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\begin{align*}f'd(a-x^*x+xx^*-b) &=f'a^{1/2}-f'fa^{\frac 1 2}+ f'dxx^*\\&=f'dxx^*\\&=0.\end{align*} | |
\begin{align*}\begin{array}{c}\widetilde{\tau}=\cosh\omega\,\tau-\sinh\omega\,\sigma,\hfill\\\widetilde{\sigma}=\cosh\omega\,\sigma-\sinh\omega\,\tau.\hfill\end{array}\end{align*} | |
\begin{align*}\begin{tabular}{|c||c|c|} \hline&${\bf 28}$ &$\overline{\bf 28}$\\\hline\hline${\bf 63}$&${\bf 36}+{\bf 420}$&$\overline{\bf 36}+\overline{\bf 420}$\\\hline${\bf 70}$&$\overline{\bf 420}$&${\bf 420}$\\\hline\end{tabular}\end{align*} | |
\begin{align*}\begin{array}{lll}V & =\frac{1}{2}\int_{M} d^{2}x \Omega & \left[ \frac{1}{2}F_{ij}F^{ij}+D_{i}\phi\overline{D^{i}\phi} \right. \\ & & \left. + \frac{1}{4}(|\phi|^{2}-1)^{2}\right]\end{array}\end{align*} | |
\begin{align*}\begin{array}{l}S_{+}\rightarrow d^2 S_{+} +2cd S_0 -c^2 S_{-}\,, \\ \\S_{0}\rightarrow bdS_{+}+(1+2bc)S_0 -acS_{-}\,, \\ \\S_{-}\rightarrow -b^2 S_{+}-2abS_0 +a^2 S_{-}\,.\end{array}\end{align*} | |
\begin{gather*}a+b = \max \{ a+b, c+d, e+f,A -a, A -c, A -f,B -b, B -d, B -e\}, \\\ell = \min \{ A, B, \ell \}.\end{gather*} | |
\begin{align*}\begin{array}{cc}{\rm Tr}[A,\phi]^2&=v^2{\rm Tr}[X(\theta)^{-1}AX(\theta),\left( \begin{array}{ccc} 1&0&0\\ 0&1&0\\ 0&0&-2\end{array}\right)]^2\\&=18 v^2 ((A^1 \sin\theta-A^2\cos\theta)^2 +(A^3)^2),\end{array}\end{align*} | |
\begin{align*}\begin{array}{l}-{\left(1-{2M\over r}\right)}^{-1}\partial_t^{\,2}{f\over r}+{\left(1-{2M\over r}\right)}^{-1}r^{-2}\partial_{r^*}\left(\left(1-{2M\over r}\right)r^2{\left(1-{2M\over r}\right)}^{-1}\partial_{r^*}{f\over r}\right)\\\\-\frac{l\left(l+1\right)}{r^2}{f\over r}-m^2{f\over r}=0\;,\end{array}\en... | |
\begin{align*}m_{\lambda}(\nu+\rho -w(\mu+\rho)) & = m_{\lambda} (w(\nu+\rho) -(\mu+\rho)) \\&= m_{\lambda} (w \cdot \nu - \mu).\end{align*} | |
\begin{align*}e^{\mu}_3(k)=\left \{ \begin{array}{ll}\frac{k^0\vec k}{\sqrt{k^2}|\vec k|}, & if \;\;\mu=1,2,3; \\ \frac{|\vec k|}{\sqrt{k^2}}, & if\;\;\mu=0\end{array}\right.\end{align*} | |
\begin{align*}H_{\pm} = \left(\begin{array}{cc} m & i\partial_{x}\pm\partial_{y}-\frac{eB}{2}(y\pm ix)\\ i\partial_{x}\mp\partial_{y}-\frac{eB}{2}(y\mp ix)& -m\end{array}\right)\end{align*} | |
\begin{align*}U=\left(\begin{array}{c} 0 \\ p_0 \\ 0 \end{array}\right)+\tilde{U}u\end{align*} | |
\begin{align*}\begin{array}{rl}\delta e& =-\partial_\tau(ae)-{i}\alpha(\tau)\psi,\\\delta\psi&=-a(\tau)\dot\psi-{3\over 2}\dot a \psi-{1\over4}\alpha(\tau)\dot e-{1\over 2}\dot\alpha(\tau)e.\end{array}\end{align*} | |
