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\begin{align*}h_{pq}=\frac{1}{\tau_2}\left(\begin{array}{cc}|\tau| ^2& \tau_1 \\\tau_1 & 1\end{array}\right) .\end{align*} | |
\begin{align*} \tilde\nu_v\cdot \psi&= \left(\frac{g_u\times\psi}{|g_u\times\psi|}\right)_v \cdot \psi\\ &= \left(\frac{g_{uv}\times\psi+g_u\times\psi_v}{|g_u\times\psi|}\right) \cdot\psi + \bigl((g_u\times\psi)\cdot\psi\bigr)\left(\frac{1}{|g_u\times\psi|}\right)_v\\ &=-\frac{\det(g_u,\psi,\psi_v)}{|g_u\times\psi|}\ne... | |
\begin{align*}0 ~=~ \det \left(\begin{array}{cc}2z-r(\alpha-1)s(\alpha-1) & 2z \\4z & - {\mbox{\small{$\frac{1}{N}$}}} p(\beta)q(\beta) - (\Pi_{B2} + \Pi_{C2})\\\end{array}\right)\end{align*} | |
\begin{align*}G_{\mu\nu} = \left(\begin{array}{cc}G_{ab} & A_{ai} \\A_{bj} & \hat{G}_{ij} + A_{ic}A_{jd}G^{cd} \\\end{array}\right)\end{align*} | |
\begin{align*}(II) \hspace{3mm} \left(\begin{array}{c}x_1+{\rm i}y_1 \\x_2+{\rm i}y_2 \\x_3+{\rm i}y_3 \\x_4+{\rm i}y_4\end{array}\right) \hspace{3mm} \leftrightarrow \hspace{3mm}\frac{1}{\sqrt{2}}\left(\begin{array}{c}-(y_2+y_4) +{\rm j}(y_2-y_4) \\(y_1 + y_3) -{\rm j}(y_1-y_3) \\(x_2 + x_4) -{\rm j}(x_2 - x_4) \\-(x_... | |
\begin{align*} 1 &= f_u(u,0)\cdot f_u(u,0) = \tilde f_u(u,0)\cdot \tilde f_u(u,0)\\ &= \left|\xi_u(u,0)\check f_{\xi}\bigl(\xi(u,0),\eta(u,0)\bigr) +\eta_u(u,0)\check f_{\eta}\bigl(\xi(u,0),\eta(u,0)\bigr) \right|^2\\ & =|\xi_u(u,0)|^2. \end{align*} | |
\begin{align*}i(CP^1)=\left\{ \frac{1}{\sqrt{1+|u|^2}} \left( \begin{array}{cc} 1 & iu \\ i\bar{u} & 1 \\ \end{array} \right) |u \in \bf C \right\} . \end{align*} | |
\begin{align*}\left\{\begin{array}{l}\varphi=ux+vy\\\theta=\alpha x+\beta y \quad. \end{array}\right.\end{align*} | |
\begin{align*} 0<\kappa_\nu(u) &=\det(f_u(u,0),f_{vv}(u,0),f_{uu}(u,0)) \\ &=\det(T \circ \tilde f_u(u,0),T \circ \tilde f_{vv}(u,0), T \circ \tilde f_{uu}(u,0))\\ &=\det(\tilde f_u(u,0),\tilde f_{vv}(u,0),\tilde f_{uu}(u,0))\\ &=\xi_u(u,0) \det(\check f_\xi(\xi(u,0),0), \check f_{\eta\eta}(\xi(u,0),0), \check f_{\xi\x... | |
\begin{align*}\bar S_{ij}^{-1}(\vec x,\vec y)=\eta_{ij}\,\left(\begin{array}{cc}0&1\\-1&0\end{array}\right)\,\frac{1}{\nabla^2}\,\delta(\vec x-\vec y)\end{align*} | |
\begin{align*}V^{(+1)}(\alpha)=\left\{ \begin{array}{l}-2\cosh\alpha,\;\;\;\;\;\;K=+1 \\-2\sinh\alpha,\;\;\;\;\;\;K=-1\\\;\;\;\;\;-e^{\alpha},\;\;\;\;\;\;\;\;\;\;\;K=0 \end{array}\right.\end{align*} | |
\begin{align*}X = \xi \tanh u, \\\qquad v \equiv e^{\Omega} \equiv e^{u + i \varphi}.\end{align*} | |
