harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\intd^{6}\xi\;F\;\left(\xi\right)\;=\;c\int_{0}^{\infty}dq\int_{0}^{\infty}dp\int_{0}^{4\pi}d\varphi\,frac{q\,p}{\sqrt{q^{2}\,+\,p^{2}\,}}\;F\;\left(q,p,\varphi\right)\,
A_{i_{1}\ldotsi_{p}}^{(g)}(x_{0},{\bfx})=x_{0}^{\frac{d}{2}}\int{\frac{d^{d}k}{(2\pi)^{d}}}e^{-i{\bfk}\cdot{\bfx}}a_{i_{1}\ldotsi_{p}}({\bfk})K_{\nu}(|{\bfk}|x_{0}).
\begin{array}{rcl}{\displaystyle\sum_{k}K_{k}^{j}(z)\phi^{k}(z)|\mathrm{vac}\rangle_{i}}&{=}&{\nu^{(i)}(z)\phi^{j}(-z)|\mathrm{vac}\rangle_{i},}\\{{}_{i}\langle\mathrm{vac}|\displaystyle\sum_{k}\phi_{k}(z)K_{j}^{k}(z)}&{=}&{\nu^{(i)}(z){}_{i}\langle\mathrm{vac}|\phi_{j}(-z),}\\\end{array}
r_{+}=\left(\begin{array}{cccc}{\frac{1}{2}}&{0}&{0}&{0}\\{0}&{-\frac{1}{2}}&{2}&{0}\\{0}&{0}&{-\frac{1}{2}}&{0}\\{0}&{0}&{0}&{\frac{1}{2}}\\\end{array}\right)
k_{2}^{\nu}T_{\rho\sigma\mu\nu}^{(2)abcd}(p_{1},p_{2};k_{1},k_{2})=C_{2}k_{1\mu}S_{\rho\sigma}^{(2)}+C_{2}\Gamma_{\sigma\mu\rho}^{(2)}(p_{2},k_{2},q_{2})
{\calL}_{m}^{(1)}=-\vec{E}\partial_{0}\vec{A}-\frac{1}{2m^{2}}(\vec{\nabla}\vec{E}_{L}-\rho)^{2}-\frac{1}{2}[\vec{E}^{2}+(\vec{\nabla}\times\vec{A})^{2}-m^{2}\vec{A}^{2}]+\vec{A}\vec{j}
\begin{array}{rcl}{T_{c}}&{=}&{\displaystyle+[3D_{m}(k_{1})D_{m}(k_{2})+D(k_{1})D(k_{2})]D_{m}(k_{1}+k_{2})(1+K_{12})}\\{}&{}&{\displaystyle+4(N-1)D(k_{1})D(k_{2})D_{m}(k_{1}+k_{2}).}\\\end{array}
\intd^{3}x\phi_{f}^{4}({\bfx},t)=\frac{3}{(2\pi)^{3}}\sum_{\alpha\neq\beta}\phi_{\alpha}^{2}\phi_{\beta}^{2}+\frac{1}{(2\pi)^{3}}\sum_{\alpha}\phi_{\alpha}^{4}.
f_{mnn^{\prime}}(v)=\frac{1}{\sqrt{v(1-v)}}\int_{0}^{v}dx\int_{0}^{1-v}dy\,\phi_{m}(x/v)\phi_{n}(y/(1-v))\frac{\phi_{n^{\prime}}(x)-\phi_{n^{\prime}}(v+y)}{(v+y-x)^{2}}
[Q,\bar{C}^{a}]_{P.B.}\stackrel{def.}{=}\tilde{K}_{a}=K_{a}+f_{ab}^{c}\bar{C}_{c}C^{b}~,~~~~[\tilde{K}_{a},\tilde{K}_{b}]_{P.B.}=f_{ab}^{c}\tilde{K}_{c}~.
\sigma_{abs}^{S}=\frac{1}{8}A_{h}(\omega^{2}-m^{2})^{2}r_{e}\frac{(e^{\frac{\omega-m\phi_{K}}{T_{H}}}-1)}{(e^{\frac{(\omega-m)}{T_{H}}}-1)(e^{\frac{1}{2}\pi(\omega+m)r_{e}}-1)}.
[G]_{ab}=\left(\begin{matrix}{G_{00}+\sum_{mn}G_{mn}v_{m}v_{n}}&{0}&{0}&{\cdots}&{0}\\{0}&{G_{11}}&{0}&{\cdots}&{\vdots}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}\\{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{0}&{0}&{0}&{\cdots}&{G_{pp}}\\\end{matrix}\right)\,
\int_{\Sigma_{g}}\omega_{z}^{j}\bar{\omega}_{\bar{z}}^{k}d^{2}z=(\bar{\Omega}-\Omega)^{jk},\quad\,\,\int_{\Sigma_{g}}\mu_{\bar{z}}^{iz}\omega_{z}^{j}\omega_{z}^{k}d^{2}z=i{\frac{\partial\Omega^{jk}}{\partialm_{i}}},\quad\,\,\int_{\Sigma_{g}}\bar{\mu}_{z}^{i\bar{z}}\bar{\omega}_{\bar{z}}^{j}\bar{\omega}_{\bar{z}}^{k}d^{2}z=-i{\frac{\partial\bar{\Omega}^{jk}}{\partial\bar{m}_{i}}},
\frac{\partial\chi(x)}{\partialx_{i}}\left[\frac{C_{i}(x)}{\chi(x)},~~\psi(x^{\prime})\right]_{t=t^{\prime}}=\frac{4\pi^{2}}{e}\left[\rho(x),~\psi(x^{\prime})\right]_{t=t^{\prime}},
\widetilde\nu(x^{\mu})=+{\frac{1}{2}}(\psi_{\downarrow}(x^{\mu})+\psi_{\uparrow}^{c}(x^{\mu}))+{\frac{\gamma^{5}}{2}}(\psi_{\uparrow}(x^{\mu})-\psi_{\downarrow}^{c}(x^{\mu}))\,.
