harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[ A \otimes_R M, U]_{A-\mathrm{Alg}/A}\simeq [A \otimes_{R-alg/A}M, U]_{_{A \setminus}R-\mathrm{Alg}_{/A}}=[M, [A, U]_{R-\mathrm{Alg}_{/A}}]_{\mathrm{Top}_+}
\mathfrak{a}_2\mathfrak{a}_3^3 \mathfrak{a}_4^3 = (1 \otimes y_2- y_2 \otimes 1) (1 \otimes y_3^3- y_3 \otimes y_3^2+ y_3^2 \otimes y_3- y_3^3 \otimes 1) (1 \otimes y_4^3- y_4 \otimes y_4^2 & + y_4^2 \otimes y_4- y_4^3 \otimes 1) &= y_1y_2y_3y_4 \otimes y_2y_3y_4 - y_2y_3y_4 \otimes y_1y_2y_3y_4.
b_j^{n-2s}H(q_j, q_j) - \sum_{i\neq j}(b_ib_j)^{\frac{n-2s}{2}}G(q_j, q_i)-\frac{2sc_1 A +c_2}{(n-4s)c_{n,s}^{n-4s}c_1}b_j^{2s}= 0 j = 1,\cdots, k.
f R_{i j k l}=\frac{1}{2(n-2)}\left(g_{i k}-\partial_i f \partial_k f\right)\left(g_{j l}-\partial_j f \partial_l f\right) -\frac{1}{2(n-2)}\left(g_{i l}-\partial_i f \partial_l f\right)\left(g_{j k}-\partial_j f \partial_k f\right) +O(r^{-\frac{1}{2}-\delta}).
D_{K_4^-}= \sum_{\left\{u,v\right\}\in E(\overline{G})}\binom{d_G(u,v)}{2}\geq e(\overline{G})\binom{\frac{1}{e(\overline{G})}\sum_{\left\{u,v\right\}\in E(\overline{G})}d_G(u,v)}{2}&=\frac{n^2}{4}\binom{n/4}{2}+o(n^4)=\frac{n^4}{128}+o(n^4) \,.
\lim_{\lambda \to +0}\lim_{V \to \infty}\omega^{0}_{\beta,\mu,\Lambda,\lambda}(\eta_{\Lambda}(b_{{0}}^{*}))=\lim_{\lambda \to +0}\lim_{V \to \infty}\frac{\partial}{\partial \lambda_{\phi}}p_{\beta,\mu,\Lambda,\lambda_{\phi}}\ ,
w(s)=\frac{1}{\gamma_2-\gamma_1}\left(e^{-\gamma_1 s}\int_{s_0}^se^{\gamma_1 t}f(t)\,dt-e^{-\gamma_2 s}\int_{s_0}^se^{\gamma_2 t}f(t)\,dt\right)+C_2(s_0)e^{-\gamma_1 s}-C_3(s_0)e^{-\gamma_2 s}
\left\{q_{( u )}^{- A}, q_{( u ) B}^{-}\right\}= 2 \delta^{A}{}_{B}\delta_{u v}H_{2 ( u )}+ i \mu \delta_{u v}\left( ( \gamma^{i j}\Pi ){}^{A}{}_{B}J_{( u )}^{i j}+ ( \gamma^{i^{\prime}j^{\prime}}\Pi^{\prime}){}^{A}{}_{B}J_{( u )}^{i^{\prime}j^{\prime}}-{\tilde{\Pi}}^{A}{}_{B}J_{( u )}^{L R}\right) \, ,
A_{\mathrm{i n t}}[ f ] = - \int d^{4}x \, \delta ( F - \tau ) \mathcal{L}_{\mathrm{i n t}}f^{\mu}\partial_{\mu}F = - \int d^{4}x \, \delta ( F - \tau ) \mathcal{L}_{\mathrm{i n t}}\delta_{f}F = 0 \; ,
S_{\mathrm{D B I}}^{( \mathrm{M 1 0})}= - \int_{R^{9 + 1}}|{\hat{k}}|^{3}\sqrt{| \mathrm{d e t}\left({\hat{\Pi}}+ |{\hat{k}}|^{- 1}{\hat{\cal F}}\right) |}\,{\hat{R}}({\hat{T}}, \partial{\hat{T}}, \dots ) \, .
\begin{array}{lcr}P(W_2)=(y^2+x^3+z^6+ \mu xyz)+ w(a_0 z^{n+3}+a_1 yz^n+a_{n-1}z^{{7 \over2}}x^{{(n-6)\over 2}}y^{{1\over 2}}) + w^2\sum\limits_{i=0}^{n-4}a_{i+1}z^{2(n-i)}x^i. \end{array}
\P\bigl( \bar{e}^{N}(T)\geq T^{1/4}\bigr) \leq\P\bigl( \bar{e}^{N}(T)\geq T^{1/4},T<\tau\bigr) +\P\bigl( \tau\leq T\bigr) \leq\P\biggl( \frac{1}{N}\sum_{i=1}^{N}(M_{T}^{i})^{2}\geq T^{-1/2}\biggr) + \P\bigl( \tau\leq T\bigr) \leq\P( A^{\complement}) +\P\bigl( \tau\leq T\bigr) \leq C'T.
K^{1_{+ +}1_{+ +}}( x , b^{\prime}) = \sum_{n = - \infty}^{\infty}\left\{- \sqrt{\alpha - 1}\cosh \left[{\frac{( r_{+}^{2}- r_{-}^{2})}{r_{+}}}\Delta t - r_{-}\Delta \phi_{n}) \right] + \sqrt{\alpha}\cosh ( r_{+}\Delta \phi_{n}) \right\}^{- 2 h_{+}}.
p\left({{d_{pF}}\leq \delta}\right) = M\int_{\mathbf{\Omega}}\prod\limits_{i = 1}^k{{{\left({{{\sin}^2}{\theta_i}}\right)}^{n - 2k}}}\,\,\, \times \prod\limits_{1 \leq i < j}^k{{{\left({{{\sin}^2}{\theta_i}-{{\sin}^2}{\theta_j}}\right)}^2}}d{{\sin}^2}{\theta_1}\cdots d{{\sin}^2}{\theta_k}.