\begin{align*}eT_{\mu\nu}={\tilde{\overline\Psi}_{L}}{\gamma_\mu}i\left(D_\nu -\frac{1}{2}\Gamma^\alpha_{\alpha\nu}\right){\tilde\Psi}_{L}-\frac{i}{2}g_{\mu\nu}\partial_\alpha({\tilde{\overline\Psi}}_{L}\gamma^\alpha{\tilde\Psi_{L}}).\\\end{align*} | |
\begin{align*}{\cal L}=\psi^\dagger i \partial_t \psi + \psi^\dagger \frac{\nabla^2}{2 M} \psi - \frac{1}{2} C_0 (\psi^\dagger \psi)^2\\-\frac{1}{2} C_2 (\psi^\dagger \nabla^2 \psi) (\psi^\dagger \psi) + h.c. + \ldots.\end{align*} | |
\begin{align*} w:=M\begin{pmatrix} 0 \\ \mu \smallskip\\ \pm\sqrt{1-\mu^2} \end{pmatrix} \end{align*} | |
\begin{align*}\Gamma ^{5}=\left( \begin{array}{cc}0 & 1_{2} \\ 1_{2} & 0\end{array}\right);\quad \Gamma ^{i}=\left( \begin{array}{cc}0 & \sigma ^{i} \\ -\sigma ^{i} & 0\end{array}\right);\quad \Gamma ^{0}=\left( \begin{array}{cc}1_{2} & 0 \\ 0 & -1_{2}\end{array}\right);\quad \Gamma ^{4}=-i\Gamma ^{5}=\left( \begin{arr... | |
\begin{align*}\langle L(u,p_{3},p_{4}) \rangle = \left[\begin{array}{ll}c_{3} c_{4} - b_{3} b_{4} u & -u(b_{3}d_{4}-c_{3}a_{4})\\\vspace{-2mm}\\d_{3} b_{4} - a_{3} c_{4} & d_{3}d_{4} - a_{3} a_{4}u\end{array}\right]. \end{align*} | |
\begin{align*}L_{eff}^{ii} = \frac{1}{2\pi} \left( \hat{\Phi}^i ~ \hat{\eta}^i \right)\left( \begin{array}{cc}A^{ii} & C^{ii}/2 \\C^{ii}/2& B^{ii}\end{array} \right) ~ \left( \begin{array}{c}\hat{\Phi}^i\\\hat{\eta}^i\end{array}\right),\end{align*} | |
\begin{align*}\left(\begin{array}{ccc}(\kappa-\ln\tilde q)\eta(\tilde\tau)^2&-\Lambda_{1,2}(\tilde\tau)&-\eta(\tilde\tau)^2\\-\Lambda_{1,2}(\tilde\tau)&0&0\\-\eta(\tilde\tau)^2&0&0\end{array}\right)\end{align*} | |
\begin{align*}\Delta _E^M (k)=3D\frac{k_\mu= (\delta_{\mu\nu}+{\bar{\sigma}}_{\mu\nu})k_\nu}{k^4} \hspace{.5cm},\hspace{.5cm} \begin{array}{lcl} {\bar{\sigma}}_{ij}& =3D & i\varepsilon_{ijk}\sigma_k= \\ {\bar{\sigma}}_{i4}&=3D& i\sigma_i=3D-{\bar{\sigma}}_{4=i} \end{array}.\end{align*} | |
\begin{align*}\bar A_{\bar z} = \left( \begin{array}{cc} 0 & {1 \over 2l} \bar T(\bar z) \\ 1 & 0 \end{array} \right)\end{align*} | |
\begin{align*}\left(\begin{array}{c}\alpha \\\beta\end{array}\right)\to\left(\begin{array}{rr}1 & 0 \\1 & 1\end{array}\right)\left(\begin{array}{c}\alpha \\\beta\end{array}\right) \ .\end{align*} | |
\begin{align*}D = \left( \begin{array}{cc} v^{-1} \partial_1 & u^{-1} \partial_2 \\ u \partial_4 & v \partial_3 \end{array} \right) \; , \quad \bar{D} =\hat{V}^{\epsilon \, -1} D = \left( \begin{array}{cc} v \partial_3 & -q^{-2} u^{-1} \partial_2 \\ -q^{-2} u \partial_4 & v^{-1} \partial_1 \end{array} \right) \; ,\end{... | |