C_{i_{1}\ldotsi_{p-2k}}=2\partial_{[i_{1}}X^{\mu_{1}}\cdots\partial_{i_{p-2k-1}}X^{\mu_{p-2k-1}}\,\bar{\theta}_{L}\Gamma_{\mu_{1}\ldots\mu_{p-2k-1}}\partial_{i_{p-2k}]}\theta_{R}\,.
S_{8}=-\frac12\intd^{8}x\sqrt{g_{(8)}}\left(R_{(8)}-\frac12\frac{\partial_{\mu}\taug^{\mu\nu}\partial_{\nu}{\bar{\tau}}}{\tau_{2}^{2}}-\frac12\frac{\partial_{\mu}\tilde{\tau}g^{\mu\nu}\partial_{\nu}{\bar{\tilde{\tau}}}}{\tilde{\tau}_{2}^{2}}\right).
V(\phi,\chi)=\frac{q_{1}^{2}q_{2}^{2}}{\kappa^{2}}[\mid\phi\mid^{2}(\mid\chi\mid^{2}-c_{2}^{2})^{2}+\mid\chi\mid^{2}(\mid\phi\mid^{2}-c_{1}^{2})^{2}]\;\;.
S=\frac{4\pi\nu}{3}\intdtd^{4}xTr\left({\bfA}\wedged{{\bfA}}\wedged{\bfA}-{\frac{3i}{2}}{\bfA}\wedge{\bfA}\wedge{\bfA}\wedged{{\bfA}}-{\frac{3}{5}}{\bfA}\wedge{\bfA}\wedge{\bfA}\wedge{\bfA}\wedge{\bfA}\right)
\begin{array}{rcl}{{\calL}_{\mathrm{g.f.}}}&{=}&{\displaystyle-\frac{1}{2\xi}\left((\partial^{\mu}Z_{\mu})^{2}+(\partial^{\mu}A_{\mu})^{2}\right)-m_{Z}\pi\partial^{\mu}Z_{\mu}-\frac{1}{2}\xim_{Z}^{2}\pi^{2},}\\{{\calL}_{\mathrm{ghost}}}&{=}&{\displaystyle{\calL}_{c\bar{c}}+{\calL}_{\phic\bar{c}}+{\calL}_{Gc\bar{c}},}\\\end{array}
{\bfH}_{1}{\bf\Phi}_{1}^{(n_{\rho},m)}={\tilde{E}}_{1}^{(n_{\rho},m)}{\bf\Phi}_{1}^{(n_{\rho},m)},\quadE_{1}^{(n_{\rho},m)}=2{\tilde{E}}^{(n_{\rho},m)},
\begin{array}{l}{\hat{a}f(z)=-z^{-1}(f(qz)-f(q^{-1}z))\,,}\\{\hat{b}f(z)=q^{-1}e^{i\varphi}f(qz)\,,}\\{\hat{c}f(z)=-q^{2}e^{-i\varphi}f(qz)\,,}\\{\hat{d}f(z)=qzf(qz)\,.}\\\end{array}
\beta_{k}^{\eta_{k}}=i\left(\varphi_{k}^{\mathrm{WKB}}\,\partial_{\eta}\varphi_{k}^{-\infty}+i\omega_{k}\varphi_{k}^{\mathrm{WKB}}\,\varphi_{k}^{-\infty}\right)\vert_{\eta=\eta_{k}}\,\,,
(x_{13}^{2})^{-2r}(x_{24}^{2})^{-2}\{(\sum_{s\inS_{r}}\prod_{a=1}^{r}\delta^{I_{a}J_{s(a)}})\delta^{I_{r+1}J_{r+1}}+(\frac{v}{u})^{2}\sum_{c,d=1}^{r}\delta^{I_{r+1}J_{d}}\delta^{I_{c}J_{r+1}}(\sum_{s\inS_{r-1}}\prod_{\substack{a=1\,a\neqc,s(a)\neqd}}^{r}\delta^{I_{a}J_{s(a)}})\}
det^{\prime}\Bigl(\astd_{\scriptscriptstyle1}+(\astd_{\scriptscriptstyle1})^{\scriptscriptstyle2}\Bigr)=(-1)^{\psi}det^{\prime}(\astd_{\scriptscriptstyle1})det^{\prime}(\mathrm{1\mkern-5muI}+\astd_{\scriptscriptstyle1})
{\calK}^{T}M{\calK}=\left(\begin{array}{cc}{{\calK}^{(1)}}&{{\tilde{\calK}}^{(1)}}\\\end{array}\right)e^{\phi}\left(\begin{array}{cc}{e^{-2\phi}+C^{(0)\,2}}&{C^{(0)}}\\{C^{(0)}}&{1}\\\end{array}\right)\left(\begin{array}{c}{{\calK}^{(1)}}\\{{\tilde{\calK}}^{(1)}}\\\end{array}\right)\,,
D_{b\beta}\lambda^{a\alpha}=\textstyle{\frac{1}{2}}\delta_{b}^{~a}\delta_{\beta}^{~\alpha}(\hat{\lambda}(z)+i\hat{\Omega}(z))-\delta_{b}^{~a}\tilde{\hat{w}}_{\beta}{}^{\alpha}(z)+\delta_{\beta}^{~\alpha}\hat{t}{}_{~b}^{a}(z)\,,
\dot{S}_{int}=\frac{\deltaS_{int}}{\delta\psi}\cdot\dot{P}_{\Lambda}\cdot\frac{\deltaS_{int}}{\delta\bar{\psi}}-\mathrm{tr}\left(\frac{\delta}{\delta\psi}\cdot\dot{P}_{\Lambda}\cdot\frac{\deltaS_{int}}{\delta\bar{\psi}}\right)
\left(\begin{array}{c}{A_{\underline{{\mu\nu}}}^{1}}\\{A_{\underline{{\mu\nu}}}^{2}\}\\\end{array}\right)=\mathcal{C}\left(\begin{array}{c}{\mathbf{A}_{\underline{{\mu\nu}}}^{1}}\\{\mathbf{A}_{\underline{{\mu\nu}}}^{2}\}\\\end{array}\right)
S_{\nu}^{\mu(j)}(w,\theta)=(1-w^{-2})^{-\mu/2}s_{\nu}^{\mu}(\theta)\sum_{r=-\infty}^{\infty}a_{\nu,r}^{\mu}(\theta)\psi_{\nu+2r}^{(j)}(2\theta^{1/2}w),\quad(j=1,2,3,4)
\vec{x}(t)=\frac{\vec{p}}{p^{0}}\left[{t+ms\int_{\varphi(0)}^{\varphi(t)}\frac{d\varphi}{(p,n)^{2}}-\varrho\int_{\varphi(0)}^{\varphi(t)}d\left(\frac{1}{pn}\right)}\right]+
\Gamma_{I}^{\phi_{S}^{0}}(-i\Omega_{m},\stackrel{\rightharpoonup}{p})=\Gamma_{I}^{\phi_{S}^{0}}(+i\Omega_{m},-\stackrel{\rightharpoonup}{p})=\Gamma_{I}^{\phi_{S}^{0}}(-i\Omega_{-m},-\stackrel{\rightharpoonup}{p}).
