harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mu_{1}=\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{-1}\\\end{array}\right)\;\;\;\;\;\mu_{2}=\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{1}\\\end{array}\right)
\{X,Y,Z\}=\left.{\frac{\partial_{r}X}{\partial\Phi^{A}}}\right|_{K}\left.{\frac{\partial_{l}^{3}W}{\partialJ_{A}\partialJ_{B}\partialJ_{C}}}\right|_{K}\left.{\frac{\partial_{l}Y}{\partial\Phi^{C}}}\right|_{K}\left.{\frac{\partial_{r}Z}{\partial\Phi^{B}}}\right|_{K}(-1)^{\varepsilon_{B}(\varepsilon(Y)+\varepsilon(Z)+1)}.
\textstyle\mathrm{sign}(\boldsymbol{\epsilon})=\mathrm{sign}(\sum_{j=1}^{8}\epsilon^{j}g^{j})\;\mathrm{sign}(\boldsymbol{g})~,\quad\mathrm{if}~~|\sum_{j=1}^{8}\epsilon^{j}g^{j}|=\|\boldsymbol{g}\|\;\|\boldsymbol{\epsilon}\|~.
S=CT=\left(\begin{array}{cc}{\frac{1}{2V_{oo}}\left[1-V_{oo}-\sqrt{(1-3V_{oo})(1+V_{oo})}\right]}&{0}\\{0}&{\frac{1}{2V_{ee}}\left[1+V_{ee}-\sqrt{(1+3V_{ee})(1-V_{ee})}\right]}\\\end{array}\right).
\langle\sigma^{2}(x)\,\sigma^{2}(y)\,\sigma^{2}(0)\rangle={\frac{1}{\pi^{6}|x||y||x-y|}}\left(4u(s)(u(s)+2)\right)^{3/4}\left({\frac{\alpha(\mu)}{(1-\alpha^{2}(\mu))(1+\alpha(\mu))^{2}}}\right)^{3/2},
I(D)=\int_{0}^{\infty}\!\!\!d\hat{T}_{1}\,d\hat{T}_{2}\int_{0}^{1}da\,db\,dc\,dd\,{\biggl[(\hat{T}_{1}+G_{Bab})(\hat{T}_{2}+G_{Bcd})-{\frac{C^{2}}{4}}\biggr]}^{-{\frac{D}{2}}}
|f_{m}|^{2}\G\left({\frac{A_{D}}{\Lambda}},{\frac{A_{D}^{\dagger}}{\Lambda}},\tau_{0},\tau_{0}^{\dagger},{\frac{MM^{\dagger}}{\Lambda^{2}}},{\frac{\tilde{M}M^{\dagger}}{\Lambda^{2}}},{\frac{\tilde{M}\tilde{M}^{\dagger}}{\Lambda^{2}}}\right)\Bigg|_{A}\+\a_{m}f_{m}G\Big|_{F}\+\\mathrm{h.c.}
E[p]=\frac{Lg^{2}}{2N}s_{\theta}[n]=\frac{Lg^{2}}{2N}\left[\sum_{j=1}^{N}\left(p_{j}-\frac{\theta}{2\pi}\right)^{2}-\sum_{j=1}^{N}\left(\frac{N+1}{2}-j-\frac{\theta}{2\pi}\right)^{2}\right]
\widehat{Q}_{B}\widehat{\Phi}(0)=\oint\frac{dz}{2\pii}j_{B}(z)\cdott\xice^{-\phi}(0)\otimes\sigma_{3}\sigma_{1}=-it\left(\frac{1}{2}\xic\partialce^{-\phi}+\etae^{\phi}\right)(0)\otimes\sigma_{2},
F_{2}({\bf{k}})=\sum_{i,{\bf{q}}}\frac{n_{F}({\bf{k}}-{\bf{q}})}{({\tilde{\omega}}_{i}(-{\bf{q}})+\frac{{\bf{k.q}}}{m}-\epsilon_{{\bf{q}}})^{2}}g_{i}^{2}(-{\bf{q}})
\widehatI=\mathrm{Tr}\\int_{X}d^{4}x\\varepsilon^{\mu\nu\rho\sigma}\partial_{\mu}\left(\widehat{\omega}_{\nu}\ast\partial_{\rho}\widehat{\omega}_{\sigma}+\frac{2}{3}\widehat\omega_{\nu}\ast\widehat\omega_{\rho}\ast\widehat\omega_{\sigma}\right).
A^{-}A^{+}(A^{-}\psi_{-}^{(n)})=E_{-}^{(n)}(A^{-}\psi_{-}^{(n)})\RightarrowH_{+}(A^{-}\psi_{-}^{(n)})=E_{-}^{(n)}(A^{-}\psi_{-}^{(n)}).
\phi_{n,m}^{\pm}=r^{-(n+\gammam)}\sum_{p=0}^{[n/2]}C_{i_{1}....i_{n-2p}}^{\pmm}x^{i_{1}}...x^{i_{n-2p}}(x^{+}x^{-})^{p}(x^{\pm})^{\gammam}~~~,
\psi_{(1)i}^{\gamma}({\bf{r}})=\,f^{\alpha\beta\gamma}\,{\cal{X}}^{\alpha}({\bf{r}})\;{\cal{Q}}_{(1)i}^{\beta}({\bf{r}})=\,f^{\alpha\beta\gamma}\,{\cal{X}}^{\alpha}({\bf{r}})\,[a_{i}^{\beta}({\bf{r}})+{\textstyle\frac{1}{2}}x_{i}^{\beta}({\bf{r}})]\;,
{\hat{H}_{D}^{00}}(\xi)=-{\frac{Q_{B}}{8\pi}}\int{d^{2}}\xi^{\prime}\sqrt{\hat{g}(\xi^{\prime})}\hat{R}(\xi^{\prime}){\partial_{\hat{n}}}{\hat{G}_{D}}(\xi,\xi^{\prime})+{\frac{\bar{Q}_{B}}{8\pi}}{k_{\hat{g}}}(\xi).