\begin{align*}T^{-1}\,=\,\left( \begin{array}{cc} a^{-1}+a^{-1}bd^{-1}ca^{-1} & -a^{-1}bd^{-1} \\ -d^{-1}ca^{-1} & d^{-1}+d^{-1}ca^{-1}bd^{-1} \end{array} \right)\end{align*} | |
\begin{align*}\begin{array}{l}(L_n-\bar{L}_{-n})\mid B_{\pm }\rangle =0 \\ (S_r\pm i\bar{S}_{-r})\mid B_{\pm }\rangle =0\end{array}\end{align*} | |
\begin{align*}\Leftrightarrow\begin{array}{ccl}\partial_+\partial_-\left(\rho-\Phi\right)&=&0\\\rho-\Phi&=&f_+\left(x^+\right)+f_-\left(x^-\right)\;.\end{array}\end{align*} | |
\begin{align*}T^{t}\left (\begin{array}{cc}0 & 1\\ 1 & 0 \end{array}\right )T=\left (\begin{array}{cc}0 & 1\\ 1 & 0 \end{array}\right ).\end{align*} | |
\begin{align*}\begin{array}{ccccccc}& \circ_{0} & \left(\frac{1}{g}\right) &&&&\\& \mid &&&&&\\\left(\frac{R_1}{g}\right) \;\; \circ_{1} - & \circ_{2} &-&\circ_{3} &- \dots-&\circ_{d-1} \;\; \left(\frac{1}{R_d}\right) \\\end{array}\end{align*} | |
\begin{align*}\begin{array}{rcl} \hat{M}_{a-1}&=&-R_5P_a\, , \ \ \ (a=0,...,4) \\ \\ \tilde{M}_{6a}&=&R_5P_a\, , \ \ \ (a=5,...,9)\, . \end{array}\end{align*} | |
\begin{align*}{\psi}^{D-K}=\left[ \begin{array}{c} \sum_{j,m}\xi^{j+}_m<jm|{\psi}_{-}^{(j)}>\\ \sum_{j,m}\xi^{j-}_m <jm|{\psi}_{+}^{(j)}> \end{array} \right].\end{align*} | |
\begin{align*}A= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & \cosh 2\phi & 0 \\ 0 & \cosh 2\phi & -1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.\end{align*} | |
\begin{align*}\left(\begin{array} {c c c}j_{1}& j_{2}& j_{3}\\-m_{1}& -m_{2}& -m_{3}\\\end{array}\right)_{q}=\left(\begin{array} {c c c}j_{2}& j_{1}& j_{3}\\m_{2}& m_{1}& m_{3}\\\end{array}\right)_{q}=(-)^{j_{1}+j_{2}+j_{3}}\left(\begin{array} {c c c}j_{1}& j_{2}& j_{3}\\m_{1}& m_{2}& m_{3}\\\end{array}\right)_{q^{-1}}... | |
\begin{align*}D=e^{-\gamma _5\phi (r)\ }i\not \! \partial \ e^{-\gamma_5\phi (r)}=\left(\begin{array}{cc}0 & \sigma ^{-1}(\partial _r+B) \\\sigma \ (-\partial _r+B) & 0\end{array}\right),\end{align*} | |
\begin{align*}T_k^{AB} \equiv T_k =\left( \begin{array}{cc} C_k & D_k \\ \bar D_k & \bar C_k \end{array} \right)\end{align*} | |
\begin{align*}\fbox{$\begin{array}{lccc}\cal{U} \; =& <\frac{1}{4}(1+e_{3}+e_{5}+e_{6}), &\frac{1}{4}(1+e_{3}+e_{5}+e_{6}), &\frac{1}{4}(-1+e_{3}+e_{5}+e_{6})+\frac{1}{2}e_{7}>. \\ \\&\mbox{even}&\mbox{even}&\mbox{odd} \\ \\& \mbox{with } 1 & \mbox{with } 1 & \mbox{with } 1 \\\end{array}$}\end{align*} | |
\begin{align*}\delta {\cal L}^{[\frac{1}{2}]} = [\kappa \ \bar{\kappa}] {\cal P}_{+} \left[\begin{array}{c}- \sqrt{-g} v^{i} \tilde{\cal H}^{j} \gamma_{i j} \bar{P} \\- \sqrt{-g} v^{i} \bar{\tilde{\cal H}}^{j} \gamma_{i j} P\end{array}\right] \end{align*} | |