{\cal{D}}_{\epsilon_{M}}[\lambda]=\frac{e^{-\frac\pi2\epsilon_{M}}e^{-i\pi/8}}{(2QE)^{1/4}\Gamma(\frac12-i\epsilon_{M})}e^{+\fraci2\lambda^{2}QE}\int_{0}^{\infty}dv\e^{-i\lambda\sqrt{2QE}v+i\frac{v^{2}}2}v^{-i\epsilon_{M}-\frac12}
\mathcal{D}_{n}^{\Lambda,\Lambda_{0}}(q\,;p_{1},\cdots,p_{n})=\mathcal{L}_{(1)\,n}^{\Lambda,\Lambda_{0}}(q\,;p_{1},\cdots,p_{n})-\mathcal{L}_{n+1}^{\Lambda,\Lambda_{0}}(q,p_{1},\cdots,p_{n}).
\left(\partial_{i}-\frac{D-2}{2}\partial_{i}\sigma\right)^{2}=\partial^{2}-(D-2)(\partial_{i}\sigma)\partial^{i}-\frac{D-2}{2}(\partial^{2}\sigma)+\frac{(D-2)^{2}}{4}(\partial_{i}\sigma)(\partial^{i}\sigma)
\frac{p_{r}^{2}}{g_{rr}}-\frac{{g^{\prime}}_{tt}}{-\calD}\left(p_{\phi}+\frac{g_{t\phi}+\Omega_{0}g_{\phi\phi}}{{g^{\prime}}_{tt}}E\right)^{2}=-\left(\frac{E^{2}}{{g^{\prime}}_{tt}}+V\right),
\langle1\rangle_{\mathrm{``exact^{\prime\prime}}}=\sum_{g}\lambda_{c}^{g-1}\langle1\rangle_{g}=\frac{1}{\lambda_{c}}\mathrm{Tr}\left(W^{-1}\cdot\frac{1}{1-\lambda_{c}W}\right)=\frac{4}{1-16\lambda_{c}^{2}\beta^{2}}.
A_{1}=\left({\frac{\omega-m\Omega_{H}}{4\piT_{H}^{BTZ}}}\right)^{2},B_{1}=-{\frac{\rho_{-}^{2}}{\rho_{+}^{2}}}\left({\frac{\omega-m\Omega_{H}\rho_{+}^{2}/\rho_{-}^{2}}{4\piT_{H}^{BTZ}}}\right)^{2}.\nonumber
dq^{k}dF_{k}=\frac{\partialF_{k}}{\partialq^{j}}dq^{k}\wedgedq^{j}+\frac{\partialF_{k}}{\partial\dot{q}^{j}}dq^{k}\wedged\dot{q}^{j}+\frac{\partialF_{k}}{\partial\ddot{q}^{j}}dq^{k}\wedged\ddot{q}^{j}
\partial_{t}\left(\begin{array}{c}{J_{0}}\\{J_{1}}\\\end{array}\right)={\calD}_{1}\left(\begin{array}{c}{{\frac{\deltaH_{3}}{\deltaJ_{0}}}}\\{{\frac{\deltaH_{3}}{\deltaJ_{1}}}}\\\end{array}\right)={\calD}_{2}\left(\begin{array}{c}{{\frac{\deltaH_{2}}{\deltaJ_{0}}}}\\{{\frac{\deltaH_{2}}{\deltaJ_{1}}}}\\\end{array}\right)
v_{\uparrow}^{^{MR}}(p^{\mu})=-i[u_{\downarrow}^{^{MR}}(p^{\mu})]^{\ast}\quad,\quadv_{\downarrow}^{^{MR}}(p^{\mu})=+i[u_{\uparrow}^{^{MR}}(p^{\mu})]^{\ast}\quad,
\alpha_{n}^{\prime}=\mu^{-n/d}{\overline{\gamma}}_{-n}^{\prime},\qquad\beta_{n}^{\prime}=\mu^{(d-n)/d}{\overline{\delta}}_{-n}^{\prime},\qquad\beta_{n}^{\prime}+\delta_{n-2d}^{\prime}=0.
\left\langle\Psi\left|h_{k^{\prime}l^{\prime}}\left(y^{\prime}\right)h_{k^{\prime\prime}l^{\prime\prime}}\left(y^{\prime\prime}\right)\right|\Psi\right\rangle=K_{\left(k^{\prime}l^{\prime}\right)\left(k^{\prime\prime}l^{\prime\prime}\right)}\left(y^{\prime},y^{\prime\prime}\right)\left\langle\Psi|\Psi\right\rangle.
g_{ij}=\frac{1}{2}\sum_{i=1}^{D}R_{i}e_{i}^{I}R_{j}e_{i}^{I},\;\;g_{ij}^{*}=2\sum_{i=1}^{D}\frac{1}{R_{i}}e_{i}^{*I}\frac{1}{R_{j}}e_{J}^{*I}
\sum_{m,m^{\prime};\,m\pmm^{\prime}\mathrm{even}}A_{M}^{B,m}\biggl\{\widetilde\Pi_{MN}^{+BC}(m,\frac{m-m^{\prime}}{2})+\widetilde\Pi_{MN}^{-BC}(m,\frac{m+m^{\prime}}{2})\biggr\}A_{N}^{C,m^{\prime}}.