\begin{array}{rcl}{(\check{R}^{V_{z_{1}},V_{z_{2}}^{*}})_{jl^{*}}^{i^{*}k}}&{=}&{(\check{R}^{V_{z_{2}},V_{z_{1}}})_{ij}^{kl}\displaystyle\prod_{p=2}^{n-1}\frac{h(-z_{1}+z_{2}+pw)}{h(w)},}\\{(\check{R}^{V_{z_{1}}^{*},V_{z_{2}}})_{i^{*}j}^{kl^{*}}}&{=}&{(\check{R}^{V_{z_{2}},V_{z_{1}+nw}})_{jl}^{ik}\displaystyle\prod_{p=1}^{n-2}\frac{h(-z_{1}+z_{2}-pw)}{h(w)}.}\\\end{array}
\frac{H_{QM}[\rho,\Pi]}{N^{2}}~=~\frac{\gamma}{8}\intd\theta\left\{\rho(\theta)\left(\frac{\partial}{\partial\theta}\frac{\delta~S}{\delta\rho(\theta)}\right)^{2}+\frac{\pi^{2}}{3}\rho^{3}(\theta)\right\}+\frac{1}{N^{2}}V[\rho]~=~\frac{E}{N^{2}}\;.
\frac{S\left(\beta\right)}{2\pir\widetilde{E}\left(\beta\right)}\approx\frac{\frac{\delta^{-3}}{180}-\frac{\mu_{eff}^{2}}{8\pi^{2}}\delta^{-1}}{\left[\frac{1}{240}\left(\delta^{-4}-1\right)-\frac{\mu_{eff}^{2}}{8\pi^{2}}\left(\delta^{-2}-\frac{\pi^{2}}{6}\right)\right]},\,\,\,\,\delta\ll1.
+\frac{3\alpha}{4\pi}\int_{0}^{\Lambda^{2}}dq^{2}\frac{q^{2}}{q^{2}+\Sigma^{2}(q^{2})}\left[\frac{\theta(p^{2}-q^{2})}{p^{2}}+\frac{\theta(q^{2}-p^{2})}{q^{2}}\right]\Sigma(q^{2}),
\pi^{ij}=-2\gamma^{1/2}e^{-\frac{3}{2}(\phi-\tilde{\phi})}\left[\frac{1}{2}\gamma^{ij}H+\gamma^{-1/3}\left(\frac{\partialH}{\partialh_{ij}}-\frac{1}{3}\frac{\partialH}{\partialh_{kl}}h_{kl}h_{ij}\right)\right]~~~,
V_{i}\cdot{\calN}_{\overline{{\alphaV}}}+f_{i}({\overline{{\alphaV}}},{\bfn})+{\frac{\alpha^{*}t^{*}}{2t_{i}}}{\vec{Q}}^{2}({\vec{P}}_{\overline{{\alphaV}}})=\sum_{j}k_{ij}\alpha_{j}+s_{i}-V_{i}\cdot({\overline{{\alphaV}}}-W({\overline{{\alphaV}}}))\pmod{1}~.
\oint{\calD}x\;F[x]=\int\!\!d^{D}\!y\int_{x(0)=y}^{x(S)=y}\!\!\!\!\!\!{\calD}x\;F[x]=\int\!\!d^{D}\!y\int_{x(0)=y}^{x(S)=y}\!\!\!\!\!\!{\calD}x\;F[x^{-1}]=\oint{\calD}x\;F[x^{-1}]\,
S=-\frac{1}{2\pi}\intd^{2}\sigma~~e~~\left[h^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X_{\mu}-i\bar{\Psi}^{\mu}\rho^{\alpha}\partial_{\alpha}\bar{\Psi}_{\mu}+2\bar{\chi}_{\alpha}\rho^{\beta}\rho^{\alpha}\Psi^{\mu}\partial_{\beta}\chi^{\mu}+\frac{1}{2}\bar{\Psi}^{\mu}\Psi_{\mu}\bar{\chi}_{\beta}\rho^{\beta}\rho^{\alpha}\chi_{\alpha}\right]
(-)^{{\bar{\alpha}}+{\bar{\beta}}+\bar{\alpha}\bar{\beta}}\;\frac{{\bar{\vartheta}}[_{\bar{\beta}}^{\bar{\alpha}}]}{{\bar{\eta}}}\rightarrow\frac{{\bar{\vartheta}[_{\bar{\beta}}^{\bar{\alpha}}]}^{13}}{{\bar{\eta}}^{13}}.
U^{(1)}=U^{(0)}\mid_{\Omega_{1}=0}={\frac{1}{2}}\vec{\pi}^{2}-{\frac{1}{2}}c^{2}(\nabla\times\vec{u})^{2}+\nu\vec{\pi}.\nabla^{2}\vec{u}-\vec{u}.\vec{J}+\nu\vec{u}.\nablan.
H=-{\frac{1}{2}}\sum_{i=1}^{N}\partial_{i}^{2}+{\frac{1}{2}}\sum_{i=1}^{N}x_{i}^{2}+{\frac{g^{2}}{2}}\sum_{{i,j}\atop{i\nej}}^{N}{\frac{1}{(x_{i}-x_{j})^{2}}}\,\,\,\,.
1+\gamma_{m}^{\mathrm{var}}=\left(\alpha+{\frac{2\pi}{3}}\right){\gamma_{m}^{\mathrm{var}}}^{\prime}-{\frac{3\pi}{8}}\alpha^{2}\left({\frac{1}{1+\gamma_{m}^{\mathrm{var}}}}\right)^{\prime}\;\;\;.
{\frac{d^{2}x^{i}}{d\tilde{\lambda}^{2}}}=\left[nn^{\prime}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}-aa^{\prime}\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}-2{\frac{a^{\prime}}{a}}\right]{\frac{dx^{i}}{d\tilde{\lambda}}}.