\frac{1}{\sqrt{1-\rho^{2}}}exp(-\frac{\rho^{2}(x^{2}+y^{2})-2\rho xy}{2(1-\rho^{2})})=\sum_{n=0}^{\infty}\frac{\rho^{n}}{n!}He_{n}(x)He_{n}(y).
\left(\begin{array}{c}v_a v_b t_a t_b\end{array}\right)\to M^\epsilon_{K3}\left(\begin{array}{c}v_a v_b t_a t_b\end{array}\right) =\left(\begin{array}{rrrr}1 0 0 0 0 1 0 0 0 -1 -1 -1 1 1 1 0\end{array}\right)\left(\begin{array}{c}v_a v_b t_a t_b\end{array}\right)
{\cal R}_{\alpha \beta}{\eta}^{\alpha}{\eta}^{\beta}= \Gamma_{n + 1 \alpha , n + 1}^{\alpha}- \Gamma_{n + 1 \, n + 1 , \alpha}^{\alpha}+ \Gamma_{n + 1 \, \alpha}^{\beta}\Gamma_{\beta \, n + 1}^{\alpha}- \Gamma_{n + 1 \, n + 1}^{\alpha}\Gamma_{\alpha \beta}^{\beta}= \kappa^{2}- \dot{h}
\|\partial_k \varphi_1\|_{H^m}=\|\partial_k (-\alpha \Delta +2\omega +i\mathbf{c}\cdot \nabla)^{-1}\{(\nabla\cdot \varphi_3)\varphi_2\}\|_{H^m}\lesssim \|(\nabla\cdot \varphi_3)\varphi_2\|_{H^m}\lesssim \|\varphi_3\|_{H^{m+1}}\|\varphi_2\|_{H^m}<\infty
{\mathfrak q}_+^{Sp(p,q)\times Sp(1)}={\mathfrak l}_+^{Sp(p,q)\times Sp(1)}+{\mathfrak u}_+^{Sp(p,q)\times Sp(1)}\subset{\mathfrak s}{\mathfrak p}(n,{\mathbb C}) \times{\mathfrak s}{\mathfrak p}(1,{\mathbb C}),
p_{s_{ij}, u_{ij}}= p_{j,u_{jj}}= v^{\ell(w_0) - 2(j+1)}+ v^{\ell(w_0) - 2(j+2)}+ \cdots + v^{\ell(w_0) - 2n}, p_{s_{ij}, w_{ij}}= v^{-j+i}\cdot p_{e,w_{i-1,j}}, p_{t_{ij}, u_{ij}}= v^{-2i}\cdot p_{j,u_{jj}}, p_{t_{ij}, w_{ij}}= p_{j,u_{jj}}
\left(\begin{array}{c}\phi_1 \phi_5 \end{array}\right) = A \mathbf{U}^H_+(z; p_{z*},0,\omega_*) +B \left[ \frac{z}{b\lambda}\mathbf{U}^H_+(z; p_{z*},0,\omega_*) + \mathbf{Q}^H(z; p_{z*}, \omega_*)\right] +\left(\begin{array}{c}\widetilde{\phi}_1 \widetilde{\phi}_5 \end{array}\right),
\Gamma_t^{(4)}({\bf{q}}_1 ,{\bf{q}}_2 ,{\bf{q}}_3 ,{\bf{q}}_4 ) - g_t \sim g_t^2 \gamma^{(4)}({\bf{q}}_1 ,{\bf{q}}_2 ,{\bf{q}}_3 ,{\bf{q}}_4 ) + O ( g_t^3 ) \; .
S_{d3}=-\int d^4 x \,{1\over 4g^2_{{\rm YM}}}\,{\rm Tr}\Bigl(F_{ab}F^{ab}+{2\over (2\pi\alpha')^2}D_a X^{\mu}D^a X_{\mu}+{1\over (2\pi\alpha')^4}[X^{\mu}, X^{\nu}]^2 \Bigr) + \cdots,
\begin{cases}\lambda \sqrt{\det h}X_z=-\partial_y(h_{11}X_x+h_{12}X_y)+\partial_x(h_{22}X_y+h_{12}X_x) -\lambda \sqrt{\det h}X_y=\partial_xX_z \lambda \sqrt{\det h}X_x=\partial_yX_z\,,\end{cases}
S_n =\sum_{(j,k)\in\Gamma_n}\sum_{r, s\in \mathbb Z}a_{r,s}\xi_{j-r, k-s}&= \sum_{(j,k)\in\Gamma_n}\sum_{r, s\in \mathbb Z}a_{j+r,k+s}\xi_{-r, -s}&= \sum_{(j,k)\in\Gamma_n}\lim_{m \to \infty}\sum_{r, s\in [-m, m]}a_{j+r,k+s}\xi_{-r, -s}=\lim_{m \to \infty}\sum_{(j,k)\in\Gamma_n}\sum_{r, s\in [-m, m]}a_{j+r,k+s}\xi_{-r, -s},
\binom{-a}{b}=(-1)^b\binom{a+b-1}{a-1},\quad\frac{(t+1)^{(\ell-1)}(t+\ell)^{(\ell-t-\mu)}}{(t+1)^{(\ell-t)}}=\frac{(2\ell-\mu-1)!}{\ell!},
\sum_{\substack{a,b r=s+t-a+b}}\binom{a}{2}\binom{s}{a}\binom{r-s}{b}=\binom{s}{2}\sum_{\substack{a,b r=s+t-a+b}}\binom{s-2}{a-2}\binom{r-s}{b}=\binom{s}{2}\sum_b \binom{s-2}{r-t-b}\binom{r-s}{b}, =\binom{s}{2}\binom{r-2}{r-t}.
p^{(id,id,\rho_0)}=\left[\begin{array}{ccccccc}1 3 3 1 1 2 2 1 2 3 3 1 2 3 2 \end{array}\right],p^{(\varphi,\psi,\rho_0)}=\left[\begin{array}{ccccccc}2 2 2 3 3 1 3 1 2 1 3 1 3 1 2 \end{array}\right].