\bigg(\langle\langleF_{\mu\nu}(z_{1})F_{\rho\sigma}(z_{1}^{\prime})\rangle\rangle-\langle\langleF_{\mu\nu}(z_{1})\rangle\rangle\,\langle\langleF_{\rho\sigma}(z_{1}^{\prime})\rangle\rangle\bigg)^{\mathrm{LR}}=0.
v_{0}=u-w_{1}a^{-\gamma}-(\mathrm{~\frac{1}{12}~}u^{\prime\prime}+u^{2})a^{-2\gamma}+2(\mathrm{~\frac{1}{6}~}w_{1}^{\prime\prime}+uw_{1})a^{-3\gamma}+\mathrm{O}(a^{-4\gamma}).
g_{L}(\sigma^{+})=\left(\begin{array}{cc}{\tilde{a}(\sigma^{+})}&{\tilde{u}(\sigma^{+})}\\{-\tilde{v}(\sigma^{+})}&{\tilde{b}(\sigma^{+})}\\\end{array}\right),\;\;\;\;\;\;\;\;\;g_{R}(\sigma^{-})=\left(\begin{array}{cc}{a(\sigma^{-})}&{u(\sigma^{-})}\\{-v(\sigma^{-})}&{b(\sigma^{-})}\\\end{array}\right)
\frac{\vec{k}^{2}}{m_{f}}\phi_{\sigma}(\vec{k})+\sum_{\sigma^{\prime}}\frac{m_{f}}{(2\pi)^{3}}\int\frac{d^{3}k^{\prime}}{\sqrt{E_{k}E_{k^{\prime}}}}v_{OBE}(\sigma,\sigma^{\prime},\vec{k},\vec{k}^{\prime})\phi_{\sigma^{\prime}}(\vec{k}^{\prime})=\frac{M_{full}^{2}-4m_{f}^{2}}{4m_{f}}\phi_{\sigma}(\vec{k})\;,
W_{1}^{\prime}={X^{\prime}}_{1}^{12}+{X^{\prime}}_{2}^{12}+{X^{\prime}}_{3}^{12}+{X^{\prime}}_{4}^{12}+{{X^{\prime}}_{5}^{2}}{{X^{\prime}}_{6}}+{{X^{\prime}}_{5}}{{X^{\prime}}_{6}^{2}}.
\langle\phi_{1}(z_{1},\bar{z}_{1})\phi_{2}(z_{2},\bar{z}_{2})\phi_{3}(z_{3},\bar{z}_{3})\phi_{4}(z_{4},\bar{z}_{4})\rangle=G\left(\eta,\bar{\eta}\right)\prod_{i<j}^{4}z_{ij}^{h/3-h_{i}-h_{j}}\bar{z}_{ij}^{\bar{h}/3-\bar{h}_{i}-\bar{h}_{j}},
\psi_{1\xi}^{\dagger}(\epsilon)\psi_{1\eta}(-\epsilon)=e^{-\frac{i}{2}[\theta(\eta)-\theta(\xi)+\frac{1}{2}R_{1}(\epsilon,\xi,\eta)]}e^{ie\intd^{2}yf_{i}(y)A_{i}(y)+\frac{i}{2e}\intd^{2}ya_{L}^{1}(y)\partial_{i}E_{i}-a_{T}^{1}(y)\epsilon_{ij}\partial_{i}E_{j}(y)}
H_{0}=\frac{1}{2}\sum_{i=1}^{4}\mathrm{tr}M_{i}^{2}=(1+a)^{2}\frac{L_{1}^{2}}{2}+(1-a^{2})\frac{L_{2}^{2}}{2}+(1-a)^{2}\frac{L_{3}^{2}}{2}+(1-a^{2})\frac{L_{4}^{2}}{2}.
I_{3}^{(1)}=\int\frac{d^{D}k}{(k^{2}+qk)(k^{2}-qk)k^{2}}=-i\pi^{D/2}\left(\frac{4}{q^{2}}\right)^{1+\epsilon}\frac{1}{2\epsilon}\frac{\Gamma(1+\epsilon)}{1+2\epsilon}.
\frac{x^{{\binom{N-1}{1}}+{\binom{N-1}{3}}+\ldots}}{|x|^{{\binom{N-1}{2}}+{\binom{N-1}{4}}+\ldots}}=\frac{x^{2^{N-2}}}{|x|^{2^{N-2}-1}}=|x|\;,\qquad\forallN>2\;.