L^{d=4}=\frac{1}{4e^{2}}\int\left(W^{2}d^{2}\theta+\bar{W}^{2}d^{2}\bar{\theta}\right)+\frac{1}{e^{2}}\sum_{r,s,t=0}^{\infty}a_{rst}\intd^{4}\theta\left(W^{2}\bar{W}^{2}\right)^{r}X^{s}Y^{t}
\vec{\beta}=\vec{\beta}_{(n_{1}^{\prime},n_{1})(n_{2}^{\prime},n_{2})(n_{3}^{\prime},n_{3})}=\sum_{a=1}^{3}(\frac{1-n_{a}^{\prime}}{2}\alpha_{-}+\frac{1-n_{a}}{2}\alpha_{+})\vec{\omega}_{a}
L_{m}=\frac{1}{2}:\sum_{n\in\mathbb{Z}}g_{MN}\alpha_{n}^{M}\alpha_{m-n}^{N}:~,\quad\tilde{L}_{m}=\frac{1}{2}:\sum_{n\in\mathbb{Z}}g_{MN}\tilde{\alpha}_{n}^{M}\tilde{\alpha}_{m-n}^{N}:~,
c\left(\begin{matrix}{b_{i}}\\{b_{j}}\\\end{matrix}\right)=c\left(\begin{matrix}{b_{i}}\\{S}\\\end{matrix}\right)=c\left(\begin{matrix}{b_{i}}\\{I,X}\\\end{matrix}\right)=-c\left(\begin{matrix}{{\bf1}}\\{I,X}\\\end{matrix}\right)=-c\left(\begin{matrix}{I}\\{X}\\\end{matrix}\right)=-c\left(\begin{matrix}{{\bf1}}\\{{\bf1}}\\\end{matrix}\right)=-1,
\begin{array}{l}{\theta_{m,3}({\frac{\tau}{2}},{\frac{z}{3}})\stackrel{z\toz+{\frac{\tau}{2}}}{\longrightarrow}q^{-1/24}y^{-1/6}~\theta_{m+1,3}({\frac{\tau}{2}},{\frac{z}{3}})\stackrel{z\toz+{\frac{\tau}{2}}}{\longrightarrow}q^{-1/6}y^{-1/3}~\theta_{m+2,3}({\frac{\tau}{2}},{\frac{z}{3}}),}\\\end{array}
E^{\underline{\alpha}}(X^{\underline{m}}(z),\Theta^{\underline{\mu}}(z))={\frac{1}{2}}(1-\bar{\Gamma})_{~\underline{\beta}}^{\underline{\alpha}}\,e^{a}E_{a}^{\underline{\beta}}+e^{\alphaq}v_{\alphaq}^{~\underline{\alpha}}(z),
\begin{array}{c}{g_{0}+g_{0}=g_{0},\g_{0}+g_{1}=g_{1},\g_{0}+g_{2}=g_{2},}\\{g_{1}+g_{1}=g_{2},\g_{1}+g_{2}=g_{0},\g_{2}+g_{2}=g_{1},}\\\end{array}
\omega_{n}^{1}=\mu+\sqrt{\mu^{2}+\left(\frac{n}{l_{s}^{2}p^{+}}\right)^{2}}\qquad\omega_{n}^{2}=-\mu+\sqrt{\mu^{2}+\left(\frac{n}{l_{s}^{2}p^{+}}\right)^{2}}
F^{\mu\nu}=\left(\begin{array}{}{h]{cccc}0}&{-E_{x}}&{-E_{y}}&{-E_{z}}\\{E_{x}}&{0}&{-B_{z}}&{B_{y}}\\{E_{y}}&{B_{z}}&{0}&{-B_{x}}\\{E_{z}}&{-B_{y}}&{B_{x}}&{0}\\\end{array}\right).
\underbrace{(\hat{D}(k)+\ldots)}_{I_{0}}\underbrace{(\hat{D}(m)e^{i(p+s)\phi(w)}+\ldots)}_{F_{p+s}}=\frac{1}{(z-w)^{\alpha}}\big(\underbrace{e^{i(p+s)\phi(w)}+\ldots}_{F_{p+s}^{\prime}}\big)
Q\left(\begin{array}{c}{\phi}\\{v_{j}}\\{\lambda_{k}}\\{\psi^{l}}\\{E^{m}}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{\lambda_{j}}\\{-\phi(v_{k})}\\{E^{l}}\\{-\phi(\psi^{m})}\\\end{array}\right)_{(\phi_{i},v_{j},\lambda_{k},\psi^{l},E^{m})}.