\begin{align*}C=\left(\begin{array}{cccc} q & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 0 & q^{-1} \end{array}\right) \end{align*} | |
\begin{align*}D_0=\frac{1}{2}\left(\begin{array}{cc}1&0 \\ \\ 0&-1\end{array}\right) \; ,\;\;\;D_+=\left(\begin{array}{cc}0&\sigma_1 \\ \\ 0&0\end{array}\right) \; ,\;\;\;D_-=\left(\begin{array}{cc}0&0 \\ \\ \sigma_1&0\end{array}\right);\end{align*} | |
\begin{align*}&\bullet (s=t)[u,u]=D(s)D(t)u&\\*&\bullet (s=t)[u,v]=(s=t)[(s*t)u,(s\neq t)v] &\end{align*} | |
\begin{align*}&\bullet(P\wedge Q)[x,y]:=P[Q[x,y],y]&\\*&\bullet (P\vee Q)[x,y]:=P[x,Q[x,y]] &\\&\bullet(\lnot P)[x,y]:=P[y,x]&\end{align*} | |
\begin{align*}\sigma_B^{k_1(x)} (h(\sigma_A(x))) & = \sigma_B^{l_1(x)}(h(x)) x \in X_A, \\\sigma_A^{k_2(y)} (h^{-1}(\sigma_B(y))) & = \sigma_A^{l_2(y)}(h^{-1}(y)) y \in X_B \end{align*} | |
\begin{align*}X\Rightarrow\left(\begin{array}{c}\vert f_a(x)\rangle \\ \vert f_b(x)\rangle\end{array}\right)\end{align*} | |
\begin{align*}A=\begin{bmatrix}1 & 1 \\1 & 1\end{bmatrix},B=\begin{bmatrix}1 & 1& 0 \\1 & 0& 1 \\1 & 0& 1 \end{bmatrix}. \end{align*} | |
\begin{align*}\mathbf{E}^\prime(\mathbf{x}) =\frac{1}{4\pi}\int\rho_e^\prime(\mathbf{x}')\frac{\mathbf{r}}{r^3}\,d^3x'\\-\frac{1}{4\pi}\int\mathbf{J}_m^\prime (\mathbf{x}')\,{\bf\times}\,\frac{\mathbf{r}}{r^3}\,d^3x',\end{align*} | |
\begin{align*}M^{-1}_{ij} =\left(\begin{array}{ccc}-(2k^{2}+\ell^{2})^{2}/4k^{2} & -\ell^{2}/2k^{2} &-(2k^{2}\ell+\ell^{3})/2k^{2} \\& & \\-\ell^{2}/2k^{2} & -1/k^{2} & -\ell/k^{2} \\& & \\-(2k^{2}\ell+\ell^{3})/2k^{2} & -\ell/k^{2} & -(k^{2}+\ell^{2})/k^{2} \\\end{array}\right) \, .\end{align*} | |
\begin{align*}\sigma_B^{k_1(x)} (h(\sigma_A(x))) & = \sigma_B^{l_1(x)}(h(x))x \in X_A, \\ \sigma_C^{k_2(y)} (g(\sigma_B(y))) & = \sigma_C^{l_2(y)}(g(y))y \in X_B. \end{align*} | |
\begin{align*}\begin{pmatrix} A & B \\ C &D \end{pmatrix} \end{align*} | |
\begin{align*}\begin{array}{ll}{}~~~[D,P_{\mu}]=-iP_{\mu}\,,~~~~&~~~~[D,K_{\mu}]=iK_{\mu}\,,\\{}&{}\\{}~~~[D,Q^{a}]=-i\textstyle{\frac{1}{2}}Q^{a}\,,~~~~&~~~~{[D,S^{a}]=i\textstyle{\frac{1}{2}}S^{a}}\,,\\{}&{}\\\multicolumn{2}{c}{[D,D]=[D,M_{\mu\nu}]=[D,A_{a}{}^{b}]=0\,.}\end{array}\end{align*} | |
\begin{align*}& \sum_{i=0}^{m-1} \{\sum_{i'=0}^{l_2 (\sigma_B^i(h(x)))-1} f(\sigma_C^{i'}(g(\sigma_B^i(h(x)))))-\sum_{j'=0}^{k_2 (\sigma_B^i(h(x)))-1} f(\sigma_C^{j'}(g(\sigma_B^{i}(h(x))))) \}\\=&\sum_{i'=0}^{l_2^m (h(x))-1} f(\sigma_C^{i'}(gh(x)))-\sum_{j'=0}^{k_2^m (h(x))-1} f(\sigma_C^{j'}(g(\sigma_B^{m}(h(x))))).\... | |