\begin{array}{c}{e_{i}=A_{i}^{-}A_{i+1}^{+},~~~1\lei\len+1}\\{f_{i}=A_{i}^{+}A_{i+1}^{-},~~~1\lei\len+1}\\{k_{i}=q^{-N_{A_{i}}+N_{A_{i+1}}},~~~1\lei\len+1}\\{e_{0}=A_{n+1}^{-}A_{1}^{+},}\\{f_{0}=A_{n+1}^{+}A_{1}^{-},}\\{k_{0}=q^{N_{A_{n+1}}+N_{A_{1}}}}\\\end{array}
a_{S}(k)\,a_{S}^{\dagger}(k^{\prime})-\sum_{l,l^{\prime}\in{\calI}}{\hat{\calR}}\left(k^{\prime},l;k,l^{\prime}\right)a_{S}^{\dagger}(l^{\prime})\,a_{S}(l)=\delta_{k\;k^{\prime}}\;,
\frac{(p+q)_{\rho}}{(p+q)^{2}}(g_{\mu\alpha}-\frac{\tilde{p}_{\mu}\tilde{p}_{\alpha}}{\tilde{p}^{2}})(g_{\nu\beta}-\frac{\tilde{q}_{\nu}\tilde{q}_{\beta}}{\tilde{q}^{2}})\Gamma_{\alpha\beta\rho}^{1,sing}(p,q)=0
(D_{j})_{i}^{l}=\left(\begin{array}{ccc}{d_{j(1)}}&{\dots}&{0}\\{0}&{\ddots}&{0}\\{0}&{\dots}&{d_{j(k)}}\\\end{array}\right)\,,\qquad(\hat{D}_{j})_{\alpha}^{\beta}=\left(\begin{array}{ccc}{a_{j(1)}\cdot{\bf1}}&{\dots}&{0}\\{0}&{\ddots}&{0}\\{0}&{\dots}&{a_{j(N)}\cdot{\bf1}}\\\end{array}\right)
\begin{array}{ccc}{(0~0~0~0~-\frac{7}{2})~\rightarrow}&{}&{(0~0~0~-\frac{7}{2})(\frac{7}{2})\oplus(0~0~0~-\frac{5}{2})(-\frac{1}{2}),}\\{(0~0~0~0~-\frac{1}{2})~\rightarrow}&{}&{(0~0~0~-\frac{1}{2})(\frac{1}{2})\oplus(0~0~0~+\frac{1}{2})(-\frac{7}{2}).}\\\end{array}
U_{DQRS}={\frac{N}{g}}tr\left({\frac{1}{2}}S^{2}+{\frac{1}{2}}D^{2}+{\frac{1}{4(1-c^{*})^{2}}}S^{4}+{\frac{1}{4(1+c^{*})^{2}}}D^{4}+{\frac{1}{2(1-c^{*})(1+c^{*})}}(SDSD+2S^{2}D^{2})\right).
=\frac1{z{}^{*}-v{}^{*}}|_{C^{*}}\frac1{w{}^{*}-u{}^{*}}|_{B^{*}}=\delta(z{}^{*}-v{}^{*})\;\delta(w{}^{*}-u{}^{*}),\;\;(|z^{*}|>|v^{*}|,|w^{*}|>|u^{*}|).
\nabla_{i}V_{j}={\frac{1}{2}}P_{AB,i}P_{CD,j}P_{,k}^{ABCD}g^{k\bar{l}}f_{\bar{l}}={\frac{1}{4}}\epsilon_{ABCDEF}P_{AB,i}P_{CD,j}P_{EF,k}g^{k\bar{l}}f_{\bar{l}}.
2\delta_{0,0+2}\approx\sqrt{n_{0}^{2}+n_{2}^{2}}~{\frac{1}{4\pi^{2}}}~({\frac{l_{s}}{b}})^{4}~{\frac{\sqrt{1-v^{2}}}{v}}~({\frac{1}{\sqrt{1+({\frac{n_{2}}{n_{0}}}})^{2}}}-{\frac{1}{\sqrt{1-v^{2}}}})^{2}
\sigma_{ij}(x^{-},y^{-};x^{+})=\int\!\frac{dP^{-}}{4\pi}\frac{dp^{+}}{4\pi}\frac{dk^{+}}{4\pi}e^{-\frac{i}{2}P^{-}x^{+}}e^{-\frac{i}{2}(p^{+}x^{--}k^{+}y^{-})}~\sigma_{ij}(p^{+},k^{+};P^{-}),
S_{nm,1}=s\left(\bar{b}_{\alpha}^{*\dot{\alpha}}b_{\dot{\alpha}}^{\alpha}+\bar{b}_{\alpha}^{\dot{\alpha}}b_{\dot{\alpha}}^{*\alpha}\right)=\bar{b}_{\alpha}^{*\dot{\alpha}}\beta_{\dot{\alpha}}^{\alpha}+\bar{\beta}_{\alpha}^{\dot{\alpha}}b_{\dot{\alpha}}^{*\alpha}\,,
\delta^{2}H=\frac{1}{4\pigP}\int\frac{dR}{R^{\prime}}\frac{R{R^{\prime}}^{2}(1+D^{2}{R^{\prime}}^{2})}{(1+{R^{\prime}}^{2})^{3/2}}\bigg(\frac{d}{dz}\frac{\deltaR}{R^{\prime}}\bigg)^{2}.
h_{L}^{\prime}(\ell+1)=(a_{1}^{\prime},a_{2}^{\prime})\left(\begin{array}{ll}{h_{L}(\ell)}&{h_{L\bullet}(\ell)}\\{h_{L\bullet}(\ell)}&{h_{\bullet}(\ell)}\\\end{array}\right)\left(\begin{array}{c}{a_{1}^{\prime}}\\{a_{2}^{\prime}}\\\end{array}\right)
Y_{1+}{\cdot\sigma}=Y_{1-}{\cdot\sigma}-4i\Phi_{1a}\bar{\Phi}{}_{1}^{a}=-(x_{\bar{2}1}{\cdot\sigma})^{-1}x_{\bar{2}3}{\cdot\sigma}(x_{\bar{1}3}{\cdot\sigma})^{-1}=(Y_{1-}{\cdot\sigma})^{\dagger}
\left(\begin{array}{cc}{\left(\frac{\partial}{\partialx}\right)^{2}}&{\frac{\partial}{\partialx}\frac{\partial}{\partialy}}\\{\frac{\partial}{\partialx}\frac{\partial}{\partialy}}&{\left(\frac{\partial}{\partialy}\right)^{2}}\\\end{array}\right)q(x_{E})\;.