{}\sum_{j=1}^{q}B_{ij}(t)\cdot z_{j}^{(k_{r})}(t)\leq c_{i}(t)+\epsilon+\sum_{j=1}^{q}\int_{0}^{t}K_{ij}(t,s)\cdot z_{j}^{(k_{r})}(s)ds\mbox{a.e. in $[0,T]$}.
\frac{\kappa p+1}{2(\kappa+1)^p}\|u u_h^{\kappa}\|_{1,p}^p &\leq \left[\frac{\kappa p+1}{(\kappa+1)^p}+M_1(\kappa p+ 1) L(\kappa, u)\right]\|u u_h^{\kappa}\|_{p}^p +M_2G(\kappa, u) \| u u_h^{\kappa}\|_{p,\partial\Omega}^p +M_3\kappa,
S_G= \int d^3x\left[-\epsilon\frac{1}{2}\left(D_\mu Q^AD^{\mu}Q^B\eta_{AB}\right) -W_G(Q^A)-\kappa\epsilon^{\mu\nu\rho}\mbox{Tr}\left(\partial_\mu A_\nu A_\rho+\frac{2}{3}A_\mu A_\nu A_\rho\right)\right],
\left| \mathrm{Ces}(tP ,N) - \left| tP \cap \mathbb{Z}^d \right| \right| \le \int_{[-h,h]^d}\left| f(x) - \left| tP \cap \mathbb{Z}^d \right| \right| F_N (x) \, \mathrm{d}x + \int_{\left[ - \frac{1}{2}, \frac{1}{2}\right]^d \backslash [-h,h]^d}\left| f(x) - \left| tP \cap \mathbb{Z}^d \right| \right| F_N (x) \, \mathrm{d}x .
n\,\mbox{{\rm Var}}\left( f_{n}(x)\right) =h_{n}^{-2}{\bf \,E\,}K^{2}\left(\frac{x-X}{h_{n}}\right) -\left\{h_{n}^{-1}{\bf \,E\,}K\left( \frac{x-X}{h_{n}}\right) \right\}^{2}.
h^\chi_m \! =\! \sqrt{\frac{\rho^\chi_0}{(d^\chi_m)^{\alpha}}}\Big( \sqrt{\frac{K^{\chi}_h}{K^{\chi}_h\!+\!1}}h^{\rm{LoS},\chi}_m \!+\! \sqrt{\frac{1}{K^\chi_h\!+\!1}}h^{\rm{NLoS},\chi}_m \Big),
I = \frac{k}{\pi}\int dt d\theta \left[ \partial_- \beta \partial_\theta \beta+ \partial_\theta \alpha \partial_- \gamma \exp{(2 \beta)}\right]+ \frac{k}{\pi}\int dt d\theta Tr\left(h^{-1}K(t) \partial_- h \right)- \frac{k}{4}\int dt (K^3)^2
\Lambda^* = \max \; \lim_{N\rightarrow \infty}\mathbb{E}_{\mathbf{H}}\!\left[\!\frac{1}{N}\sum_{t=1}^N g(s_t, a_t(s_t))\!\right] = \max \; \lim_{N\rightarrow \infty}\mathbb{E}_{\mathbf{H}}\!\left[\!\frac{1}{N}\sum_{t=1}^N \bar{g}(s_t, a_t(s_t))\!\right]
u_{s}^{\dag}( \vec{k}) u_{s^{\prime}}( \vec{k}) = \delta_{s s^{\prime}}, v_{s}^{\dag}( \vec{k}) v_{s^{\prime}}( \vec{k}) = \delta_{s s^{\prime}}, v_{s}^{T}( \vec{k}) u_{s^{\prime}}( - \vec{k}) = 0 ,
\begin{array}{cl}A_{\lambda}^{(m)}(x|z^{(n)}|\cdots|z^{(1)}) |_{z_m^{(k)}=a\tau^{2n-2k}~~(k=1, \cdots, n)}=(x-a\tau^{n-1}) \times \displaystyle\sum_{\rho =0}^{n-1}(-1)^{\rho}\sigma_{\rho}(a\tau, a\tau^3 , \cdots , a\tau^{2n-3}) A_{\lambda -\rho}^{(m-1)}(x|z'^{(n)}|\cdots|z'^{(1)}).\end{array}
r_{\ell}\leq e^{40\varepsilon q_n}\frac{e^{-q_nL}}{\max(|\ell|,1)}\max(r_{\ell-1}, r_{\ell+1})\times\begin{cases}\max(|\ell|, e^{\delta_n q_n}, 1), \beta_n\geq \delta_n+200\varepsilon e^{\beta_n q_n}, \beta_n<\delta_n+200\varepsilon\end{cases}.
