ÄÎ Ç ÓÑ ( µ 266590) (E-mail: jgzhu980@yahoo.com.cn) Ð ( Æ (Í ), µ 266555) (E-mail: bbhao981@yahoo.com.cn) Þ» ½ α- Ð Æ Ä
Other Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
ON THE STRUCTURE OF PRINCIPAL SUBSPACES OF STANDARD MODULES FOR AFFINE LIE ALGEBRAS OF TYPE A
ON THE STRUCTURE OF PRINCIPAL SUBSPACES OF STANDARD MODULES FOR AFFINE LIE ALGEBRAS OF TYPE A BY CHRISTOPHER MICHAEL SADOWSKI A dissertation submitted to the Graduate School New Brunswick Rutgers, The
Second Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
The circle theorem and related theorems for Gauss-type quadrature rules
OP.circle p. / The circle theorem and related theorems for Gauss-type quadrature rules Walter Gautschi wxg@cs.purdue.edu Purdue University OP.circle p. 2/ Web Site http : //www.cs.purdue.edu/ archives/22/wxg/codes
Finite size Emptiness Formation Probability for the XXZ spin chain at = 1/2
Finite size Emptiness Formation Probability for the XXZ spin chain at = 1/2 Luigi Cantini luigi.cantini@u-cergy.fr LPTM, Université de Cergy-Pontoise (France) CFT AND INTEGRABLE MODELS Bologna, September
Note: Please use the actual date you accessed this material in your citation.
MIT OpenCourseWare http://ocw.mit.edu 6.03/ESD.03J Electromagnetics and Applications, Fall 005 Please use the following citation format: Markus Zahn, 6.03/ESD.03J Electromagnetics and Applications, Fall
Non-Abelian Gauge Fields
Chapter 5 Non-Abelian Gauge Fields The simplest example starts with two Fermions Dirac particles) ψ 1, ψ 2, degenerate in mass, and hence satisfying in the absence of interactions γ 1 i + m)ψ 1 = 0, γ
Min-max Theory, Willmore conjecture, and Energy of links
Min-max Theory, Willmore conjecture, and Energy of links André Neves (Joint with Fernando Marques) Q: What is the best way of immersing a sphere in space? A: The one that minimizes the bending energy H
Lie Algebras Representations- Bibliography
Lie Algebras Representations- Bibliography J. E. Humphreys Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer 1980 Hans Samelson, Notes on Lie Algebras B. C.
Distances in Sierpiński Triangle Graphs
Distances in Sierpiński Triangle Graphs Sara Sabrina Zemljič joint work with Andreas M. Hinz June 18th 2015 Motivation Sierpiński triangle introduced by Wac law Sierpiński in 1915. S. S. Zemljič 1 Motivation
AdS black disk model for small-x DIS
AdS black disk model for small-x DIS Miguel S. Costa Faculdade de Ciências da Universidade do Porto 0911.0043 [hep-th], 1001.1157 [hep-ph] Work with. Cornalba and J. Penedones Rencontres de Moriond, March
A Note on Characterization of Intuitionistic Fuzzy Ideals in Γ- Near-Rings
International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 1 (2011), pp. 61-71 International Research Publication House http://www.irphouse.com A Note on Characterization
D Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
The k-α-exponential Function
Int Journal of Math Analysis, Vol 7, 213, no 11, 535-542 The --Exponential Function Luciano L Luque and Rubén A Cerutti Faculty of Exact Sciences National University of Nordeste Av Libertad 554 34 Corrientes,
ΒΙΟΓΡΑΦΙΚΟ ΣΗΜΕΙΩΜΑ. Dr. ΗΛΙΑΣ A. ΖΑΦΕΙΡΗΣ
ΒΙΟΓΡΑΦΙΚΟ ΣΗΜΕΙΩΜΑ Dr. ΗΛΙΑΣ A. ΖΑΦΕΙΡΗΣ Ημερομηνία και Τόπος Γεννήσεως : Αθήνα, Ελλάς, 2 Μαΐου 1970. Υπηκοότητα : Ελληνική. Διεύθυνση : Ιωνίας 40, Άλιμος 17456, Αθήνα. Τηλέφωνο : 210-9910768, 697-4464277.