\begin{array}{c}{{\displaystyleK_{ab}^{+}~=\;-\,\frac{1}{4}\,(E_{aa}+E_{bb}+E_{\theta(a)\,\theta(a)}+E_{\theta(b)\,\theta(b)})\,+\,\frac{1}{2n}\,\mathbf{1}_{2n}^{}~,}}\\{{\displaystyleM_{ab}^{+}~=\;-\,\frac{1}{4}\,(E_{aa}+E_{bb}-E_{\theta(a)\,\theta(a)}-E_{\theta(b)\,\theta(b)})~,}}\\\end{array}
\left(v^{j}\frac{\partial}{\partial\Phi^{j}}+\bar{v}^{\bar{j}}\frac{\partial}{\partial\bar{\Phi}^{\bar{j}}}\right)G_{i\bar{k}}+G_{j\bar{k}}\frac{\partialv^{j}}{\partial\Phi^{i}}-G_{i\bar{j}}\frac{\partial\bar{v}^{\bar{k}}}{\partial\bar{\Phi}^{\bar{j}}}=0\,,
T_{\theta}^{\theta}=\varphi^{2}e^{-2gM}-\frac{A^{2}\varphi^{2}}{r^{2}}e^{-2gM}+{\calP}M^{2}-\left(\frac{A^{\prime}}{rh}\right)^{2}-{\calR}\left(\frac{G}{gr}\right)^{2}+U,
\gamma_{\alpha\gamma}^{a}\tilde{\gamma}^{b\gamma\beta}+\gamma_{\alpha\gamma}^{b}\tilde{\gamma}^{a\gamma\beta}=2\delta_{\alpha}^{\beta}\eta^{ab},\qquad\mathrm{tr}(\gamma^{a}\tilde{\gamma}^{b})=4\eta^{ab},\quad\gamma_{a\alpha\beta}\gamma_{\gamma\delta}^{a}=-2\epsilon_{\alpha\beta\gamma\delta}.
\begin{array}{rclcrcl}{\mathrm{Tr}[T_{a},\cdots]}&{=}&{0,}&{{\qquad}}&{\mathrm{Tr}[\Sigma_{\alpha},\cdots]}&{=}&{0,}\\{\mathrm{Str}[T_{a},\cdots]}&{=}&{0,}&{{\qquad}}&{\mathrm{Str}\{\Sigma_{\alpha},\cdots\}}&{=}&{0,}\\\end{array}
\Sigma(\tau^{\prime\prime},\tau^{\prime})=e^{{\calG}\tau^{\prime\prime}}[c(\tau^{\prime})-\delta(\tau^{\prime})]+\delta(\tau^{\prime\prime}-\tau^{\prime})+{\calG}\theta(\tau^{\prime\prime}-\tau^{\prime})e^{{\calG}(\tau^{\prime\prime}-\tau^{\prime})}
\begin{array}{c}{2ImT_{1}=\frac12\intd\tauT_{\mu\nu}^{abcd}T_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}b^{\prime}cd*}P^{\mu\mu^{\prime}}(k_{1})P^{\nu\nu^{\prime}}(k_{2})}\\{+\frac12\intd\tauS^{abcd}S^{a^{\prime}b^{\prime}cd*}}\\\end{array}
\left(\partial_{u}^{2}+{\frac{3+u^{2}}{4(1-u^{2})^{2}}}\right)\sqrt{1-u^{2}}\partial_{u}\phi={\frac{1}{\sqrt{1-u^{2}}}}\partial_{u}\left[(1-u^{2})\partial_{u}^{2}-{\frac{1}{4}}\right]\phi=0.
S_{E}={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-g^{E}}\left[{\calR}_{g^{E}}-{\frac{4}{D-2}}(\partial\phi)^{2}-{\frac{1}{2\cdot(p+2)!}}e^{2a_{p}\phi}F_{p+2}^{2}-{\frac{1}{2\cdotD!}}e^{2a\phi}F_{D}^{2}\right].
L_{0}[a_{n},a_{0}]=-i\sum_{n>0}a_{n}\dot{a}_{n}^{*}-m^{2}La_{0}^{2}-\sum_{n>0}\frac{m^{2}L}{4\pin}a_{n}^{*}a_{n}\equiv-i\sum_{n>0}a_{n}\dot{a}_{n}^{*}-H\;,
\sum_{\alpha=0}^{3}\eta_{\alpha\alpha}(x^{\alpha}-y^{\alpha}(t^{\prime}))\frac{dy^{\alpha}(t^{\prime})}{dt^{\prime}}=c|{\bfx}-{\bfy}(t^{\prime})|-\sum_{i=1}^{3}(x^{i}-y^{i}(t^{\prime}))\frac{dy^{i}(t^{\prime})}{dt^{\prime}}.
Z_{m}(0|J,\bar{Q})=\mathrm{exp}\Bigr\{(i/\hbar)(\sigma-z_{Q}^{-1})g^{ij}\bigr(J_{i}\bar{Q}_{j}+\mathrm{~\frac{1}{2}~}m^{2}\bar{Q}_{i}\bar{Q}_{j}\bigr)\Bigr\}\,\mathrm{exp}\Bigr\{m^{2}\bar{Q}_{i}\frac{\delta}{\deltaJ_{i}}\Bigr\}Z_{m}(0|J).
V_{\cal{C}}^{-1}({\bf{r}})\,{\textstyle\frac{\lambda^{d}}{2}}\,V_{\cal{C}}({\bf{r}})=\sum_{m=0}\sum_{n=0}{\textstyle\frac{g^{m+n}}{m!n!}}(-1)^{m+n}f_{(m)}^{\vec{\mu}ac}f_{(n)}^{\vec{\nu}cd}\,{\calR}_{(m)}^{\vec{\mu}}({\bfr})\,{\calM}_{(n)}^{\vec{\nu}}({\bfr})\,{\textstyle\frac{\lambda^{a}}{2}}\;,
\mathrm{Res}\left[I_{C}(k,q),q_{0}=|\vec{q}|\right]=\frac{1}{2|\vec{k}||\vec{q}|}\left.\left(4q\cdotu-4k\cdotu\frac{k\cdotq}{k^{2}}-k\cdotu\right)\right|_{q_{0}=|\vec{q}|},
\dot{s}_{\mathrm{mat}}=\left(3{\frac{c_{\mathrm{ph}}^{2}-c^{2}}{c^{2}c_{\mathrm{ph}}^{2}}}\hat{p}{\frac{\dot{a}}{a}}-\hat{\rho}{\frac{\dot{c}_{\mathrm{ph}}}{c_{\mathrm{ph}}}}\right){\frac{cc_{\mathrm{ph}}a^{3}}{T}}.