\psi_{n_1,\ldots,n_r}(w_1\cdots w_r) := \sum_{k=h}^N \sum_{\substack{0<i_1<\cdots<i_k<N+1 \{i_1,\ldots,i_k\}\cap P_{n_1,\ldots,n_r}= \overline{\rho}_{n_1,\ldots,n_r}(w_1\cdots w_r)}}\varphi({\bf e}_{i_1,\ldots,i_k}(n_1,\ldots,n_r)) ,
J_{t,0}=J_{s,0}+\sum_{r=1}^d\bigg[\int_s^t \frac{\partial a_i}{\partial x_r}(u,X_u) J_{u,0}\d u +\sum_{k=1}^m \int_s^t \frac{\partial b_{\cdot,k}}{\partial x_r}(s,X_s)J_{u,0}\d W^{k}_u+ \sum_{q=1}^l \int_s^t \frac{\partial c_{\cdot,q}}{\partial x_r}(s,X_s)J_{u,0}\d B^{H,q}_u\bigg],
V = q \left({{\cal A}_{0}\left({\bf y}\right) -{\cal A}_{0}\left({{\bf y}^{\prime}}\right)}\right) = - \frac{{q^{2}}}{\pi}\frac{1}{{1 + e^{2}a \left({1 - 1 2 \alpha^{2}g^{2}\partial^{2}}\right)}}K_{0}\left({\chi |{\bf y}-{\bf y}^{\prime}|}\right) .
f_{3;1}(t) =(t-\tfrac74)^++(t-\tfrac34)^++(t-\tfrac12)^++1\leq\tfrac{15}{4}, f_{3;2}(t) =(t-\tfrac74)^++(t-\tfrac34)^++1\leq\tfrac{7}{4}, f_{3;3}(t) =(t-\tfrac74)^++\tfrac34\leq\tfrac{3}{4},
\langle\deltaj^{\mu}(x)\rangle\simeq\frac{1}{2}\int\frac{d^{\,3}p}{(2\pi)^{3}}\left[\left(1,{\bfp}/\omega_{+}\right)^{\mu}\frac{-i\Sigma_{off}^{(bare)(+)}}{|Im\Sigma_{R}^{(+)}|}-\left(1,-{\bfp}/\omega_{-}\right)^{\mu}\frac{-i\Sigma_{off}^{(bare)(-)}}{|Im\Sigma_{R}^{(-)}|}\right].
\langle\Psi_{1}^{I_{1}}(x_{1},\theta_{1}^{a},\bar{\theta}_{1a})\,\cdots\,\rangle=e^{i(\kappa_{1}+\cdots+\kappa_{n})\Omega}\langle\Psi_{1}^{I_{1}}(x_{1},e^{i\frac{1}{2}\Omega}\theta_{1}^{a},e^{-i\frac{1}{2}\Omega}\bar{\theta}_{1a})\,\cdots\,\rangle\,,
\begin{aligned}{T_{Dp-brane}}&{{}=T_{\frac{p}{2},\frac{p}{2}-1}=\frac{(2\pi)^{\frac{p}{2}}\,R^{\frac{p}{2}-1}}{l_{p}^{\frac{3p}{2}}}\,=\,\frac{1}{(2\pi)^{p}\,g_{s}\,l_{s}^{p+1}}\,,}\\{T_{NSp-brane}}&{{}=T_{\frac{p+3}{4},\frac{5-p}{4}}=\frac{(2\pi)^{\frac{9-p}{4}}\,R^{\frac{5-p}{4}}}{l_{p}^{\frac{2p+9}{4}}}\,=\,\frac{1}{(2\pi)^{p}\,g_{s}^{\frac{p-1}{2}}\,l_{s}^{p+1}}\,.}\\\end{aligned}
R_{A}^{~B}(z;g)=\frac{\partialx^{\primeB}}{\partialx^{A}}-i\bar{\theta}_{i}^{\prime}\gamma^{B}\frac{\partial\theta^{\primei}}{\partialx^{A}}-i\bar{\tilde{\theta}}{}_{\tilde{\imath}}^{\prime}\tilde{\gamma}^{B}\frac{\partial\tilde{\theta}^{\prime\tilde{\imath}}}{\partialx^{A}}
\sum_{\nu}\,^{\prime}\,\sum_{\mu}\sum_{\kappa}\intd^{2}x\intd^{2}y\intd^{2}z(\Psi_{\nu}^{*}(x)\delta\lambda_{\mu}\Psi_{\mu}(x))(\Psi_{\mu}^{*}(y)e^{-\epsilon\lambda_{\kappa}}\Psi_{\kappa}(y))(\Psi_{\kappa}^{*}(z){\frac{1}{\lambda_{\nu}}}\Psi_{\nu}(z))\,,
\hat{a}=\left(\begin{matrix}{a}&{0}\\{0}&{I_{D-d}}\\\end{matrix}\right),\\\hat{b}=\left(\begin{matrix}{b}&{0}\\{0}&{0}\\\end{matrix}\right),\\\hat{c}=\left(\begin{matrix}{c}&{0}\\{0}&{0}\\\end{matrix}\right),\\\hat{d}=\left(\begin{matrix}{d}&{0}\\{0}&{I_{D-d}}\\\end{matrix}\right).
\{\eta^{MN}\}=\left(\begin{array}{cc}{\eta^{\alpha\beta}}&{\eta^{\mu\beta}}\\{\eta^{\alpha\nu}}&{\eta^{\mu\nu}}\\\end{array}\right)=\left(\begin{array}{cc}{g^{\alpha\beta}}&{-g^{\gamma\beta}\,A_{\gamma}^{\mu}}\\{-g^{\alpha\gamma}\,A_{\gamma}^{\nu}}&{g^{\mu\nu}+g^{\gamma\delta}\,A_{\gamma}^{\mu}\,A_{\delta}^{\nu}}\\\end{array}\right),
S_{{\mathrm{\scriptsize~{gauged}}}}=\frac12\int_{M}d^{2}x\left(G_{\mu\nu}D^{a}X^{\mu}D_{a}X^{\nu}+iY_{\alpha}\epsilon^{ab}(\partial_{a}V_{b}^{\alpha}-\partial_{b}V_{a}^{\alpha})\right)\,,
B=\int_{0}^{1}\frac{2^{n}t^{n-1}}{\{r-x_{n+1}+t^{2}(r+x_{n+1})\}^{n}}dt\\sum_{j=1}^{n}(-1)^{n-1+j-1}x_{j}dx_{1}\wedge\cdots\wedge\breve{dx_{j}}\wedge\cdots\wedgedx_{n}.