\begin{align*}& \sum_{i=0}^{l_1(x)-1} \{\sum_{i'=0}^{l_2 (\sigma_B^i(h(x)))-1} f(\sigma_C^{i'}(g(\sigma_B^i(h(x)))))-\sum_{j'=0}^{k_2 (\sigma_B^i(h(x)))-1} f(\sigma_C^{j'}(g(\sigma_B^{i}(h(x))))) \}\\=&\sum_{i'=0}^{l_2^{l_1(x)} (h(x))-1} f(\sigma_C^{i'}(gh(x)))-\sum_{j'=0}^{k_2^{l_1(x)}(h(x))-1} f(\sigma_C^{j'}(g(\sigm... | |
\begin{align*}& \sum_{i=0}^{l_1(x)-1} \Psi_g(f)(\sigma_B^i(h(x)) )\\=&\sum_{i'=0}^{l_2^{l_1(x)} (h(x))-1} f(\sigma_C^{i'}(gh(x)))-\sum_{j'=0}^{k_2^{l_1(x)}(h(x))-1} f(\sigma_C^{j'}(g(\sigma_B^{l_1(x)}(h(x))))).\end{align*} | |
\begin{align*}& \sum_{j=0}^{k_1(x)-1} \Psi_g(f)(\sigma_B^j(h(\sigma_A(x)))) \\=&\sum_{i'=0}^{l_2^{k_1(x)} (h(\sigma_A(x)))-1} f(\sigma_C^{i'}(gh(\sigma_A(x))))-\sum_{j'=0}^{k_2^{k_1(x)}(h(\sigma_A(x)))-1} f(\sigma_C^{j'}(g(\sigma_B^{k_1(x)}(h(\sigma_A(x)))))).\end{align*} | |
\begin{align*}& \sum_{i'=0}^{k_3(x)-1} f(\sigma_C^{i'}(gh(\sigma_A(x)))) \\-& \{\sum_{i'=l_2^{k_1(x)}(h(\sigma_A(x)))}^{k_3(x)-1} f(\sigma_C^{i'}(gh(\sigma_A(x)))) +\sum_{j'=0}^{k_2^{k_1(x)}(h(\sigma_A(x)))-1} f(\sigma_C^{j'}(g(\sigma_B^{k_1(x)}(h(\sigma_A(x)))))) \}.\end{align*} | |
\begin{align*}F^{(L)}_{p;r}(q) =q^{-r(r-1)} \times\left\{ \begin{array}{ll}F^{(L)}_{p-2} \left[ \begin{array}{c}{\bf Q}_{r,1} \\ {\bf 0} \end{array} \right]({\bf e}_r | q) &\mbox {if $L \not\equiv r-1$ mod $2$}, \\F^{(L)}_{p-2} \left[ \begin{array}{c}{\bf Q}_{p-r,p} \\ {\bf 0} \end{array} \right]({\bf e}_{p-r} | q) &\m... | |
\begin{align*}\kappa=-\epsilon\frac{m\pm 1/2}{k\pm 1/2},\quad s=\left\{\begin{array}{cc}k\pm 1/2+\epsilon (m\pm 1/2),&{\rm conventional}\\- (k\pm 1/2)-\epsilon (m\pm 1/2),&{\rm exotic}\end{array}\right. .\end{align*} | |
\begin{align*}\sigma_B^{k_1(x)} (h(\sigma_A(x))) & = \sigma_B^{l_1(x)}(h(x)),x \in X_A, \\\sigma_A^{k_2(y)} (h^{-1}(\sigma_B(y))) & = \sigma_A^{l_2(y)}(h^{-1}(y)),y \in X_B, \\\intertext{and}l_1(x) - k_1(x) = & 1 + b_1(x) - b_1(\sigma_A(x)), x \in X_A, \\l_2(y) - k_2(y) = & 1 + b_2(y) - b_2(\sigma_B(y)), y \in X_B. \en... | |
\begin{align*}\pi_B(\bar{h}(\bar{x})) & = \sigma_B^{f_1(x)}(h(x)) \bar{x} \in \bar{X}_A, \\\pi_A(\bar{h}^{-1}(\bar{y})) & = \sigma_A^{f_2(y)}(h^{-1}(y)) \bar{y} \in \bar{X}_B\end{align*} | |
\begin{align*}H=\frac{1}{2}(\tilde{ \phi}, \phi)= \frac{1}{2} \left( \begin{array}{cc} \phi^{0*} & \phi^{+} \\ -\phi^{-} & \phi^{0} \end{array} \right),\end{align*} | |
\begin{align*}\check{R}(z_1 -z_2 ) \phi _{ad}(z_1 )\otimes \phi _{dc}(z_2 )=\sum_{b} W_{z_1 -z_2 }\left( \begin{array}{cc} a & b \\ d & c \end{array} \right)\phi _{ab}(z_2 )\otimes \phi _{bc}(z_1 ). \end{align*} | |