\left\{J_{H}^{-1}\left(J_{\Sigma}^{-1}(z)\right),z\right\}-\left\{J_{\Sigma}^{-1}(z),z\right\}=\left\{J_{H}^{-1}\left(J_{\Sigma}^{-1}\right),J_{\Sigma}^{-1}\right\}\left(\partial_{z}J_{\Sigma}^{-1}(z)\right)^{2}.
L=-{\frac{1}{g^{2}}}\!\int\!d^{2}x\Bigl({\frac{1}{4}}F_{\mu\nu}*F^{\mu\nu}+D_{\mu}\phi*(D^{\mu}\phi)^{\dagger}+{\frac{\lambda}{2}}(\phi*\phi^{\dagger}\!-\!v^{2})^{2}\Bigr)
B_{l}^{2}=-\frac{4q^{2}r}{(r^{2}+q^{2})^{n}}\lbrace\int^{r}\frac{(s^{2}+q^{2})^{n}}{s^{2}}ds-\int^{q}\frac{(s^{2}-q^{2})^{n}}{s^{2}}ds-\alpha\rbrace.
D_{x}={\frac{\partial}{\partialx}}+u_{x}{\frac{\partial}{\partialu}},\;\;\;D_{y}={\frac{\partial}{\partialy}}+u_{y}{\frac{\partial}{\partialu}},\;\;\;D_{z}={\frac{\partial}{\partialz}}+u_{z}{\frac{\partial}{\partialu}}.
{_3F}_{2}\left\{\begin{array}{l}{s,s^{\prime},-N}\\{t^{\prime},1-N-t}\\\end{array}\biggr|1\right\}=\frac{(t+s)_{N}}{(t)_{N}}{_3F}_{2}\left\{\begin{array}{l}{s,t^{\prime}-s^{\prime},-N}\\{t^{\prime},t+s}\\\end{array}\biggr|1\right\},
\prod_{i=1}^{m_{a}^{\prime}}\psi_{2}^{(a)}({y^{\prime}}_{i}^{(a)})\prod_{i=1}^{n_{a}}\overline{{\psi}}_{1}^{(a)}(x_{i}^{(a)}\!+\hat{\tau})\prod_{i=1}^{m_{a}}\psi_{2}^{(a)}(y_{i}^{(a)}\!+\hat{\tau})\prod_{i=1}^{n_{a}^{\prime}}\overline{{\psi}}_{1}^{(a)}({x^{\prime}}_{i}^{(a)})\Big\rangle_{0}\;.
g{\epsilon^{N}}\epsilon^{\nu\mu}A_{\mu}\partial_{\nu}a+g\left({\frac{N\epsilon^{N-1}\phi(x_{\mu})}{M^{N}}}+...\right)\epsilon^{\mu\nu}\partial_{\mu}a(g_{\nu\rho}-{\frac{\partial_{\nu}\partial_{\rho}}{\partial^{2}}})A^{\rho},
{\calH}_{\mathrm{int}}=-\frac{1}{4}\mathrm{Tr}\left[\left(\frac{\partial}{\partial\Phi}\right)U^{\prime}(\Phi)\right]=-\frac{1}{4}\sum_{i,j}\frac{U^{\prime}(x_{i})-U^{\prime}(x_{j})}{x_{i}-x_{j}}.
\left(\begin{array}{cc}{2A_{[{\tti}}{}^{[{\ttI}}\delta_{{\ttj}]}{}^{{\ttJ}]}}&{\Sigma_{\ttijKL}}\\{\Sigma^{\ttklIJ}}&{2A_{[{\ttk}}{}^{[{\ttK}}\delta_{{\ttl}]}{}^{{\ttL}]}}\\\end{array}\right)
[\delta(\varepsilon_{1}),\delta(\varepsilon_{2})]A_{\mui}=D_{\mui}^{j}(P^{mn}{}_{,j}\varepsilon_{1n}\varepsilon_{2m})-(\epsilon^{\nu\rho}D_{\rho}X^{j})\epsilon_{\nu\mu}P^{mn}{}_{,ji}\;\varepsilon_{1n}\varepsilon_{2m}
\mathrm{tr}[a_{1}^{\dagger}(\frac{P^{+}}{3})a_{2}^{\dagger}(\frac{2P^{+}}{3})]|0\rangle+\mathrm{tr}[a_{2}^{\dagger}(\frac{P^{+}}{3})a_{1}^{\dagger}(\frac{2P^{+}}{3})]|0\rangle,
G_{\mu\nu}[\xi|s]=4\pi{\tilde{g}}\epsilon_{\mu\nu\rho\sigma}\dot{\xi}^{\rho}(s)[{\bar{\psi}}(\xi(s))\gamma^{\sigma}T^{i}\psi(\xi(s))]\Omega_{\xi}^{-1}(s,0)\tau_{i}\Omega_{\xi}(s,0),
S_{\sigma,s}^{(1)}(t_{s}^{b},t^{\prime})=Q_{\Gamma_{b,s}}^{-2}(t)\left(S_{\sigma,s}^{(1)}(t,t^{\prime})+\sum_{N_{s}}\hat{Y}_{\sigma,N_{s}}^{(1)}(t)\Psi_{\sigma,N_{s}}^{(1)}(t^{\prime})\right)
q_{b}(a,r)=-\frac{1}{2\pia^{3}r^{n}S_{D}\sigma^{D-1}}\sum_{l=0}^{\infty}D_{l}\int_{ma}^{\infty}dz\,z^{3}\frac{\bar{I}_{\nu_{l}}(z)}{\bar{K}_{\nu_{l}}(z)}
S=\int\,d^{D}\,x\,\sqrt{g}\left[R(g)-\partial_{\mu}\,\varphi\,\partial^{\mu}\,\varphi-\sum_{p}\\frac{1}{2}\,\frac{1}{(p+1)!}\e^{\lambda_{p}\,\varphi}\,(d\,A_{p})^{2}\right]\,.