\langleT_{0}^{0}\rangle_{\mathrm{sub}}^{(R)}\sim\langleT_{i}^{i}\rangle_{\mathrm{sub}}^{(R)}\sim-2m\xi\langleT_{1}^{1}\rangle_{\mathrm{sub}}^{(R)}\sim\frac{(4\zeta-1)m^{(D+1)/2}}{2^{D}\pi^{(D+3)/2}\xi^{(D+1)/2}}e^{-2m\xi},\quadi=2,...,D,\quadm\xi\to\infty.
h^{p,q}(X)=\quad\begin{array}{ccccccc}{}&{}&{}&{1}&{}&{}&{}\\{}&{}&{0}&{}&{0}&{}&{}\\{}&{0}&{}&{1}&{}&{0}&{}\\{1}&{}&{21}&{}&{21}&{}&{1}\\{}&{0}&{}&{1}&{}&{0}&{}\\{}&{}&{0}&{}&{0}&{}&{}\\{}&{}&{}&{1}&{}&{}&{}\\\end{array}
\bar{\Pi}_{\alpha}{}^{\beta}\equiv(\bar{\gamma}^{1}\gamma^{2}\bar{\gamma}^{3}\gamma^{4})_{\alpha}{}^{\beta}\,,\qquad(\bar{\Pi}^{\prime})_{\alpha}{}^{\beta}\equiv(\bar{\gamma}^{5}\gamma^{6}\bar{\gamma}^{7}\gamma^{8})_{\alpha}{}^{\beta}\,.
P=\frac{1}{2}\left[\begin{array}{c}{E(\vec{p}_{\mathrm{B}})+E(\vec{p}_{\mathrm{T}})}\\{\vec{p}_{\mathrm{B}}+\vec{p}_{\mathrm{T}}}\\\end{array}\right]=\left[\begin{array}{c}{\frac{1}{2}[E(\vec{p}_{\mathrm{B}})+E(\vec{p}_{\mathrm{T}})]}\\{\vec{0}}\\\end{array}\right]=\left[\begin{array}{c}{\frac{1}{2}\sqrt{s}}\\{\vec{0}}\\\end{array}\right]
\begin{array}{rcl}{Q}&{=}&{S_{+}C^{+}+S_{-}C^{-}+A_{1}C^{3}+A_{2}\bar{C}_{-}C^{-}C^{3}+A_{3}\bar{C}_{+}C^{+}C^{3}+A_{4}\bar{C}_{3}C^{+}C^{-}}\\{}&{}&{+A_{5}\bar{C}_{+}\bar{C}_{-}C^{+}C^{-}C^{3}~,}\\\end{array}
g=\left({\frac{R_{\natural}}{\ell_{11}}}\right)^{\!3/2}=R_{\natural}^{3/4}\left(E_{f}^{\prime}\right)^{\!3/4}\rightarrow{0}\quad{,}\qquad\ell_{s}=\left({\frac{\ell_{11}^{3}}{R_{\natural}}}\right)^{\!1/2}=R_{\natural}^{1/4}\left(E_{f}^{\prime}\right)^{\!-3/4}\rightarrow{0}\quad{,}
\int\frac{d^{d}k_{1}}{[i\pi^{d/2}]}~(qk_{1})~P_{k_{1},m_{1}}^{\nu_{1}+1}P_{k_{1}-q,m_{2}}^{\nu_{2}}=\nu_{2}q^{2}I_{\nu_{1}+1~\nu_{2}+1}^{(d+2)}.
V\left(t,\bar{t},(E,M)\right)\sim|Z^{\mathrm{local}}(t,\bar{t},(q,p))|^{2}=\left|e^{\frac{K(t,\bar{t})}{2}}[X^{\Lambda}(t)q_{\Lambda}-F_{\Lambda}(t)\,p^{\Lambda}]\right|^{2}\,
p_{A}=\bar{p}_{A}(\psi,\psi^{*})\equivm\frac{(\widetilde{\psi}\Gamma_{AB}\psi)(\widetilde{\psi}\Gamma^{B}\psi)^{*}+c.c.}{2\left|(\widetilde{\psi}\Gamma_{C}\psi)\right|^{2}}
|0\rangle=\Psi_{\frac{\beta}{2}}^{\dagger}|-\beta\rangle=\Psi_{\frac{\beta}{2}}^{\dagger}\Psi_{\frac{3\beta}{2}}^{\dagger}\cdot\cdot\cdot|-\infty\rangle=\prod_{i=0}^{\infty}\Psi_{-m_{i}^{0}}^{+}|-\infty\rangle
\tilde{M}_{F_{s},F_{r}^{\prime}}(\{\sigma_{q}\})=\int_{C_{v_{r}}}\Phi_{\sigma,F_{s}}^{(0)}(z)\frac{dz}{2\pii}P_{F_{r}^{\prime}}(z)-\frac{[1-(-1)^{2l_{1_{r}}}]}{2}\int_{\tilde{C}_{r}}\Phi_{\sigma,F_{s}}^{(0)}(z)\frac{dz}{2\pii}P_{F_{r}^{\prime}}^{(a)}(z).
R_{\mu\nu\lambda\rho}=g_{\mu\rho}R_{\nu\sigma}+g_{\nu\sigma}R_{\mu\rho}-g_{\nu\rho}R_{\mu\sigma}-g_{\mu\sigma}R_{\nu\rho}-\frac12(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho})R~~~.
\begin{array}{c}{3(3,2)_{(1,-1,0,0,0)}+3(\overline{{3}},1)_{(-1,0,1,0,0)}+3(\overline{{3}},1)_{(-1,0,0,1,0)}}\\{+3(1,2)_{(0,1,0,0,-1)}+3(1,1)_{(0,0,-1,0,1)}+3(1,1)_{(0,0,0,-1,1)}.}\\\end{array}
{\calE}[A_{0}^{\mathrm{(II)}},A_{1}^{\mathrm{(II)}}]=\frac{2\piR\Lambda^{4}}{g^{2}}-\int_{0}^{2\piR}dy\frac{g^{2}}{16}(\xi(y))^{4}\leq{\calE}[A_{0}^{(\mathrm{I})},A_{1}^{(\mathrm{I})}].