\begin{align*}V_{a}(r) = \left\{ \begin{array} {ll} - \lambda/a^{2} & \mbox{for$r<a$} \\ -\lambda/r^{2} & \mbox{for $r\geq a$}\end{array}\right.\; ,\end{align*} | |
\begin{align*}\psi = {\psi}_- ( \vec{x}) \left(\begin{array}{c} 0 \\ 1 \end{array}\right) e^{-i E_f t } ,\end{align*} | |
\begin{align*}(a_1,~\ldots~,~a_N)\left(\begin{smallmatrix}f(w_1)\\ \vdots \\ f(w_N)\end{smallmatrix}\right)=c.\end{align*} | |
\begin{align*}{ \left( \gamma^a \right)_\alpha}^\beta = \left[ \left( \begin{array}{cc} i & 0 \\ 0 & -i \end{array}\right) , \, \left(\begin{array}{cc} 0 & i \\ i & 0 \end{array} \right) , \, \left( \begin{array}{cc} 0 & i \\ -i & 0 \end{array}\right) \right] \, .\end{align*} | |
\begin{align*}U^{\prime}_{[4]\times[N]} = - \left( \! \begin{array}{cccc} U^{\prime}_{11} & U^{\prime}_{12} & \cdots & U^{\prime}_{1N} \\U^{\prime}_{21} & U^{\prime}_{22} & \cdots & U^{\prime}_{2N} \\U^{\prime}_{31} & U^{\prime}_{32} & \cdots & U^{\prime}_{3N} \\U^{\prime}_{41} & U^{\prime}_{42} & \cdots & U^{\prime}_{... | |
\begin{align*}i{\bf \Gamma}_n(t,\vec x)=\left\{\begin{array}{cc} -i \Gamma_n(t,\vec x)&0 \\0 & i \Gamma_n(t,\vec x)\end{array} \right\} \; .\end{align*} | |
\begin{align*}\begin{array}[t]{c}{\displaystyle {\mathop{{\sum}'}}}\\{\scriptstyle 1\leq {n_1} < {n_2} < \cdots < {n_m} \leq D-1}\end{array}=\begin{array}[t]{c}{\displaystyle {\mathop{{\sum}}}}\\{\scriptstyle 1\leq {n_1} < {n_2} < \cdots < {n_m} \leq D-1}\\{\scriptstyle {l_j} {\not{|}} {n_i}}\end{array}\,\,\,.\end{alig... | |
\begin{align*}\left(\begin{array}{c}p'_{0} \\ p'_{1}\end{array}\right)={1\over\sqrt{1-B_{01}^{2}}} \left(\begin{array}{cc}1 & 0 \\-B_{01} & 1-B_{01}^{2}\end{array}\right) \left(\begin{array}{c}p_{0}-B_{01}p_{1} \\p_{1}\end{array}\right) =\left(\begin{array}{c}\gamma p_{0}-\gamma vp_{1} \\-v\gamma p_{0}+\gamma p_{1}\end... | |
\begin{align*}[a_{0},a_{n}]={1 \over A}{1 \over n}a_{0}a_{n}\hbar +{1 \over 2A}\sum _{k \ne 0,n}{\epsilon(n) \over \sqrt{|kn(n-k)|}}a_{k}a_{n-k}\hbar+O(\hbar^{2}), \\ n \ne 0,\end{align*} | |
\begin{align*}\begin{array}{l} \omega^{\alpha\beta}=\{\eta^\alpha,\eta^\beta\}=\tilde{J}_{\alpha\beta}=-J_{\alpha\beta}\\ \chi_{\alpha\beta}=J_{\alpha\beta} \end{array} \end{align*} | |
\begin{align*}\sigma_s : \left\{\begin{array}{l}y \mapsto y+2\pi sR, \\ \tilde{\theta} \mapsto \tilde{\theta}+2\pi s/n.\end{array}\right. \end{align*} | |