\begin{align*}S_i^A e_x^A =\begin{cases}e_{ix}^A & ix \in X_A,\\0 & \end{cases}\end{align*} | |
\begin{align*}A=\begin{bmatrix}1 & 1 \\1 & 1\end{bmatrix},B=\begin{bmatrix}1 & 1 \\1 & 0 \end{bmatrix}.\end{align*} | |
\begin{align*}A=\begin{bmatrix}1 & 1 \\1 & 1\end{bmatrix},B=\begin{bmatrix}1 & 1& 0 \\1 & 0& 1 \\1 & 0& 1 \end{bmatrix}.\end{align*} | |
\begin{align*}\left(\begin{array}{cc}a&-1\\1&0\end{array}\right)=T^{a}S\end{align*} | |
\begin{align*}j=\left(\begin{array}{cccccc} \theta_1 &&&&&\\&\ddots&&&&\\ &&\theta_{q+1}&&& b\\ &&&\ddots &&\\&&&&\theta_p &\\&&c^t&&&0\end{array}\right).\end{align*} | |
\begin{gather*}\Psi(1,1) =(\alpha, \alpha), \Psi(2,1) =(\beta, \beta, \alpha), \Psi(3,1) =(\beta, \alpha, \alpha), \\\Psi(1,2) =(\alpha, \beta), \Psi(2,3) =(\beta, \beta, \beta), \Psi(3,3) =(\beta, \alpha, \beta)\end{gather*} | |
\begin{align*}l_2(y)& ={\begin{cases}3 & (y_1, y_2)=(1,1), (1,2), \\4 & (y_1, y_2)=(2,1), (2,3), (3,1), (3,3),\\\end{cases}} \\k_2(y)& ={\begin{cases}2 & (y_1, y_2)=(1,1), (2,1), (3,1), \\3 & (y_1, y_2)=(1,2), (2,3), (3,3), \\\end{cases}}\end{align*} | |
\begin{align*}\psi(\phi(\alpha,x_1,x_2,\dots ))& = (x_1,x_2,\dots ), \\\psi(\phi(\beta,x_1,x_2,\dots ))& = (x_1,x_2,\dots ) \end{align*} | |
\begin{align*}\phi(\alpha, \psi(y_3,y_4, \dots )) &= (y_3, y_4,\dots ) y_2 = 1, \\\phi(\beta, \psi(y_3,y_4, \dots )) &= (y_3, y_4,\dots ) y_2 = 2, 3.\end{align*} | |
\begin{align*}A=\begin{bmatrix}1 & 1 & 1\\1 & 1 & 1\\1 & 0 & 0\end{bmatrix},B= A^{t}=\begin{bmatrix}1 & 1 & 1\\1 & 1 & 0\\1 & 1 & 0\end{bmatrix}.\end{align*} | |
\begin{align*}\left(\begin{array}{ll}\quad \tilde{d} & -\tilde{b}/q\\- q \tilde{c} & \quad \tilde{a}\end{array}\right)=\left(\begin{array}{ll}a^{*} & c^{*}\\b^{*} & d^{*}\end{array}\right)\;.\end{align*} | |
\begin{align*}\Gamma = \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right),\end{align*} | |
\begin{align*}\sum_{\vec{V}_r(L)}\cdot\prod_{i=1}^{\nu-2}\left[\begin{array}{c}\frac{1}{2}(K_{\nu-2}\cdot\vec{m}+\vec{u}_r(L))_i+\frac{1}{2}L\delta_{i,\nu-2} \\ m_i \end{array}\right]=\end{align*} | |
\begin{align*}y^i_1\otimes 1 & 0 & \dots & 0 \\\end{align*} | |
\begin{align*}0 & y_2^i \otimes 1 & \dots & 0 \\\end{align*} | |
\begin{align*}\dots & \dots & \dots & \dots \\\end{align*} | |
\begin{align*}0 & \dots & \dots & y^i_N \otimes 1 \\\end{align*} | |