A_{\mu}(x;y)=\eta_{\mu\mu}4\pim_{y}G\intE(x-x_{1})\frac{dy^{\mu}(\frac{x_{1}^{0}}{c})}{dx_{1}^{0}}\delta({\bfx}_{1}-{\bfy}(\frac{x_{1}^{0}}{c}))d^{4}x_{1}
\tilde{\psi}_{a_{1}\cdotsa_{2k}}\;=\;-(2k)(2k-1)\:N_{[a_{1}}^{(\beta)}\:(N^{+}N)^{-1}\:D_{a_{2}}^{(\beta)\:\dagger}\tilde{\psi}_{a_{3}\cdotsa_{2k}]}\:+\:\tilde{\psi}_{a_{1}\cdotsa_{2k}}^{(h)}\;\;\;,
M{\frac{\partialM}{\partial\phi_{\infty}^{c}}}+G_{ab}\Sigma^{a}\nabla_{c}\Sigma^{b}-{\frac{1}{2}}{\frac{\partialV}{\partial\phi_{\infty}^{c}}}=4S^{2}T{\frac{\partialT}{\partial\phi_{\infty}^{c}}}\.
(F^{(1)})^{2}=\frac{\kappa^{2}}{36}H_{\nu\rho}^{bc}\left\{\frac{\partial^{2}W^{(0)}}{\partialp_{\nu}\partialp_{\rho}},\sigma_{bc}\right\}H_{\kappa\epsilon}^{gh}\left\{\frac{\partial^{2}W^{(0)}}{\partialp_{\kappa}\partialp_{\epsilon}},\sigma_{gh}\right\}
T_{00}={\frac{1}{2}}\left(1-{\frac{e^{2}}{\kappa^{2}}}|\Phi|^{2}\right)\left(B^{2}+|\vec{E}|^{2}\right)+{\frac{1}{2}}\left[|\nabla_{0}\Phi|^{2}+\left(\nabla_{i}\Phi\right)^{\ast}\nabla_{i}\Phi\right]+{\frac{1}{2}}m^{2}|\Phi|^{2}\,.
{\calL}=\frac{1}{2gl_{s}}\mathrm{Tr}\;\left[\dot{X}^{a}\dot{X}^{a}+\frac{1}{2}[X^{a},X^{b}]^{2}+\theta^{T}(i\dot{\theta}-\Gamma_{a}[X^{a},\theta])\right]\,.
\pi^{A^{\prime}}\pi^{B^{\prime}}\pi^{C^{\prime}}\pi^{D^{\prime}}(-\tilde{\Psi}_{A^{\prime}B^{\prime}C^{\prime}D^{\prime}}+\nabla_{A^{\prime}}^{A}\tilde{\Theta}_{B^{\prime}C^{\prime}AD^{\prime}})=0.
\left(\begin{matrix}{s}\\{v}\\{t}\\{a}\\{p}\\\end{matrix}\right)(4,2\,;\,3,1)=-\,\frac{1}{4}\left(\begin{matrix}{1}&{1}&{1}&{1}&{1}\\{4}&{-2}&{0}&{2}&{-4}\\{6}&{0}&{-2}&{0}&{6}\\{4}&{2}&{0}&{-2}&{-4}\\{1}&{-1}&{1}&{-1}&{1}\\\end{matrix}\right)\;\;\left(\begin{matrix}{s}\\{v}\\{t}\\{a}\\{p}\\\end{matrix}\right)(4,1\,;\,3,2)
\frac{1}{2}\left(m_{B}^{2}t-\frac{\lambda_{B}v^{2}}{t+1}\right)=\int^{\Lambda}\frac{d^{d}q}{(2\pi)^{d}}\frac{g_{B}^{2}v^{2}}{q^{2}(q^{2}+g_{B}^{2}v^{2})},
{\calL}_{\mathrm{M}}=\tau\left[1-\frac{\phi}{2}-\frac{1}{2\phi}\left(1-\kappa^{2}B_{n}B^{n}\right)\right]-\tau\kappa^{2}\epsilon^{mnp}B_{m}\partial_{n}A_{p}+\frac{m}{2}\epsilon^{mnp}A_{m}\partial_{n}A_{p},
{\bfA}^{\mu}=\left(\begin{matrix}{A_{1}^{\mu}}\\{A_{2}^{\mu}}\\\end{matrix}\right)\,,\qquad{\bf\Psi}=\left(\begin{matrix}{\psi_{1}}\\{\psi_{2}}\\\end{matrix}\right)\,,\qquad\bar{{\bf\Psi}}=\left(\begin{matrix}{\bar{\psi}_{1}}&{\bar{\psi}_{2}}\\\end{matrix}\right)\,.
\chi^{(1)}=a^{(1)}+\sum_{s=1}^{k+1}\frac{p_{s}^{F}}{|\vec{x}-\vec{x}_{s}|}\quad,\quad\chi^{(2)}=a^{(2)}+\sum_{s=1}^{k+1}\frac{p_{s}^{H}}{|\vec{x}-\vec{x}_{s}|}
M={\frac{y_{L}+y_{R}}{y_{L}y_{R}{\bfV}^{2}}}-{\frac{R\sigma_{3}}{(y_{L}+y_{R})^{2}-R^{2}}}\equiv{\frac{y_{L}+y_{R}}{y_{L}y_{R}{\bfV}^{2}}}-g\sigma_{3}
\delta{\calL}=-(Z_{\Phi}-1)\left[\frac{1}{2}\left(\partial_{\mu}\Phi_{\nu}^{\dagger}-\partial_{\nu}\Phi_{\mu}^{\dagger}\right)\left(\partial^{\mu}\Phi^{\nu}-\partial^{\nu}\Phi^{\mu}\right)+\partial_{3}\Phi_{\mu}^{\dagger}\partial^{3}\Phi^{\mu}\right].