ds^{2}=-\partial_{r}g_{\tilde{t}\tilde{t}}|_{r=r_{H}}(r-r_{H})d\tau^{2}+\frac{1}{\partial_{r}g^{rr}|_{r=r_{H}}(r-r_{H})}dr^{2}+\cdots=\rho^{2}d\Theta^{2}+d\rho^{2}+\cdots
M=\left(\begin{matrix}{G^{-1}}&{-G^{-1}B}\\{BG^{-1}}&{G-BG^{-1}B}\\\end{matrix}\right)=\left(\begin{matrix}{1\!\!1}&{0}\\{B}&{1\!\!1}\\\end{matrix}\right)\left(\begin{matrix}{G^{-1}}&{0}\\{0}&{G}\\\end{matrix}\right)\left(\begin{matrix}{1\!\!1}&{-B}\\{0}&{1\!\!1}\\\end{matrix}\right),
\left(\lambda^{a}\right)^{\beta\gamma}\left(\lambda^{a}\right)_{\delta}^{\alpha}+\left(\lambda^{a}\right)^{\alpha\gamma}\left(\lambda^{a}\right)_{\delta}^{\beta}=\frac{C_{\rho}}{d-1}\left(2\eta^{\alpha\beta}\delta_{\delta}^{\gamma}-\eta^{\alpha\gamma}\delta_{\delta}^{\beta}-\eta^{\beta\gamma}\delta_{\delta}^{\alpha}\right)
\langleE_{\perp}(r)E_{\perp}(r^{\prime})\rangle_{eff}=\frac{1}{\pir^{2}a^{2}}\int_{0}^{x_{0}}\frac{xdx}{n}\sum_{l=1}^{\infty}\frac{2l+1}{4\pi}\{A_{F}(x)s_{l}^{2}(nx\tilde{r})-A_{G}(x)[s_{l}(nx\tilde{r})]^{2}\},
{\frac{3}{b^{2}}}{\frac{a^{\prime}}{a}}\left({\frac{a^{\prime}}{a}}+{\frac{n^{\prime}}{n}}\right)-{\frac{3}{n^{2}}}\left[{\frac{\dot{a}}{a}}\left({\frac{\dot{a}}{a}}-{\frac{\dot{n}}{n}}\right)+{\frac{\ddot{a}}{a}}\right]-{\frac{3k}{a^{2}}}=\kappa_{5}^{2}\left[{\frac{4}{3}}n^{-2}\dot{\phi}^{2}+{\frac{4}{3}}b^{-2}\phi^{\prime\,2}-e^{-2\alpha\phi}\Lambda\right],
\int_{C_{\rho}(\sigma_{k})}{\frac{\bar{H}_{\nu}^{(1)}(z)}{\bar{J}_{\nu}(z)}f(z)dz}-\int_{C_{\rho}(-\sigma_{k})}{\frac{\bar{H}_{\nu}^{(2)}(z)}{\bar{J}_{\nu}(z)}f(z)dz}=2\pii{\mathrm{Res}}_{z=\sigma_{k}}\frac{\bar{H}_{\nu}^{(1)}(z)}{\bar{J}_{\nu}(z)}f(z)
i\Pi_{(a)+(b)}^{\mu\nu}\simi\frac{g^{2}}{16\pi^{2}}\left\{g^{\mu\nu}\frac{24}{-\tilde{q}^{2}}+\frac{10}{3}\left(g^{\mu\nu}q^{2}-q^{\mu}q^{\nu}\right)\mathrm{ln}(q^{2}\tilde{q}^{2})+32\frac{\tilde{q}^{\mu}\tilde{q}^{\nu}}{\tilde{q}^{4}}-\frac{4}{3}\frac{q^{2}}{\tilde{q}^{2}}\tilde{q}^{\mu}\tilde{q}^{\nu}\right\}\,.
W=W^{\mathrm{cl}}\frac{H}{H_{0}}\frac{3n\left(1-\beta^{2}\right)}{10}\left(\frac{1+\zeta}{2}\frac{n+\mu-1}{n+\mu}+\frac{1-\zeta}{2}\right)\left(\frac{5}{6}l_{2}^{2}+\frac{1}{6}l_{3}^{2}\right)\;.
M={\frac{1}{2}}\left(\begin{matrix}{0}&{ime^{(q_{1}-q_{2})/2}}&{ime^{(-q_{1}-q_{2})/2}}&{0}\\{i\,me^{(q_{1}-q_{2})/2}}&{0}&{i\,me^{q_{2}}}&{ime^{(-q_{1}-q_{2})/2}}\\{ime^{(-q_{1}-q_{2})/2}}&{i\,me^{q_{2}}}&{0}&{i\,me^{(q_{1}-q_{2})/2}}\\{0}&{ime^{(-q_{1}-q_{2})/2}}&{i\,me^{(q_{1}-q_{2})/2}}&{0}\\\end{matrix}\right).
\mathrm{J_{0}\equiv\frac{1}{r+s}(a_{1}^{\dagger}a_{1}-a_{2}^{\dagger}a_{2})\;,\;\mathrm{J_{+}\equiv(a_{1}^{\dagger})^{s}a_{2}^{r}\;,\;\mathrm{J_{-}=a_{1}^{s}(a_{2}^{\dagger})^{r},}}}
G_{L}\left(z\right)=\left(\begin{array}{cc}{1}&{0}\\{\theta_{L}}&{1}\\\end{array}\right)\left(\begin{array}{cc}{H_{L}}&{0}\\{0}&{g_{L}}\\\end{array}\right)\left(\begin{array}{cc}{1}&{\tilde{\theta}_{L}}\\{0}&{1}\\\end{array}\right)
\left(\begin{matrix}{m^{i}}\\{n^{i}}\\{p^{i}}\\\end{matrix}\right)=\left(\begin{matrix}{m}&{j}&{k}&{m^{4}}&{m^{5}}&{\cdots}\\{0}&{n}&{l}&{n^{4}}&{n^{5}}&{\cdots}\\{0}&{0}&{p}&{p^{4}}&{p^{5}}&{\cdots}\\\end{matrix}\right)
\Gamma[V]=\int_{0}^{\infty}{\frac{\mathrm{d}T}{T}}[4\piT]^{-d/2}\mathrm{e}^{-m^{2}T}{\sum_{n=1}^{\infty}}{\frac{(-T)^{n}}{n!}}\!\!\int\mathrm{d}x_{0}\;\mathrm{tr}\;O_{n}\;.
S\supset\intd^{4}x\intdz\sqrt{-G}R\supset\intd^{4}x\intdzz^{3/2}~{\frac{\hat{h}^{\mu\nu}(x,z)\partial_{\mu}\hat{h}_{\alpha\beta}(x,z)\partial_{\nu}\hat{h}^{\alpha\beta}(x,z)}{M_{Pl}}}.