\begin{align*}\begin{array}{r c l l c c c c}\mbox{\underline{sector}}&{}&\mbox{\underline{state}}&{}&{}&\mbox{\underline{$R_1$}}&\mbox{\underline{$R_2$}}&\mbox{\underline{$R_3$}} \\\mbox{NS}&{}&|2s_3,2s_4\rangle,&s_3=s_4&{}&+&+&-2is_3\\\mbox{R}&{}&|-,2s_1,2s_2\rangle,&s_1=s_2&{}&+&+&+2is_1.\\\end{array}\end{align*} | |
\begin{align*}= i \int_0^P \left(\begin{array}{cc} \psi_1 \bar{\psi}_2 & -\bar{\psi}_2^2 \\\psi_1^2 & -\psi_1\bar{\psi}_2 \end{array}\right)dz +\left(\begin{array}{cc} \bar{\psi}_1 \psi_2 & \bar{\psi}_1^2 \\-\psi_2^2 & -\bar{\psi}_1 \psi_2 \end{array}\right)d\bar{z} =\end{align*} | |
\begin{align*}\widetilde{U} = U -i \left(\left(\begin{array} {cc} 0 & 1 \\ 0 & 0 \end{array}\right) G \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right)+ \left(\begin{array} {cc} 0 & 0 \\ 1 & 0 \end{array}\right) G\left(\begin{array} {cc} 0 & 0 \\ 0 & 1 \end{array}\right)\right),\end{align*} | |
\begin{align*}X_0=\left (\begin{array}{cccc}-a_1&-a_2&\dots&-a_N\\1&0&\dots&0\\0&1&\dots&0\\0&\dots&1&0\\\end{array}\right ).\end{align*} | |
\begin{align*}\psi({\bf x},0) = \left(\frac{\epsilon_p + m}{2\epsilon_p}\right)^\frac{1}{2} \left( \begin{array}{c} {\displaystyle -\frac{p}{\epsilon_p + m}}\chi_1 \\ \chi_1 \end{array} \right) e^{ipz}\end{align*} | |
\begin{align*} \{V,f\} &= V(f) \\ \{U,V\} &= [U,V],\end{align*} | |
\begin{align*} S_\rho[Z_{ij}] &= \int d^2x\Big(\pi^i(x)\check{\pi}^j(x) - \pi^j(x)\check\pi^i(x)\Big) \\ S_\rho[Z_{Ni}] &= \int d^2x \sigma(x)\check\pi^i(x).\end{align*} | |
\begin{align*} d_1S + \frac{1}{2}\{S,S\} &= \frac{1}{2}\{S^0,S^0\} + \Big(d_1S^0 - \{I^\nabla, S^0\}\Big) + \Big(-d_1I^\nabla + \frac{1}{2}\{I^\nabla,I^\nabla\}\Big)\\ &= S_R,\end{align*} | |
\begin{align*}S(\phi) = \left(\begin{array}{cc} \exp(\frac{i}{2}\phi) & 0\\0 & \exp(-\frac{i}{2}\phi)\end{array}\right)\end{align*} | |
\begin{align*}\Delta_{IJ}=\left(\begin{array}{cc}I_{4} & 0 \\0 & -I_{4} \end{array}\right)_{IJ} ~ . \end{align*} | |
\begin{align*}\begin{aligned} R_\ell\pi^i(x) & = \pi^i(\ell x) \\ R_\ell\check{\pi}^i(x) & = \ell^2 \check\pi^i(\ell x), \ell > 0\end{aligned} \end{align*} | |
\begin{align*}\begin{array}{rclrcl}\delta k & = & \frac{1}{2} \theta k\ell\, ,\hspace{1cm}&\delta A^{(1)}&=&-\theta A^{(2)}\, ,\\& & & & & \\\delta e^{\phi}&=&-\frac{7}{4}\theta \ell e^{\phi}\, ,&\delta A^{(2)} & = & \theta A^{(1)}\, ,\\& & & & & \\\delta \ell&=&\theta \left( 1 +\ell^{2} -2k e^{-2\phi}\right)\, ,\hspac... | |
\begin{align*}f(x) = \left\{\begin{array}{ll}1 & \mbox{if } x \in A_1 \\1-\frac{d(x,A_1)}{r} & \mbox{if } x \in A_1(r) \setminus A_1 \\0 & \mbox{if } x \notin A_1(r)\end{array}\right.\end{align*} | |