{\calF}_{0}=\frac{1}{2\betaV}\sum_{n}\sum_{\bfp}\mathrm{ln}\left(\frac{\betaM^{2}}{\rho}\right)+2T\int\frac{d^{2}p}{(2\pi)^{2}}\left[\beta\omega+ln(1-e^{-\beta\omega})\right]\;,
ds^{2}=-(1-{\frac{\lambda}{r^{d-2}}}+{\frac{r^{2}}{l^{2}}})dt^{2}+(1-{\frac{\lambda}{r^{d-2}}}+{\frac{r^{2}}{l^{2}}})^{-1}dr^{2}+r^{2}d\Omega_{d-1}^{2}\,,
{\calF}_{\mu\nu}(x)=-\frac{1}{4\pi^{2}}\left[\frac{\partial}{\partialx_{\mu}}\intd^{4}y\frac{j_{\nu}(y)}{(x-y)^{2}}-\frac{\partial}{\partialx_{\nu}}\intd^{4}y\frac{j_{\mu}(y)}{(x-y)^{2}}\right]
a=a_{M}\left(\left[A_{i}^{a}\right],\left[\partial_{i}B_{a}^{0i}\right],\left[\pi\right],\left[\mathcal{P}_{\Gamma}\right]\right)e^{M}\left(\eta_{2a}^{i},\partial^{\left(j\right.}\eta_{2a}^{\left.i\right)}\right)+\gamma_{I}b,
\begin{array}{rcl}{L_{g}:}&{G\longrightarrow}&{G}\\{}&{h\longrightarrow}&{L_{g}h:=gh\,,}\\\end{array}\qquad\begin{array}{rcl}{R_{g}:}&{G\longrightarrow}&{G}\\{}&{h\longrightarrow}&{R_{g}h:=hg\,.}\\\end{array}
L^{+}=\{\partial_{J}^{\,I}\;,\I<J\}\;,\quadL^{0}=\{\partial_{I}^{\,I}\;,\\sum_{I=1}^{N}\partial_{I}^{\,I}=0\}\;,\quadL^{-}=\{\partial_{J}^{\,I}\;,\I>J\}\;.
\left(\begin{array}{c}{\psi_{\pm}^{B,(0)}}\\{\psi_{\pm}^{B,(1)}}\\\end{array}\right)={\calT}_{\pm}^{BA}\left(\begin{array}{c}{\psi_{\pm}^{A,(0)}}\\{\psi_{\pm}^{A,(1)}}\\\end{array}\right)\;.
\underline{{hs_{\bfC}(2,2)}}:\;\;\;\;f=i\alpha+\alpha_{i\bar{j}}a^{i}\bar{a}^{\bar{j}}+i\alpha_{i_{1}i_{2}\bar{j}_{1}\bar{j}_{2}}a^{i_{1}}a^{i_{2}}\bar{a}^{\bar{j}_{1}}\bar{a}^{\bar{j}_{2}}+\ldots
{\calL}=-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}\\,\\F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}-gf^{abc}A_{\mu}^{b}A_{\nu}^{c}\,
\hat{M}_{0}(z)=\displaystyle{\left(\begin{array}{ccc}{\hat{w}(z)+\textstyle{\frac{1}{2}}\hat{\lambda}(z)}&{0}&{0}\\{ib{\cdot\gamma}}&{\hat{w}(z)-\frac{1}{2}\hat{\lambda}(z)}&{\sqrt{2}\hat{\rho}{}^{b}(z)}\\{-\sqrt{2}\hat{\bar{\rho}}_{a}(z)}&{0}&{\hat{t}{}_{a}{}^{b}(z)}\\\end{array}\right)}\,.
\frac{\langles^{\prime}|e^{-\hat{H}_{s}\tau_{f}}\hat{O}_{1}^{s}(\tau_{1})\hat{O}_{2}^{s}(\tau_{2})\ldots\hat{O}_{n}^{s}(\tau_{n})e^{\hat{H}_{s}\tau_{i}}|s\rangle}{\langles^{\prime}|e^{-\hat{H}_{s}(\tau_{f}-\tau_{i})}|s\rangle}\,.
L=\intd^{10}x\sqrt{g}\left(-2e^{-2\phi}T^{2}+\sum_{n=1,3,\pm}{\frac{1}{2}}(1\pmT+{\frac{T^{2}}{2}})|F_{(n)}^{\pm}|^{2}+(1+T+{\frac{T^{2}}{2}})|F_{(5)}|^{2}\right)
\lambda\to\frac{a\lambda+b}{c\lambda+d}\;,\;\;\;\quad\left(\begin{matrix}{C_{\alpha\beta}}\\{B_{\alpha\beta}}\\\end{matrix}\right)\to(\Omega^{T})^{-1}\left(\begin{matrix}{C_{\alpha\beta}}\\{B_{\alpha\beta}}\\\end{matrix}\right)\;,
\begin{array}{c}{\left(\partial_{T}-i\lambdae^{-\omegaT}\right)A(T)=i\epsilonB(T),\quad\left(\partial_{T}+i\lambdae^{-\omegaT}\right)B(T)=i\epsilonA(T)}\\{\left(\partial_{T}+i\lambdae^{-\omegaT}\right)C(T)=\epsilonD(T),\quad\left(\partial_{T}-i\lambdae^{-\omegaT}\right)D(T)=-\epsilonC(T).}\\\end{array}\quad
a_{2}({\tilde{\calM}}_{\beta})=\int_{{\tilde{\calM}}_{\beta}}\left({\frac{1}{180}}R_{\mu\nu\lambda\rho}R^{\mu\nu\lambda\rho}-{\frac{1}{180}}R_{\mu\nu}R^{\mu\nu}+{\frac{1}{72}}R^{2}\right)~~~,
{\calI}(t)=\frac{4}{3\pib}\frac{1}{(a^{2}+1)^{7/2}}[-(6a^{5}-6a^{3}+3a)ln(\sqrt{a^{2}+1}-a)+(2a^{6}-4a^{4}+9a^{2})\sqrt{a^{2}+1}],
R_{\mu\nu}n_{i}^{\mu}n_{i}^{\nu}=-{\frac{2r_{-}}{r_{+}^{3}}}~~~,~~~R_{\mu\nu\lambda\rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}={\frac{2r_{+}-4r_{-}}{r_{+}^{3}}}~~~
{L_{\Lambda}}^{\Sigma}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{L_{0}}&{f_{0}^{k}{(g^{-{\frac{1}{2}}})_{k}}^{j}}\\{0}&{0}&{L_{i}}&{f_{i}^{k}{(g^{-{\frac{1}{2}}})_{k}}^{j}}\\\end{matrix}\right),
L=\pi^{ij}\dot{g}_{ij}-N^{\mu}T_{\mu}+P_{\mu}\dot{N}^{\mu}+\bar{\tilde{C}}_{\mu}\dot{\tilde{C}}^{\mu}+\bar{C}_{\mu}\{\dot{C}^{\mu}-U_{\alpha\beta}^{\mu}N^{\alpha}C^{\beta}-\tilde{C}^{\mu}\}.