\begin{align*}\left( \begin{array}{cc}a\gamma & a\delta \\b\gamma & b\delta \end{array} \right) \: = \:\left( \begin{array}{cc}0 & 0 \\0 & 0 \end{array} \right)\end{align*} | |
\begin{align*}a_l=\sum_{n\neq 0}\frac{1}{(an)^{l+6}}\quad\left\{\begin{array}{l} =0,\ l\ \mathrm{odd}\\ =\frac{2\zeta(l+6)}{a^{l+6}},\ l\ \mathrm{even} \end{array}\right. ,\end{align*} | |
\begin{align*}\hat\eta = \sqrt{{N-1 \over N}}\left(\begin{array} {cccc}\eta & & & \\ & -{\eta \over N-1} & & \\ & & \ddots & \\ & & & -{\eta \over N-1} \end{array}\right)\ .\end{align*} | |
\begin{align*}\begin{array}{ll}\hbox{Sector}\ \ &\hbox{Field content} \\\hbox{even} & [1,1],\, [3,3],\, [5,5],\, [7,7],\, [9,9],\, [11,11],\, [13,13] \\\hbox{odd} & [2,2],\, [4,4],\, [6,6],\, [8,8],\, [10,10],\, [12,12] \\\hbox{twisted} & [1,13],\, [3,11],\, [5,9],\, [7,7],\, [9,5],\, [11,3],\, [13,1] \\\end{array}\end... | |
\begin{align*}f(z)&=Z(z)F(p(z))\\&=Z(z)\left(\frac{V^*v^*}{vVV^*v^*}c+\left(I_N+(p(z)-1)\frac{V^*v^*vV^*}{{vVV^*v^*}}G(p(z))\right)\right).\end{align*} | |
\begin{align*}\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)\end{align*} | |
\begin{align*}\begin{array}{rcl}m_0^2&=&\displaystyle\frac{1}{2a^2}L^{\alpha\beta}L_{\alpha\beta}\\ &=&E^2-{\bf P}^2-\displaystyle\frac{{\bf L}^2}{a^2}~,\end{array}\end{align*} | |
\begin{align*}\left( \hat{\epsilon}_{\hat{\mu}}{}^{\hat{a}} \right)=\left(\begin{array}{cc}e_{\mu}{}^{a} & k^{-1} B_{\mu} \\& \\0 & k^{-1} \\\end{array}\right)\, ,\hspace{1cm}\left( \hat{\epsilon}_{\hat{a}}{}^{\hat{\mu}}\right)=\left(\begin{array}{cc}e_{a}{}^{\mu} & -B_{a} \\& \\0 & k \\\end{array}\right)\, .\end{align... | |
\begin{align*}g_{\mu\nu} = g^{^{\rm (0)}}_{\mu\nu} + h_{\mu\nu}, \\ \ \phi = \varphi +\delta \phi ,\end{align*} | |
\begin{align*}Q_{-}^{\lambda} = \left\{ \begin{array}{ll} (-Q)^{\lambda} & Q<0 \\ 0 & Q\geq 0 \end{array} \right.\end{align*} | |
\begin{align*}\|a\|_\sharp+\|x^\natural-a\|_\sharp&\geq\|x^\natural\|_\sharp\\&\geq\|x^\star\|_\sharp\\&=\|x^\natural+(x^\star-x^\natural)\|_\sharp\\&=\|a+(x^\natural-a)+z_1+z_2\|_\sharp\\&\geq\|a+z_1\|_\sharp-\|(x^\natural-a)+z_2\|_\sharp\\&\geq\|a+z_1\|_\sharp-\|x^\natural-a\|_\sharp-\|z_2\|_\sharp\\&=\|a\|_\sharp+\|... | |
\begin{align*}S=\,{8\pi^2\over g^2} +2\pi^2\rho^2(|v_+|^2+|v_-|^2)-{i\over \sqrt{2}} \left(\begin{array}{llll}\bar{v}_+ & \\& \bar{v}_-\end{array}\right)_{f}^{\,\,\dot\beta}\mu_{\dot\beta}\, ({\cal K}_f+\tilde{\cal K}_f) \ ,\end{align*} | |
\begin{align*}\beta^{-2}d\Omega_{2}^2\equiv \left\{ \begin{array}{ll} \beta^{-2}d\theta^2 +\beta^{-2}\sin^2\!\theta\,d\varphi^2\ \ & {\mbox{if }}\beta\neq 0 \\ dx^2+dy^2 & {\mbox{if }}\beta = 0. \end{array}\right.\end{align*} | |