=\frac{1}{2}\int_{0}^{L}(\vec{l}-i\partial_{x}\vec{n})^{2}\;dx\,=\frac{1}{2}\int_{0}^{L}(\vec{l}^{2}-(\partial_{x}\vec{n})^{2})\;dx\,-i\int_{0}^{L}(\vec{l}\cdot\partial_{x}\vec{n})\;dx
Z(\beta)=\pi^{\zeta/2}\,e^{-\frac{i\pi}{4}\eta}\,\lambda^{-\zeta/2}\,det(\widetilde{T}^{2})^{-1/4}\,\prod_{k=0}^{N}\Bigl(det(\widetilde{T}_{k+1}^{*}\widetilde{T}_{k+1})det(\Phi_{k}^{*}\Phi_{k})^{-1}\Bigr)^{\frac{1}{2}(-1)^{k}}\,.
\begin{array}{rcccl}{[{\calH},\,Q]\!\!}&{=}&{\!\!0\,,\qquad\quad\;\;[{\calH},\,S]\!\!}&{=}&{\!\!-i\,Q\,,}\\{[{\calK},\,Q]\!\!}&{=}&{\!\!i\,S\,,\qquad\quad[{\calK},\,S]\!\!}&{=}&{\!\!0\,,}\\{[{\calD},\,Q]\!\!}&{=}&{\!\!\displaystyle{\frac{i}{2}}\,Q\,,\qquad\;\;[{\calD},\,S]\!\!}&{=}&{\!\!-\displaystyle{\frac{i}{2}}\,S\,.}\\\end{array}
(\gamma^{[A}\tilde{\gamma}^{B}\gamma^{C]})_{\alpha\beta}(\tilde{\gamma}_{[A}\gamma_{B}\tilde{\gamma}_{C]})^{\gamma\delta}=-24(\delta_{\alpha}^{~\gamma}\delta_{\beta}^{~\delta}+\delta_{\alpha}^{~\delta}\delta_{\beta}^{~\gamma})
\{\epsilon_{a}^{+}(Q_{1/2}^{a}+i\bar{Q}_{1/2}^{a}),\intd^{2}zV_{(0,0)}(\zeta_{\mu\nu}^{AST})\}=\intd^{2}z\delta_{\epsilon^{+}}V+\epsilon_{a}^{+}\zeta_{\mu\nu}^{AST}\intdxe^{\phi/2}S^{b}\partialX^{[\mu}(\gamma^{\nu]})_{b}^{a},
\{D_{\alpha}^{\mathbfi},{\bar{D}}_{\dot{\alpha}{\mathbfj}}\}=-2i\,delta_{\mathbfj}^{\mathbfi}\\partial_{\alpha\dot{\alpha}}\qquad\mathrm{and}\qquad\{D_{\alpha}^{{\mathbfi}},D_{\beta}^{{\mathbfj}}\}=0\.
E_{n}=\sqrt{{\frac{1}{2}}[Q^{2}+\sqrt{Q^{4}+4J^{2}}]}-Q=Q\left\{\sqrt{{\frac{1}{2}}+{\frac{1}{2}}\sqrt{1+{\frac{n^{2}}{Q^{4}}}}}-1\right\}={\frac{n^{2}}{8Q^{3}}}-{\frac{5n^{4}}{128Q^{7}}}+\cdots.
\zeta^{D}(s)\rfloor_{M=0}=\frac{V\mu^{2s}}{2(4\pi)^{\frac{m-1}{2}}}|S|^{m-2s-1}\frac{\Gamma\left(s-\fracm2+\frac12\right)\Gamma\left(s-\fracm2+1\right)\Gamma\left(\fracm2-s\right)}{\pi\Gamma(s)}\,.
\overline{{A}}^{\mu}(\overline{{q}})=A^{\nu}(q)\,\overline{{a}}_{\nu}^{\mu}(q)=\overline{{a}}_{\nu}^{\mu}(q)A^{\nu}(q),\quad\overline{{A}}_{\mu}(\overline{{q}})=A_{\nu}(q)\,{a}_{\mu}^{\nu}(q)={a}_{\mu}^{\nu}(q)A_{\nu}(q)
\Omega_{IJ}\;=\;\left(\begin{array}{cccc}{0}&{a}&{0}&{0}\\{a}&{0}&{0}&{0}\\{0}&{0}&{b}&{-a}\\{0}&{0}&{-a}&{0}\\\end{array}\right)\,,\,\Omega^{IJ}\;=\;\left(\begin{array}{cccc}{0}&{\frac{1}{a}}&{0}&{0}\\{\frac{1}{a}}&{0}&{0}&{0}\\{0}&{0}&{0}&{-\frac{1}{a}}\\{0}&{0}&{-\frac{1}{a}}&{-\frac{b}{a^{2}}}\\\end{array}\right)\,.
\tilde{\mu}_{s}=\mu_{s}+\sum_{j=1}^{2}X_{\mu_{s}\mu_{j}}\mu_{j}+X_{\mu_{s}\nu_{j}}\nu_{j}\quad\mathrm{and}\quad\tilde{\nu}_{s}=\nu_{s}+\sum_{j=1}^{2}X_{\nu_{s}\mu_{j}}\mu_{j}+X_{\nu_{s}\nu_{j}}\nu_{j}
S=\intd\tau\{\frac1{2e}(\dot{x}^{m}+i\theta\sigma^{m}\dot{\bar{\theta}}-i\dot{\theta}\sigma^{m}\bar{\theta}+i\psi\sigma^{m}\bar{\rho}-i\rho\sigma^{m}\bar{\psi})^{2}-\rho^{\alpha}{\dot{\theta}}_{\alpha}-{\bar{\rho}}_{\dot{\alpha}}{\dot{\bar{\theta}}}^{\dot{\alpha}}\}.
\sum_{\ell}\left(\alpha^{i\ell}\frac{\partial\alpha^{jk}}{\partialx^{\ell}}+\alpha^{j\ell}\frac{\partial\alpha^{ki}}{\partialx^{\ell}}+\alpha^{k\ell}\frac{\partial\alpha^{ij}}{\partialx^{\ell}}\right)=0.
H_{2}=p_{x}^{2}/2\gamma+\gamma(\omega_{1}^{2}+\omega_{2}^{2})x^{2}/2-\gamma\omega_{1}^{2}\omega_{2}^{2}q^{2}/2+p_{q}x+\gamma\omega_{1}^{2}\omega_{2}^{2}qp_{\lambda}.