{\widehat{\calQ}}{}^{1\,\alpha}=ip^{\alpha}{}_{\beta}{\widehatW}_{q}^{\beta}+m{\widehat{\tilde{W}}}{}_{q}^{\alpha}\,,\qquad{\widehat{\calQ}}{}^{2\,\alpha}=ip^{\alpha}{}_{\beta}{\widehatV}_{q}^{\beta}+m{\widehat{\tilde{V}}}{}_{q}^{\alpha}\,,
{\calG}=-\frac{(2N-1)V\beta^{2}}{16\piGN}\left[\frac{\Lambdae^{2b\phi_{0}}}{4N-1}r_{+}^{4N-1}+\frac{16\pi^{2}Q^{2}e^{2a\phi_{0}}}{V^{2}\beta^{4}}\frac{1}{r_{+}}\right],
\xi_{M}^{(J)}(x_{+})=e^{\tauL_{0}}\,\xi_{M}^{(J)}(\sigma)\,e^{-\tauL_{0}},\quad{\overline{\xi}_{M}^{(J)}}(x_{-})=e^{-\tau{\overline{L}}_{0}}{\overline{\xi}_{M}^{(J)}}(\sigma)e^{\tau{\overline{L}}_{0}}.
I_{L}=\intdt\Biggl[-\frac{3\pim_{P}^{2}}{4}\Biggl(\frac{a}{N}\Bigl(\frac{\partiala}{\partialt}\Bigr)^{2}-NV_{g}(a)\Biggr)+2\pi^{2}a^{3}\Biggl(\frac{1}{2N}\Bigl(\frac{\partial\phi}{\partialt}\Bigr)^{2}-NV(\phi)\Biggr)\Biggr],
\begin{array}{lll}{z^{1}z^{2}=p^{2}z^{2}z^{1}\;,\qquad}&{z^{2}z^{3}=p^{2}z^{3}z^{2}\;,\qquad}&{z^{3}z^{4}=p^{2}z^{4}z^{3}\;,}\\{z^{4}z^{1}=p^{2}z^{1}z^{4}\;,\qquad}&{z^{4}z^{2}=z^{2}z^{4}\;,\qquad}&{z^{1}z^{3}=z^{3}z^{1}\;.}\\\end{array}
{\calL}={\frac{1}{4\pi\alpha^{\prime}}}\int_{D}d^{2}\sigma\\bigg(g_{ij}\partial_{a}X^{i}\partial^{a}X^{j}-2\pii\alpha^{\prime}B_{ij}\varepsilon^{ab}\partial_{a}X^{i}\partial_{b}X^{j}\bigg)
\langle\underline{{O}}|\underline{{I}}_{p\,^{\prime}}\underline{{I}}_{p}|\underline{{O}}\rangle=Z\frac{1}{N}\delta_{\Gamma_{m}}(\vec{p},\vec{p}\,^{\prime})\chi(\vec{p})+Z^{2}\Big(1-\frac{1}{N}\Big)\chi(\vec{p})\chi(\vec{p}\,^{\prime}).
\begin{array}{lcl}{{\widehat\Sigma}(\theta)}&{=}&{\oplus_{r\leqs}{\widehat\Sigma}^{(r,s)}}\\{}&{=}&{\oplus_{r\leqs}\oplus_{m\inZ}{\widehat\Sigma}_{m}^{(r,s)}(\theta)}\\{}&{=}&{\oplus_{r\leqs}\oplus_{m\inZ}\oplus_{k=r}^{s}{\widehat\Sigma}_{m}^{(k,k)}(\theta)}\\\end{array}
\vert\beta_{k}\vert^{2}\simeq\frac{N}{\sqrt{2}}(\rho-\rho_{\mathrm{min}})^{2}\,,\quad\vert\beta_{k}\vert^{2}\simeq\frac{\pi^{4}}{16}J^{2}N^{2}(\Phi-\Phi_{\mathrm{min}})^{2}\,.
\sum_{n=1}^{N}\langlei\lambda\epsilon_{n}-\mu_{n}\frac{{\calF}^{(n)}}{\sqrt{m_{n}}}\cdotp^{(n)}+\frac{1}{2}\mu_{n}\bigg[\frac{{\calF}^{(n)}}{\sqrt{m_{n}}}\cdot(\frac{p^{(n)}}{\sqrt{m_{n}}}-\nu_{n}{\calF}^{(n)})+(\frac{p^{(n)}}{\sqrt{m_{n}}}-\nu_{n}{\calF}^{(n)})\cdot\frac{{\calF}^{(n)}}{\sqrt{m_{n}}}\bigg]\rangle.