\begin{align*}g_i(s,\tau) = \left\{\begin{array}{cl}f_i(s) & \mbox{if } \tau \leq f_i(s) \\\tau & \mbox{if } \tau > f_i(s)\end{array}\right.\end{align*} | |
\begin{align*}G_r^{(2)}(x)=\sum^{\infty}_{n_0,n_1,\ldots,n_{\nu-2}=0} x^{\frac{r-1}{2}+\sum_{i=0}^{\nu-2}n_i(\nu-1-i)}\times\prod_{i=1}^{\nu-2}\left[\begin{array}{c}n_i+\tilde{n}_i+\tilde{V}_{i,r} \\ n_i \end{array} \right] .\end{align*} | |
\begin{align*}{\bf \tau}(E)=\frac{1}{\det {\bf \tau}^{-1}} \left( \begin{array}{cc}-\frac{C}{C_2^2} - I_5 - 2 \mu E I_3 - 4 \mu^2 E^2 I(E) & I_3 + 2 \mu E \, I(E) -\frac{1}{C_2}\\ I_3 + 2 \mu E \, I(E) - \frac{1}{C_2} & -I(E)\end{array} \right),\end{align*} | |
\begin{align*}P^{(\pm 1)}=\left( \begin{array}{cc} \frac{1+\Gamma}{2} p^{(\pm)} & 0 \\ 0 & \frac{1-\Gamma}{2} p^{(\pm)} \end{array} \right). \end{align*} | |
\begin{align*}\left\{\begin{array}{c} M_{H}L_2(p) = E_t(p)L_1(p)-\hat{I}_q\left\{\left[c^{(-)}_pc_q^{(-)} +\hat{p}\cdot\hat{q}s_p^{(-)}s_q^{(-)}\right]L_1(q)\right\} \\ M_{H}L_1(p) = E_t(p)L_2(p)-\hat{I}_q\left\{\left[c^{(+)}_pc_q^{(+)} +\hat{p}\cdot\hat{q}s_p^{(+)}s_q^{(+)}\right]L_2(q)\right\}. \end{array}\right.\end... | |
\begin{align*} \begin{array}{lll} A) & J_1,J_2,J_3\mbox{ unrestricted,} \qquad & l \geq 6 + \frac{1}{2}(J_1+2 J_2+3 J_3) + 2(a_1+a_2) \\ B) & J_3 = 0, \qquad & l = 4 + \frac{1}{2} (J_1+2 J_2)+ 2(a_1+a_2) \\ C) & J_2=J_3=0, \qquad & l = 2 + \frac{1}{2} J_1 + 2(a_1+a_2) \\ D) & J_1=J_2=J_3=0,\qquad & l = 2 (a_1+a_2) \end... | |
\begin{align*}\begin{array}{lll}\mathcal{Q}_{t}(x_{1:t-1})=\rho_t\Big( \mathfrak{Q}_{t}(x_{1:t-1}, \xi_t) \Big) & \geq & \rho_t \Big( \mathfrak{Q}_{t}^{k-1}(x_{1:t-1}, \xi_t) \Big)\\& \geq & \displaystyle \sup_{p \in \mathcal{P}_t} \;\displaystyle \sum_{j=1}^{M} p_j \Phi_{t, j} \mathfrak{Q}_{t}^{k-1}(x_{1:t-1}, \xi_{t,... | |
\begin{align*}\psi=\left( \begin{array}{c} \psi_1 \\ \psi_2 \end{array}\right)\,,\qquad \psi_c = \left( \begin{array}{r} \psi_1^\dagger \\ \ -\psi_2^\dagger \end{array}\right)\,.\end{align*} | |
\begin{align*}\begin{array}{rcl} {\cal{V}}_0 &=& {\cal{V}}_1, \\ {\cal{E}}_0 &=& -[Q+{\cal{A}}_0, {\cal{V}}_1]. \end{array}\end{align*} | |
\begin{align*}\Gamma_{s}(P_{a}) =\left\{ \begin{array}{ccc}\frac{i}{2R_{7}}\Gamma^{789\, 10}\Gamma_{a}\, ,& \hspace{1cm} & a\leq 6\, ,\\& & \\-\frac{i}{R_{4}}\Gamma^{789\, 10}\Gamma_{a}\, ,& \hspace{1cm} & a> 6\, .\\ \end{array}\right.\end{align*} | |
\begin{align*}\left\{ \begin{array}{l}\kappa r = z^{2/(\beta + 2)}\\ \kappa^{-D/2} u (r) = w (z) \,z^{-\beta/2(\beta + 2)}\end{array}\right.\;\end{align*} | |