Horizontal and Vertical Recurrence Relations for Exponential Riordan Matrices and Their Applications
4th RART, July 17-20 2017, Universidad Complutense de Madrid Horizontal and Vertical Recurrence Relations for Exponential Riordan Matrices and Their Applications Ji-Hwan Jung Sungkyunkwan University, Korea
ERG for a Yukawa Theory in 3 Dimensions
ERG for a Yukawa Theory in 3 Dimensions Hidenori SONODA Physics Department, Kobe University, Japan 14 September 2010 at ERG2010 Abstract We consider a theory of N Majorana fermions and a real scalar in
SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Adachi-Tamura [4] [5] Gérard- Laba Adachi [1] 1
![Adachi-Tamura [4] [5] Gérard- Laba Adachi [1] 1](/thumbs/89/99472248.jpg "Adachi-Tamura [4] [5] Gérard- Laba Adachi [1] 1")
207 : msjmeeting-207sep-07i00 ( ) Abstract 989 Korotyaev Schrödinger Gérard Laba Multiparticle quantum scattering in constant magnetic fields - propagator ( ). ( ) 20 Sigal-Soffer [22] 987 Gérard- Laba
RS306 High Speed Fuse for Semiconductor Protection
Product Information Standard square body fuse, comply with GB13539/IEC60269/UL-248/VDE-0636 Protection: ar (GB\IEC), short circuit or backup Rated : 250-4000VAC Very low I 2 t/high current-limiting/high
Lie Algebras Representations- Bibliography
Lie Algebras Representations- Bibliography J. E. Humphreys Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer 1980 Alex. Kirillov An Introduction to Lie Groups
Homomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΙΚΑ ΜΑΘΗΜΑΤΑ
ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΙΚΑ ΜΑΘΗΜΑΤΑ Αναπαραστάσεις Αλγεβρών Lie Ενότητα 1: Αναπαραστάσεις Αλγεβρών Lie Κ. Δασκαλογιάννης Τμήμα Μαθηματικών Α.Π.Θ. (Α.Π.Θ.) Αναπαραστάσεις
1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r(t) = 3cost, 4t, 3sint
1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P, π, rt) cost, t, sint ). b) 5 points) Find curvature of the curve at the point P. Solution: a) r t) sint,,
2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
第 61 回トポロジーシンポジウム講演集 2014 年 7 月於東北大学 Kontsevich-Kuperberg-Thurston ( ) Kontsevich-Kuperberg-Thurston Kontsevich Chern-Simons E. Witten Chern-
Kontsevich-Kuperberg-Thurston ( ) Kontsevich-Kuperberg-Thurston Kontsevich Chern-Simons 3 1 1989 E. Witten Chern-Simons 3 ( ) ([14]) Witten 3 Chern-Simons M. Kontsevich [5], S. Axerod I. M. Singer [2]
Bundle Adjustment for 3-D Reconstruction: Implementation and Evaluation
3 2 3 2 3 undle Adjustment or 3-D Reconstruction: Implementation and Evaluation Yuuki Iwamoto, Yasuyuki Sugaya 2 and Kenichi Kanatani We describe in detail the algorithm o bundle adjustment or 3-D reconstruction
Kostka functions associated to complex reflection groups
Kostka functions associated to complex reflection groups Toshiaki Shoji Tongji University March 25, 2017 Tokyo Kostka functions associated to complex reflection groups March 25, 2017 Tokyo 1 / 30 Kostka
Αναπαραστάσεις οµάδων και Αλγεβρες Τελεστών
7 Ιουλίου 2015 1 χαρακτήρες χαρακτήρες Ορισµός G οµάδα, (π, H) unitary αναπαράσταση της G. Λέµε χαρακτήρα της π την συνάρτηση χ π : G, που ορίζεται χ(x) = tr π(x) Πρόταση G οµάδα, (π, H) unitary αναπαράσταση
4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION
AN ELEMENTARY APPROACH TO THE GAUSS HYPERGEOMETRIC FUNCTION TOSHIO OSHIMA Abstract. We give an introduction to the Gauss hypergeometric function, the hypergeometric equation and their properties in an
Iterated trilinear fourier integrals with arbitrary symbols
Cornell University ICM 04, Satellite Conference in Harmonic Analysis, Chosun University, Gwangju, Korea August 6, 04 Motivation the Coifman-Meyer theorem with classical paraproduct(979) B(f, f )(x) :=