\begin{align*}z_2[i]:=\left\{\begin{array}{cl}z[i]&\mbox{if }i\in\operatorname{supp}(a)\\0&\mbox{otherwise.}\end{array}\right.\end{align*} | |
\begin{align*}\left\{\begin{array}{c}z\\ \overline z\end{array}\right\}=\tau\pm ix=ix^{\pm}\;.\end{align*} | |
\begin{align*}z_j=g^{(4)}_j(0)=\left\{\begin{array}{ccc}1 & \mathrm{for} & j=1 \\i & \mathrm{for} & j=2 \\-1 & \mathrm{for} & j=3 \\-i & \mathrm{for} & j=4\end{array}\right. ,\end{align*} | |
\begin{align*}Z_1:=U\left[\begin{array}{cc}0&0\\0&Y_{22}\end{array}\right]V^*,Z_2:=U\left[\begin{array}{cc}Y_{11}&Y_{12}\\Y_{21}&0\end{array}\right]V^*.\end{align*} | |
\begin{align*}\mathrm{LHS}:=\min_{\substack{x\in X\\\|x\|_2=1}}\max_{\substack{y\in Y\\\|y\|_2=1}}\langle x,y\rangle=\min_{\substack{x\in X\\\|x\|_2=1}}\|P_Yx\|_2.\end{align*} | |
\begin{align*}\sqrt{1-\mathrm{LHS}^2}=\max_{\substack{x\in X\\\|x\|_2=1}}\|P_{Y^\perp}x\|_2=\max_{\substack{z\in\mathbb{R}^N\\\|z\|_2=1}}\|P_{Y^\perp}P_Xz\|_2=\|P_{Y^\perp}P_X\|_2.\end{align*} | |
\begin{align*}G_{ab}^{(1)} = \left\{ \begin{array}{ll} 0 & \mbox{for $a,b\ne i,j$} \\ f_if_j+k_{ij}+\delta_{ij}A\rho & \mbox{for $a,b= i,j$} \end{array}\right. \end{align*} | |
\begin{align*}\begin{array}{rcl}\psi\circ s_{h}:=\phi: &G/H& \longrightarrow {\bf C}^{n} \\ &[g]& \longrightarrow \phi([g])=\psi(s_{h}([g]))\equiv\psi(gh)\end{array}\end{align*} | |
\begin{align*}\tilde f^{(1)} = \left(\begin{array}{cccc}0 & -\delta_{ij} & 2a_i & 0 \\\delta_{ij} & 0 & 0 & 0 \\-2a_i & 0 & 0 & 0 \\0 & 0 & 0 & 0 \end{array}\right),\end{align*} | |
\begin{align*}\begin{array}{ccl}\xi _k^\alpha (x) & = & \left( \left( \frac 1{R\left( A(x)\right) R\left(A(x)\right) }\right) ^{1/4}\right) _{kk^{^{\prime }}}^{\alpha \alpha^{^{\prime }}}B_{k^{^{\prime }}}^{\alpha ^{^{\prime }}}(x) \\ & = & \frac 1{\pi \sqrt{2}}\int_0^\infty d\lambda \lambda ^{-1/4}\frac1{\lambda +R\bu... | |
\begin{align*}\left[\begin{array}{cc}1 & 0 \\0 & 1\end{array}\right],\hspace{2mm}\left[\begin{array}{cc}0 & 1 \\1 & 0\end{array}\right],\hspace{2mm}\left[\begin{array}{cc}0 & i \\-i & 0\end{array}\right],\hspace{2mm}\left[\begin{array}{cc}1 & 0 \\0 & -1\end{array}\right].\end{align*} | |
\begin{align*}S(t) = \left\{ \begin{array}{cl}1 , &\chi = 0 \\\cosh \beta t , &\chi = {\beta}^{2}\end{array} \right.\end{align*} | |
\begin{align*}\begin{array}{c}\Phi(\tau,\xi)=-v \;\;\mbox{for}\;\; \xi\leq-\xi_{1}, \\\Phi(\tau,\xi)=+v \;\;\mbox{for}\;\; \xi\geq\xi_{0}.\end{array}\end{align*} | |