\begin{align*}Z^{(1)}[\eta]={\rm Sdet}^{-\frac12}{\sf Y}[\eta]={\rm Sdet}^{-\frac12}\left(\begin{array}{cc}p^2 & \lambda\bar\eta \\\lambda\eta & ip\llap/ \end{array}\right).\end{align*} | |
\begin{align*} \left\{\begin{array}{l}\displaystyle \inf_{x_m} \;F_{\tau_m}^{k-1}(x_{[n]}, x_m, \Psi_{m}):=f_{\tau_m}(x_{[n]}, x_m, \Psi_{m}) + \mathcal{Q}_{m}^{k-1}(x_{[n]}, x_{m})\\g_{\tau_m}(x_0, x_{[n]}, x_{m}, \Psi_{m}) \leq 0,\\\mbox{[ } A_{0, m}, \ldots, A_{\tau_m, m}\mbox{]} \mbox{[}x_0; x_{[n]}; x_{m}\mbox{]}=... | |
\begin{align*}\left\{\begin{array}{ll} 1\leq \rho\leq \frac{g-2}{2},\quad 1\leq y \leq |\tilde{G}|, & \quad\mbox{$g$ even},\\ 1\leq \rho\leq g-2, \quad 1\leq y \leq \frac{g-1}{2}, &\quad\mbox{$G=A_{g-1}$, $g$ odd,}\end{array} \right.\end{align*} | |
\begin{align*}\left\vert E, p_1^{\mbox{min}},\ldots, p_{N-1}^{\mbox{min}}\right>=\left\vert\begin{array}{c} E, \left[0\right]\\ \left[ q \right] \end{array}\right>=\left\vert \begin{array}{c} E, 0, \ldots , 0 \\ q_1,\ldots,q_k,\ldots,q_{N} \end{array} \right>.\end{align*} | |
\begin{align*}\Omega (\mid x\mid_p) = \left \{ \begin{array}{ll} 1, & 0 \leq \mid x\mid_p \leq 1 ,\\0, & \mid x\mid_p>1, \end{array} \right.\end{align*} | |
\begin{align*}\begin{array}{rl} E_{8}^{+} = & \{\pm e_{a}\} \\\cup & \{(\pm e_{a}\pm e_{b}\pm e_{c}\pm e_{d})/2: a,b,c,d\mbox{ distinct}, \; e_{a}(e_{b}(e_{c}e_{d}))=\pm 1\}, \\ \\& a,b,c,d\in\{0,...,7\}. \\\end{array}\end{align*} | |
\begin{align*}D(\gamma)=\left(\begin{array}{cc}a_{\gamma}&0\\0&a_{\gamma}^{-1} \end{array}\right) \:,\qquad\qquad|a_{\gamma}|>1 \:.\end{align*} | |
\begin{align*}\begin{array}{rcl}{\cal M}^{\prime} & = & S^{T}{\cal M}S\, ,\\& & \\K^{\prime} & = & RK\, .\\\end{array}\end{align*} | |
\begin{align*} {\nabla }\,v=\left( \begin{array}{c} {\rm grad}^T\, v_1 \\ {\rm grad}^T\, v_2 \\ {\rm grad}^T\, v_3 \\ \end{array}\right)\, ,\end{align*} | |
\begin{align*}\begin{array}{l}f^{(+)}_{lk}f^{(+)}_{cd}-f^{(+)}_{ld}f^{(+)}_{ck}=\|f^{(+)}\|\varepsilon_{lcm}\varepsilon_{kdn}f^{(-)}_{nm},\\ f^{(-)}_{kl}f^{(-)}_{dc}-f^{(-)}_{dl}f^{(-)}_{kc}=\|f^{(-)}\|\varepsilon_{lcm}\varepsilon_{kdn}f^{(+)}_{mn},\end{array}\end{align*} | |
\begin{align*}Z\; =\;\prod_{j=1}^{M} \Gamma_{\, j} , E\; =\;\prod_{j=1}^{M} V_{\, j} ,V_{\, j}\; =\; \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right)_{\, j} .\end{align*} | |
\begin{align*}\left|\left (\begin{array}{cc}[m]+[\Delta (\Gamma )]\\ (m^\prime)\end{array}\right); (\Gamma )\right\rangle=\end{align*} | |