\sum_{r}\mathrm{Tr}\left(\gamma^{n_{1}}\gamma^{r}\gamma^{n_{3}}\gamma^{r}\gamma^{n_{2}}(\boldsymbol{\gamma}\cdot\boldsymbol{K})\right)=(4-D_{0})\,\mathrm{Tr}\left(\gamma^{n_{1}}\gamma^{n_{3}}\gamma^{n_{2}}(\boldsymbol{\gamma}\cdot\boldsymbol{K})\right),
\tilde{A}_{2}(p)=\gamma\,g^{2}N(N^{2}-1)\delta_{b}^{a}\delta_{J}^{I}\int\frac{d^{4}k}{(2\pi)^{4}}d^{4}\theta\,\frac{[(p-k)^{2}+p^{2}]}{k^{4}[(p-k)^{2}+m^{2}]}\left\{\Phi_{aI}^{\dagger}(p,\theta,{\overline{\theta}})\Phi^{bJ}(-p,\theta,{\overline{\theta}})\right\}.
\dot{S}=\intdV\Bigl\{{\frac{\sigma_{ik}^{\prime}}{2kT}}\Bigl({\frac{\partialv_{i}}{\partialx_{k}}}+{\frac{\partialv_{k}}{\partialx_{i}}}\Bigr)-{\frac{{\bfq}\cdot\mathrm{grad}(kT)}{(kT)^{2}}}\Bigr\}\.
\phi\left({\bfr}\right)\equiv2mV\left({\bfr}\right)=\frac{g^{2}}{4\pi}\sum_{i=1}^{N}\int\mathrm{d}^{3}{\bfr}^{\prime}\;\frac{\mathrm{e}^{-\mu_{0}\vert{\bfr}-{\bfr}^{\prime}\vert}}{\vert{\bfr}-{\bfr}^{\prime}\vert}\,\left|\,\psi_{i}\left({\bfr}^{\prime}\right)\right|^{2}\,,
D^{\frac{1}{2}}=\frac{1}{\sqrt{1+\midz\mid^{2}}}~\left(\begin{matrix}{1}&{-z^{*}}\\{z}&{1}\\\end{matrix}\right)\left(\begin{matrix}{e^{-i\frac{\eta}{2}}}&{0}\\{0}&{e^{i\frac{\eta}{2}}}\\\end{matrix}\right)
\begin{aligned}{{}[D_{\mu},\Gamma^{\nu_{1}\cdots\nu_{n}}]}&{{}=n\,\partial_{\mu}e_{r}{}^{[\nu_{1}}\Gamma^{|r|\nu_{2}\cdots\nu_{n}]}+\frac{1}{4}\omega_{\mu}{}^{rs}[\Gamma_{rs},\Gamma^{\nu_{1}\cdots\nu_{n}}]\,}\\{}&{{}=n\,D_{\mu}e_{r}{}^{[\nu_{1}}\Gamma^{|r|\nu_{2}\cdots\nu_{n}]}\,.}\\\end{aligned}
+H_{1}^{\frac{1}{4}}H_{2}^{-\frac{1}{4}}(dz_{2}^{2}+dz_{3}^{2}+dz_{4}^{2}+dz_{5}^{2})+H_{1}^{\frac{1}{4}}H_{2}^{\frac{3}{4}}(dx_{6}^{2}+dx_{7}^{2}+dx_{8}^{2}+dx_{9}^{2}).
D=tH-\frac{1}{2}\sum_{i=1}^{N}\int_{-\infty}^{\infty}dx(x-v_{i}t){\calP}_{i}(x-v_{i}t)-\sum_{i=1}^{N}\frac{v_{i}}{2}t\int_{\infty}^{\infty}dx{\calP}_{i}(x-v_{i}t)=t(H-\sum_{i=1}^{N}\frac{v_{i}}{2}P_{i})-\frac{1}{2}D_{0},
I_{1}(\mu)-I_{1}(\mu_{0})={\frac{1}{2}}(\mu^{2}-\mu_{0}^{2})I_{0}(\mu_{0})-\frac{\mu_{0}^{3}}{8\pi}\left[{\frac{1}{3}}\left(\sqrt{\frac{\mu^{2}}{\mu_{0}^{2}}}-1\right)^{2}\left(2\sqrt{\frac{\mu^{2}}{\mu_{0}^{2}}-1}\right)\right],
\sigma_{\Lambda}(k^{2})=\left(1+\frac{(1+\alpha)m^{2}M^{2}+\alpham^{4}}{\Lambda^{6}}k^{2}\right)e^{-\frac{(k^{2}+m^{2})(k^{2}+\alpham^{2})(k^{2}+M^{2})}{\Lambda^{6}}}.
\langleN\rangle_{T}^{(d)}\sim-\frac{mT}{\pi}\frac{1}{d!(2T)^{d}}\,e^{-\frac{m}{T}}\,\intd^{2}x\,\frac{q(\vec{x})}{|\vec{v}(\vec{x})|}\left[\left(\partial_{i}\hat{n}^{a}\,\partial_{i}\hat{n}^{a}+|\vec{v}(\vec{x})|\right)^{\frac{d}{2}}-\left(\partial_{i}\hat{n}^{a}\,\partial_{i}\hat{n}^{a}-|\vec{v}(\vec{x})|\right)^{\frac{d}{2}}\right]
G_{B}(x,y)=\frac1{2\pii}\left(\begin{array}{cc}{\frac{R\w\e^{\alpha(\phi(x)+\phi(y)-2\phi(R))}}{XY^{*}-R^{2}}}&{\frac{e^{\alpha(\phi(x)-\phi(y))}}{X-Y}}\\{\frac{e^{-\alpha(\phi(x)-\phi(y))}}{(X-Y)^{*}}}&{\frac{R\e^{-\alpha(\phi(x)+\phi(y)-2\phi(R))}}{w\(XY^{*}-R^{2})^{*}}}\\\end{array}\right).