\begin{align*}R=\left(\begin{array}{cc}W_{R} & 0 \\0 & r\end{array}\right) = G'~\Delta~G'\end{align*} | |
\begin{align*}\psi_1=\frac{1}{2}(r/l)^{\frac{1}{2}}\left( \begin{array}{c} 1 \\ 1 \end{array} \right)\end{align*} | |
\begin{align*}\alpha=\left( \begin{array}{ccc} \omega_{2m} & 0 & 0 \\0 & \omega_{2m}^{-1} & 0 \\ 0 & 0 & 1 \end{array} \right) ,~~~~\beta=\left( \begin{array}{ccc} 0 & i & 0 \\ i & 0 & 0 \\ 0 & 0 & 1 \end{array} \right)\end{align*} | |
\begin{align*}\begin{pmatrix}0 & t\\0 & 0\end{pmatrix}\stackrel{\phi}{\longmapsto} x\end{align*} | |
\begin{align*}\begin{array}{rcl}C & = & C^{(0)} + C^{(1)} + C^{(2)} +\ldots \\& & \\G & = & G^{(0)} + G^{(1)} + G^{(2)} +\ldots \\& & \\\Lambda^{(\cdot)} & = & \Lambda^{(0)} +\Lambda^{(1)} + \ldots\, , \\\end{array}\end{align*} | |
\begin{align*}\begin{array}{lll}V^{(1)}&=&\displaystyle{1 \over 4!} f\varphi^4+{1 \over 48} (\beta_f -4f \gamma_{\varphi}) \varphi^4\left( \log {\varphi^2 \over \mu^2 } - {25 \over 6} \right) \\&&\displaystyle-{1 \over 2}\xi R \varphi^2-{1 \over 4} (\beta_{\xi} -2\xi \gamma_{\varphi}) R \varphi^2\left( \log {\varphi^2 ... | |
\begin{align*}e^{ih} &=e^{i(h_{1,n}+h_{2,n}+h_{3,n}+h_{4,n})}e^{ic_n}\\&=e^{i(h_{1,n}+h_{2,n})}e^{ i(h_{3,n}+h_{4,n})}e^{ic_n'}\\&= e^{ih_{1,n}} e^{i h_{2,n}} e^{ih_{3,n}}e^{i h_{4,n}}e^{ic_n''},\end{align*} | |
\begin{align*}\left\{ \begin{array}{rcl}ds^{2} & = & R_{2}^{2}\, d\Pi_{(2)}^{2} -R_{2}^{2}\, d\Omega_{(2)}^{2}\, , \\& & \\F & = & -\frac{2}{R_{2}}\omega_{AdS_{2}}\, ,\\ \end{array}\right.\end{align*} | |
\begin{align*}\varphi = \frac{1}{\sqrt{2}} e^{ig\Delta} \left( \begin{array}{c}0 \\ \chi \end{array} \right) = \frac{1}{\sqrt{2}} \chi \hat{\varphi} \ .\end{align*} | |
\begin{align*}\displaystyle[\imath\partial_\mu\gamma^\mu-m\gamma^{2n+1}]{\bf f}(x)=0 \;\;\Longleftrightarrow\left\{ \begin{array}{c}[\imath\partial_\mu\Gamma^\mu_+-m]u_+(x)=0\;, \\\displaystyle[\imath\partial_\mu\Gamma^\mu_-+m]u_-(x)=0\;,\end{array}\right.\end{align*} | |
\begin{align*}A(\hat x)=\left\{\begin{array}{ll}-{eV_0/\hat am\omega^2R^2},&\mbox{for $\hat x\in(-\hat a,0)$},\\{eV_0/\hat bm\omega^2R^2},&\mbox{for $\hat x\in(0,\hat b)$},\\0, &\mbox{otherwise}.\end{array}\right.\end{align*} | |
\begin{align*}e^{-2\Phi}=e^{-2\rho}=\left\{\begin{array}{ccc}-\lambda^2x^+x^-&,&x^+<x^+_0\\{M\over\lambda}-\lambda^2x^+\left(x^-+{M\over{\lambda^3x^+_0}}\right)&,&x^+>x^+_0\;.\end{array}\right.\end{align*} | |
\begin{align*}\varepsilon = \left\{ \begin{array}{ccc}\epsilon_+\left( \sigma\right) & , & 0\leq \sigma < \pi \\\pm \epsilon_-\left( 2\pi -\sigma\right) & ,& \pi \leq \sigma < 2\pi\end{array} \right. .\end{align*} | |