\begin{align*}{\hspace{1mm}^*}\hspace{-1mm}f_{\mu \nu}(x) = - \frac{e_0}{4 \pi \varepsilon_0 c} [ \partial_\mu \vec{n}(x) \wedge \partial_\nu \vec{n}(x) ] \vec{n}(x)=\left( \begin{array}{cccc} 0 & B_x & B_y & B_z \\ -B_x & 0 & \frac{E_z}{c} & \frac{-E_y}{c} \\ -B_y & \frac{-E_z}{c} & 0 & \frac{E_x}{c} \\ -B_z & \frac{E... | |
\begin{align*}j~=~{\textstyle{1\over 2}} \; \left( \begin{array}{cc} 1 & - \mbox{Ad}(g) \\ - \mbox{Ad}(g)^{-1} & 1 \end{array} \right)~~~.\end{align*} | |
\begin{align*}g\left(\begin{array}{cc} n_1 & n_2 \\ m_1 & m_2 \end{array}\right)=\frac{G}{2^{(n_1+m_1+n_2+m_2)/2}}\sqrt{\frac{ \Gamma(n_1+m_1+1)\Gamma(n_2+m_2+1)}{\Gamma(n_1+1)\Gamma(m_1+1)\Gamma(n_2+1)\Gamma(m_2+1)}}.\end{align*} | |
\begin{align*}[\hat{\rm H}_{\rm phys}, \hat{\rm P}_{\rm phys}]_{-} = 0, \\\end{align*} | |
\begin{align*}\omega_{ab} = \left ( \begin{array}{cc} 0 & I_N \\-I_N & 0 \end{array} \right )\end{align*} | |
\begin{align*} \frac{d}{dz}\left[2\,k\,z\,\log^2\frac{1+z}{1-z}+2\,z-\log\frac{1+z}{1-z}\right]&=2\,k\,\log^2\frac{1+z}{1-z}+\frac{8kz}{1-z^2}\log\frac{1+z}{1-z}+2-\frac2{1-z^2}\\ &=2\,k\,\log^2\frac{1+z}{1-z}+\frac{2z}{1-z^2}(4\,k\,\log\frac{1+z}{1-z}-z). \end{align*} | |
\begin{align*}\begin{tabular}{llllllll}$N\ \ D_{p}-\mathrm{branes}:$ & 0 & 1\thinspace \thinspace 2 & 3$\cdots p$ &\quad $-$ & \quad $-$ & \quad $-$ & ,$\quad \mathrm{with}\ B_{12}$ \\$\quad O_{p+2}-\mathrm{plane}:$ & 0 & 1$\,-$ & 3$\cdots p$ & $p+1$ & $p+2$ &$p+3$ & .\end{tabular}\end{align*} | |
\begin{align*} \frac{\partial^2 \psi}{\partial i_1^2}(i_1,i_2)&=\frac{8\,i_2\,e^{\frac{k}{2}\log^2\frac{i_1+R}{i_1-R}}} {\sqrt{i_1^2-4\,i_2}(i_1^2-4\,i_2)(-1) (\sqrt{i_1^2-4\,i_2}-i_1)(\sqrt{i_1^2-4\,i_2}+i_1)}\cdot\\ &\qquad\cdot\left(2\,k\,R\log^2\frac{i_1+R}{i_1-R}+2\,R -i_1\log\frac{i_1+R}{i_1-R}\right), \end{align... | |
\begin{align*}\left[\begin{array}{cc}0&-\left({2\pi n\over m_l-m_r}\right)^2 b_i\\ b_i&0\end{array}\right],\;\;\;\;\;\left[\begin{array}{cc}0&-\left({2\pi n\over m_l-m_r}\right)^2 b_i^*\\ b_i^*&0\end{array}\right]\end{align*} | |
\begin{align*} \begin{array}{c} B^-B^+-QB^+B^-~=~Q^{-N_B}\\ B^-B^+-Q^{-1}B^+B^-~=~Q^{N_B}\\ Q^{N_B}B^+Q^{-N_B}~=~QB^+\\ Q^{N_B}B^-Q^{-N_B}~=~Q^{-1}B^-\\ Q^{N_B}Q^{-N_B}~=~Q^{-N_B}Q^{N_B}~=~1 \end{array}\end{align*} | |
\begin{align*}\left\{\begin{array}{l}\partial_t h = G(h) \phi\\\partial_t \phi = - g h + \sigma \dfrac{\partial_x^2 h}{ (1+h_x^2)^{3/2} } - \dfrac{1}{2} {|\phi_x|}^2 + \dfrac{{\left( G(h)\phi + h_x \phi_x \right)}^2}{2(1+{|h_x|}^2)}\end{array}\right.\end{align*} |
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