F_{B}^{bosonic}={\frac{\Gamma(2-2k_{0}^{2})\Gamma(\vec{k}\cdot\vec{p}+k_{0}^{2}-1)}{\Gamma(\vec{k}\cdot\vec{p}-k_{0}^{2}+2)}}\bigg[\vec{k}\cdot\vec{p}\epsilon_{ij}^{1}\epsilon_{ij}^{2}-2\epsilon_{ij}^{1}\epsilon_{il}^{2}p^{j}k^{l}(1-k_{0}^{2})\bigg]
S_{{\mathrm{\scriptsize~{int}}}}^{(2)}=\frac12\int_{\calM}\xiw^{\prime\prime}(\bar{X})\xi+\int_{\partial{\calM}}(-V^{\prime}(\bar{X})\xi\partial_{N}\xi-\frac12\xiV^{\prime\prime}(\bar{X})\xi\partial_{N}\bar{X}+\frac12\xi\hat{w}^{\prime\prime}(\bar{X})\xi)
\chi(R)=\frac{1}{e\theta}\left(1-\left(1-\frac{\widetilde{\theta}}{R^{2}}\right)^{1/2}\right)=\frac{1}{2}\frac{e}{\pi\kappa}\frac{1}{R^{2}}\left(1+\frac{1}{4}\tilde{\theta}\frac{1}{R^{2}}+0(\theta^{2})\right)\,,
T_{\mu\nu}^{(\alpha)}(k)=-i\int\frac{d^{2}p}{(2\pi)^{2}}\int_{0}^{1}d\xi_{1}\int_{0}^{1}d\xi_{2}\;\delta(1-\xi_{1}-\xi_{2})\left[\frac{d}{dz}F^{(\alpha)}(z)\right]_{z=\sum\xi_{j}p_{j}^{2}},
{\calL}_{s}=\frac{F_{\pi}^{2}}{16}Tr\,U_{\mu}^{+}U^{\mu}-\frac1{16e^{2}}Tr\left\{U_{\mu}^{+}U_{\nu}U^{+\mu}U^{\nu}-U_{\mu}^{+}U_{\nu}U^{+\nu}U^{\mu}\right\}.
{\tilde{F}}_{\mu}(L)\;=\;\frac{1}{2}\epsilon_{\mu\nu\lambda}F_{\nu\lambda}(L)\;\;\;,\;\;\;F_{\mu\nu}(L)\;=\;\partial_{\mu}L_{\nu}-\partial_{\nu}L_{\mu}\,+\,g^{\frac{1}{2}}[L_{\mu},L_{\nu}]\;.
G_{\perp}\left(\vec{x}\right)=\vec{\nabla}_{\perp}\vec{E}_{\perp}\left(\vec{x}\right)+g\epsilon^{abc}\frac{\tau^{a}}{2}\vec{A}_{\perp}^{b}\left(\vec{x}\right)\left(\vec{E}_{\perp}^{c}\left(\vec{x}\right)-\vec{\eta}\left(\vec{x}_{\perp}\right)\delta_{c,3}\right)+g\rho^{m}(\vec{x})\.
\varepsilon\cdotp=\left(\begin{array}{cccc}{0}&{\varepsilon^{01}}&{\varepsilon^{02}}&{\varepsilon^{03}}\\{-\varepsilon^{01}}&{0}&{\varepsilon^{12}}&{\varepsilon^{13}}\\{-\varepsilon^{02}}&{-\varepsilon^{12}}&{0}&{\varepsilon^{23}}\\{-\varepsilon^{03}}&{-\varepsilon^{13}}&{-\varepsilon^{23}}&{0}\\\end{array}\right)\left(\begin{array}{c}{\omega}\\{0}\\{0}\\{\omega}\\\end{array}\right)=0
\begin{array}{lc}{}&{(0)(0~-\frac{3}{2})}\\{\mid\mathrm{ground}>}&{(0)(0~-\frac{3}{2})}\\{\mid\mathrm{1st}>}&{(0)(1~-\frac{3}{2})}\\{\mid\mathrm{2nd}>}&{(0)(0~-\frac{1}{2}),}\\\end{array}
F_{\lambda}[x^{+},x^{-},x_{i}]={\frac{1}{\sqrt{2}}}\frac{1}{2\pi}e^{ik_{+}x^{+}}e^{ik_{-}x^{-}}\Pi_{i=1}^{8}\left(\frac{\sqrt{\omega}}{2^{n_{i}}n_{i}!\sqrt{\pi}}\right)^{\frac{1}{2}}H_{n_{i}}(\tilde{x}_{i})e^{-\frac{1}{2}\tilde{x}_{i}^{2}}
{\calL}=\frac{1}{2}\partial^{\mu}\phi_{1}\partial_{\mu}\phi_{1}-\alpha_{1}({\phi_{2}}^{2}+\epsilon_{1})(1-cos\phi_{1})+\frac{1}{2}\partial^{\mu}\phi_{2}\partial_{\mu}\phi_{2}-\alpha_{2}({\phi_{1}}^{2}+\epsilon_{2})(1-cos\phi_{2})
\langles(x_{1})s(x_{2})\dotss(x_{2n})\rangle\;=\;\sum_{\sigma\in{\calS}_{ext}}(-1)^{|\Gamma\cap\sigma^{*}|}\;\;t^{2|\sigma|}\;\Bigg/\;\sum_{\sigma\in{\calS}_{ext}}t^{2|\sigma|}\;,
\begin{array}{rcl}{R^{V_{z_{1}},V_{z_{2}+nw/2}^{*}}}&{=}&{(I\otimesC)^{-1}R^{V_{z_{1}},V_{z_{2}+(n-1)w}\otimes\cdots\otimesV_{z_{2}+w}}(I\otimesC),}\\{R^{V_{z_{1}+nw/2}^{*},V_{z_{2}}}}&{=}&{(C\otimesI)^{-1}R^{V_{z_{1}+(n-1)w}\otimes\cdots\otimesV_{z_{1}+w},V_{z_{2}}}(C\otimesI).}\\\end{array}
\frac{d^{2}f}{dx^{2}}-\frac{(5x^{3}+1)}{2x(1-x^{3})}\frac{df}{dx}-\frac{l(l+6)}{4x^{2}(1-x^{3})}f+\frac{\lambda^{2}}{(1-x^{3})^{2}}\left(1+\frac{\rho_{0}^{6}}{(\omegaR)^{6}x^{3}}\right)f=0\,,
\begin{array}{ccccccc}{\mathcal{L}}&{\supset}&{\mathcal{H}^{n}}&{\supset}&{\!\pi(\Omega^{n})\!}&{\supset}&{\!\pi(\mathcal{J}^{n})\!}\\{}&{}&{\!\cup\,}&{}&{\cap}&{}&{}\\{}&{}&{\mathcal{C}^{n}}&{}&{\mathcal{B}}&{\subset}&{\mathcal{L}}\\\end{array}
r_{1,2}^{2}=\frac{l^{2}}{4}\left(1-\frac{12\tilde{\alpha}}{l^{2}}\right)\left(1\mp\sqrt{1-\frac{16\tilde{\alpha}}{l^{2}}\left(1-\frac{12\tilde{\alpha}}{l^{2}}\right)^{-2}}\right).