S^{(1)}=\int_{M}[-e(A-a)d\alpha-\frac{e^{2}}{4\pim}ada+2\pi\alphad\beta-ebd\beta-\frac{e^{2}}{4\pi(2p_{1})}bdb],\;\;m\in2{\bfZ}+1,\;\;p_{i}\in{\bfZ}.
Z_{W}(S^{3};N,k)=\sqrt{\frac{N^{\prime}}{N}}{N^{\prime}}^{-\frac{N^{\prime}}{4}}{\bfe}\left[\rho_{N^{\prime},N}\right]\prod_{j=1}^{N^{\prime}-N-1}\left(1-{\bfe}\left[{\textstyle\frac{N+j}{N^{\prime}}}\right]\right)^{\frac{j}{2}}\prod_{j=1}^{N-1}\left(1-{\bfe}\left[{\textstyle\frac{N-j}{N^{\prime}}}\right]\right)^{\frac{j}{2}}\,,
B=\frac{4\kappa}{8\pi^{3}n}\left[1-z\int\frac{d^{2}q}{4\pi^{2}}\frac{1}{(\frac{2\kappa}{\pin}q^{2}D(q)+2z)^{2}}\right]^{-1}\int\frac{d^{2}q}{4\pi^{2}}\frac{q^{2}}{(\frac{2\kappa}{\pin}q^{2}D(q)+2z)^{2}}
\left\langleU_{\lambda}\left(w\right)\right\rangle_{w^{\prime}}^{\left({\itG}_{1}\right)_{X_{i^{\prime}}^{\prime}}}=\left\{\prod_{j^{\prime}\inJ_{X_{i^{\prime}}^{\prime}}}e^{q_{1}(L^{d}-1)r(L^{\beta}t_{j^{\prime}}^{\prime})}\right\}^{-1}\times
S=\sum_{\mu\nu}\frac{\beta_{\mu\nu}}{2N}\sum_{\{p_{\sigma}\mu\nu\}}ReTr(U_{p_{\sigma}\mu\nu}+U_{p_{\sigma}\mu\nu}^{+})=\sum_{\mu\nu}\frac{\beta_{\mu\nu}}{2N}\sum_{\{p_{\sigma}\mu\nu\}}Tr(U_{p_{\sigma}\mu\nu}+U_{p_{\sigma}\mu\nu}^{+}),
H=\frac{1}{2m}\left(p_{x}^{2}+p_{y}^{2}\right)+\frac{1}{2}mw^{2}\left[\left(x-\frac{\theta}{2\hbar}p_{y}\right)^{2}+\left(y+\frac{\theta}{2\hbar}p_{x}\right)^{2}\right],
L_{q}^{QCD}=g\,\int\,d^{3}xd^{3}x^{\prime}\,\bar{q}_{\alpha\alpha^{\prime}}(x,x^{\prime})\gamma_{\alpha^{\prime}\beta^{\prime}}^{\mu^{\prime}}A_{\mu^{\prime}}^{a}(x,x^{\prime})\frac{\lambda^{a}}{2}q_{\alpha\beta^{\prime}}(x,x^{\prime}),
\hat{h}_{m}^{II}(y)=\sqrt{-|y-y_{*}|+y_{*}+y_{0}}\left\{c_{1}J_{0}[m(-|y-y_{*}|+y_{*}+y_{0})]+c_{2}Y_{0}[m(-|y-y_{*}|+y_{*}+y_{0})]\right\}\,.
ds^{2}=e^{-2k\midy\mid}\left[-\left(1-\frac{U_{T}^{4}}{k^{4}}e^{4k\midy\mid}\right)dt^{2}+\sum_{i=1}^{3}dx^{i}dx^{i}\right]+\frac{dy^{2}}{1-\frac{U_{T}^{4}}{k^{4}}e^{4k\midy\mid}}
\begin{array}{c}{a\bar{a}+\bar{a}a=1+c_{1}N,\quadc_{1}\in{\bfC},}\\{\varphi(n)=(-)^{n-1}\{(1-c_{1}q\frac{d}{dq})\frac{q^{-n}-1}{q^{-1}-1}\}_{q=-1}}\\{=(-)^{n-1}\{S_{0}(n,-1)+c_{1}S_{1}(n,-1)\}=\frac{1}{2}c_{1}n+[n]_{-1}(1-\frac{1}{2}c_{1}).}\\\end{array}
\begin{array}{c}{a_{i}a_{j}=q^{sgn(j-i)}a_{j}a_{i}}\\{a_{i}a_{j}^{+}=qa_{j}^{+}a_{i},\quadi\neqj}\\{a_{i}a_{i}^{+}=1+(q^{2}-1)\sum_{j}\theta_{ij}a_{j}^{+}a_{j}+q^{2}a_{i}^{+}a_{i}.}\\\end{array}
\begin{aligned}{U(1)_{a}:}&{{}(\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4})\sim(e^{i\theta_{a}}\Phi_{1},\Phi_{2},e^{i\theta_{a}}\Phi_{3},\Phi_{4})\,,}\\{U(1)_{b}:}&{{}(\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4})\sim(\Phi_{1},e^{i\theta_{b}}\Phi_{2},\Phi_{3},e^{i\theta_{b}}\Phi_{4})\,.}\\\end{aligned}
I(\nu)=\frac{2W(1)}{(\nu^{2}-\frac{1}{4})}-\int_{1}^{\infty}d\alpha(\frac{d}{d\alpha}[W(\alpha)\alpha^{\frac{3}{4}}])[{\frac{\alpha^{\frac{\nu}{2}+\frac{1}{4}}}{\frac{\nu}{2}+\frac{1}{4}}}-{\frac{\alpha^{\frac{-\nu}{2}+\frac{1}{4}}}{\frac{\nu}{2}-\frac{1}{4}}}].