\begin{align*}\begin{array}{rcl}\overline{K}(z+1) & = & -g\overline{K}(z)g, \\\overline{K}(z+\tau ) & = & -h\overline{K}(z)h \times \exp \left\{ -2\pi \sqrt{-1}\left( z+\frac{\tau}{2}+c \right)\right\}, \end{array}\end{align*} | |
\begin{align*}v^{(1)}=\left(\begin{array}{ccc}0 \\a_i \\-1\end{array}\right),\end{align*} | |
\begin{align*}\theta^{1,\alpha}=\left(\begin{array}{c}\theta^{1,\alpha_{1}} \\\\0 \\\end{array}\right)\, , \hspace{1cm}\theta^{2,\alpha}=\left(\begin{array}{c}\theta^{2,\alpha_{1}} \\\\0 \\\end{array}\right)\, .\end{align*} | |
\begin{align*}X_{A \dot{A}}=X^i\sigma^i_{A\dot{A}},\qquad \sigma^i_{A\dot{A}}=(i\tau^1,i\tau^2,i\tau^3,{\bf 1}_2),\\\end{align*} | |
\begin{align*}\left\{\begin{array}{lcl}G^{\pm}_{(2n+1)} & = & dC^{\pm}_{(2n)} -HC^{\pm}_{(2n-2)} +F^{(2)} C^{\pm}_{(2n-1)}\, ,\\& & \\G^{\pm}_{(2n)} & = & dC^{\pm}_{(2n-1)} -HC^{\pm}_{(2n-3)} +F^{(1)} C^{\pm}_{(2n-2)}\, .\\\end{array}\right.\end{align*} | |
\begin{align*}z\frac{d}{dz}+\left( \begin{array}{cccccc}b_{1}(z)&b_{2}(z)&b_{3}(z)&\ldots&b_{n-1}(z)&b_{n}(z)\\1&0&0&\ldots&0&0\\0&1&0&\ldots&0&0\\0&0&1&\ldots&0&0\\.&.&.&.&.&.\\0&0&0&\ldots&1&0 \end{array}\right)\leftrightarrow \frac{d^n}{d\tau^n}+\bar{b}_{1}(\tau)(\frac{d^{n-1}}{d\tau^{n-1}})+\cdots+\bar{b}_{n}(\tau)... | |
\begin{align*}e^{2\pi i h(t)}=\begin{pmatrix}e^{-2\pi i \omega(t)} & & &\\& e^{-2\pi i \omega(t)} & &\\& & \ddots & \\ &&&e^{2\pi i (n-1)\omega(t)}\end{pmatrix}.\end{align*} | |
\begin{align*}K=\left(\begin{array}{lcccc}&u_1&u_2&u_3&u_4\\m_1 \, &1&0&0&0\\m_2 \, &0&1&0&0\\m_3 \, &0&0&1&0\\m_4 \, &0&0&0&1\\m_5 \, &1&1&-1&0\\m_6 \, &1&1&0&-1\\\end{array}\right).\end{align*} | |
\begin{align*}A^{\left( *\right) }(p)=\sum_{m}v(p,m)a^{*}(p,m) \\ \end{align*} | |
\begin{align*}B^{\left( *\right) }(p,)=\sum_{m}v(p,m)b^{*}(p,m) \\ \end{align*} | |
\begin{align*}A(p)=\sum_{m}u(p,m)a(p,m) \\ \end{align*} | |
\begin{align*}u&=\prod_{j=1}^{k+1} u_j^*e^{ih_j'}u_j \prod_{j=1}^{k+2} (v_j',w_j')e^{ic}\\&=u_1^*(\prod_{j=1}^{k+1} e^{i h_j'})u_1\prod_{j=1}^{k} (v_j'',w_j'')\prod_{j=1}^{k+2} (v_j',w_j')e^{ic'}.\end{align*} | |
\begin{align*}{\cal{U}} =\left( \begin{array}{cc} \cos{\frac{\theta}{2}} &\sin{\frac{\theta}{2}} \\ \\ -\sin{\frac{\theta}{2}} &\cos{\frac{\theta}{2}} \end{array} \right)\end{align*} | |
\begin{align*}e^{ih} &=(u,v)e^{i[x,x^*]}e^{ia}e^{-ib}\\&=(u,v)e^{ic}e^{i[x,x^*]+ia}e^{-ib}\\&=(u,v)e^{ic}e^{i([x,x^*]+a-b)}.\end{align*} |
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