{\calI}=\frac{1}{2i}\left(\partial_{T}-\frac{2}{(T-{\bar{T}})}\right)\{(\partial_{U}-\frac{2}{U-{\bar{U}})})f(T,U)+(\partial_{\bar{U}}+\frac{2}{U-{\bar{U}}}){\bar{f}}(T,{\bar{U}}\}+c.c.
\partial_{z}^{2}U=\frac{1}{4\pi}\int_{D_{\eta}}\frac{1}{z-z^{\prime}}[e^{U+v-\phi_{B}}\partial_{z^{\prime}}(U+v)-\partial_{z^{\prime}}\beta-\beta\partial_{z^{\prime}}\phi_{B}](z^{\prime})e^{\phi_{B}(z^{\prime})}d^{2}z^{\prime}-
\frac{d^{2}F(t)}{dt^{2}}=-e^{-t\Phi_{0}}\Phi_{0}Q_{0}(\Phi_{0})e^{t\Phi_{0}}+e^{-t\Phi_{0}}Q_{0}(\Phi_{0})\Phi_{0}e^{t\Phi_{0}}=e^{-t\Phi_{0}}[Q_{0}(\Phi_{0}),\Phi_{0}]e^{t\Phi_{0}}\.
S^{\prime}=S+\frac{1}{2}\oint_{-\pir_{c}}^{\pir_{c}}dy\intd^{4}x\sqrt{-G}\left(-G^{55}\Lambda_{0}^{2}\delta^{2}(y)+G^{55}\Lambda_{\pi}^{2}\delta^{2}(y-\pir_{c})\right)\.
\psi_{N}^{\{r,\lambda\}}(z_{q})=<Nr+\lambda\mid\prod_{q=1}^{N}U^{r}(z_{q})\mid\lambda>=\prod_{l>l^{\prime}}(z_{l}-z_{l^{\prime}})^{r^{2}}\prod_{q=1}^{N}z_{q}^{r\cdot\lambda}
\Xi^{2p}=\left[\oplus_{j=0,(even)}^{2p}\Lambda^{j}(M)\right]\oplus\left[\oplus_{j=0,(even)}^{2p}\Lambda^{j}(M)\right]\oplus\left[\oplus_{j=1,(odd)}^{2p-1}\Lambda^{j}(M)\right]\oplus\left[\oplus_{j=1,(odd)}^{2p-1}\Lambda^{j}(M)\right]
Z=e^{-\frac{N}{2}S_{eff}(m^{2},g)}\int({\cal{D}}\lambda)e^{-\frac{1}{2}\int\mathrm{d}^{d}x\,\mathrm{d}^{d}y\bigl[\lambda(x)D(x-y)\lambda(y)\bigl]+O(1/\sqrt{N})}\,,
A=\left\langle\sum_{X_{i}\in({\itG}_{1})_{X_{i^{\prime}}^{\prime}}}\sum_{\stackrel{i_{\alpha_{1}}<j_{\alpha_{1}}<k_{\alpha_{1}}}{\left\{i_{\alpha_{1}},j_{\alpha_{1}},k_{\alpha_{1}}\right\}\inJ_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}t_{k_{\alpha_{1}}}{\bf1}_{(w(t_{i_{\alpha_{1}}})=w(t_{j_{\alpha_{1}}})=w(t_{k_{\alpha_{1}}}))}\right.\times
{\calL}=-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}\;\;\;,\;\;\;G_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c}
\begin{array}{l}{e^{+2}\nabla\psi_{+2q}^{2-}+e^{-2}\nabla\psi_{-2q}^{2-}+e^{+}\nablah_{+q}^{2-}+e^{-}\nabla\chi_{q}^{2}+T^{+2}\psi_{+2q}^{2-}+T^{-2}\psi_{-2q}^{2-}+T^{+}h_{+q}^{2-}+T^{-}\chi_{q}^{2}}\\{=\frac12\gamma_{q\dot{p}}^{i}\Omega^{-2i}\pi_{\dot{p}}^{2+},}\\\end{array}
I_{E}=-2\piPT_{2}\intdz\int_{0}^{R_{+}^{2}}dR^{2}\bigg\{\sqrt{\frac{R_{+}^{2}+D^{2}P^{2}R^{2}}{R_{+}^{2}-R^{2}}}-\frac{h}{2}\frac{R^{2}}{\sqrt{R_{+}^{2}-R^{2}}}\bigg\}
e^{-2\rho}+\frac{\kappa}{2}\rho=e^{\lambda(w^{+}-w^{-})}-\frac{\kappa\lambda}{4}(w^{+}-w^{-})-\frac{m}{\lambda^{2}x_{0}^{+}}\left(e^{\lambdaw^{+}}-e^{w_{0}^{+}}\right)