harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calI}_{1}^{reg}(n,L;s)={\frac{1}{2}}\left[(n+{\frac{1}{2}}{\frac{\pi}{L}}\right]^{3-2s}\;{\frac{\Gamma(s-3/2)}{\Gamma(s-1/2)}}={\frac{1}{(2s-3)}}\left[(n+{\frac{1}{2}}{\frac{\pi}{L}}\right]^{3-2s}
\left(\begin{array}{cc}{u}&{i\alpha}\\{i\beta}&{\bar{u}}\\\end{array}\right)=\frac{1}{\sqrt{r_{+}^{2}+r_{-}^{2}}}\left(\begin{array}{cc}{e^{i(r_{+}\phi+r_{-}t)}\sqrt{r^{2}+r_{-}^{2}}}&{ie^{r_{-}\phi-r_{+}t}\sqrt{r_{+}^{2}-r^{2}}}\\{ie^{-r_{-}\phi+r_{+}t}\sqrt{r_{+}^{2}-r^{2}}}&{e^{-i(r_{+}\phi+r_{-}t)}\sqrt{r^{2}+r_{-}^{2}}}\\\end{array}\right).
{\calA}_{m_{1},m_{2},...,m_{N}}^{j_{1},j_{2},...,j_{N}}(z_{1},z_{2},...,z_{N})=\int\prod_{i=1}^{N}d^{2}z_{i}\left\langle\prod_{i=1}^{N}V_{j_{i},m_{i}}(z_{i})\right\rangle.
{\frac{m_{p}}{m_{e}}}={\frac{137\pi}{<x(1-x)><{\frac{1}{y}}>}}={\frac{137\pi}{({\frac{3}{14}})[1+{\frac{2}{7}}+{\frac{4}{49}}]({\frac{4}{5}})}}=1836.15\{\bf1497\ldots}
W_{L}=\sum_{s,r=1,2}({\lambda}_{r,s}M+{m_{Q}}_{r,s})V^{2r-1,2s}+(\mathrm{Pf}V-{\Lambda_{L}}^{4})Y+\sum_{k=2}^{N_{c}}g_{r}{u^{cl}}_{r},\nonumber\,
\tilde{\tau}_{0}=2\frac{2m^{2}+L^{2}}{\sqrt{m^{2}+L^{2}}}K(\frac{m^{2}}{m^{2}+L^{2}})-2\sqrt{m^{2}+L^{2}}\;E(\frac{m^{2}}{m^{2}+L^{2}}),
\langlea_{out}({\bfp}_{1})\dotsa_{in}^{*}({\bfk}_{n})\rangle=\int\prod_{i=1}^{m}F_{{\bfp}_{i}}(x_{i})d^{5}x_{i}\prod_{j=1}^{n}F_{{\bfk}_{j}}^{*}(y_{j})d^{5}y_{j}\Box_{x_{1}}\dots\Box_{y_{n}}G(x_{1},\dots,y_{n})+...
S[g]=-\frac{1}{8\pi}\int\!d^{2}x\,\mathrm{Tr}(g\partial_{\mu}g^{-1}g\partial^{\mu}g^{-1})-\frac{1}{12\pi}\int_{\Gamma}\!d^{3}x\,\epsilon_{\mu\nu\rho}\mathrm{Tr}(g\partial^{\mu}g^{-1}g\partial^{\nu}g^{-1}g\partial^{\rho}g^{-1})\quad.
\left[M^{+i},F^{-j}(x)\right]=-i\left(x^{+}\partial^{i}-x^{i}\partial^{+}\right)F^{-j}(x)-iF^{ij}(x)-ig^{ij}\left(F^{+-}(x)+\frac12[\partial_{-}]^{-1}*G(x)\right)
\begin{array}{l}{\Phi_{3}\times\Phi_{\overline{3}}=\Phi_{1}+\Phi_{8}\,\\Phi_{6}\times\Phi_{\overline{6}}=\Phi_{1}+\Phi_{8}\,\\Phi_{10}\times\Phi_{\overline{{10}}}=\Phi_{1}}\\{\Phi_{8}\times\Phi_{8}=\Phi_{1}+2\Phi_{8}+\Phi_{10}+\Phi_{\overline{{10}}}\,\\Phi_{15}\times\Phi_{\overline{{15}}}=\Phi_{1}+\Phi_{8}}\\\end{array}
\int\frac{\delta\hat{\Gamma}_{cl}}{\delta\hat{\varphi}_{0}}\;,\;\int\left\{A\frac{\delta}{\deltaA}+c\frac{\delta}{\deltac}\right\}\hat{\Gamma}_{cl}\;,\;m_{H}\partial_{m_{H}}\hat{\Gamma}_{cl}\;,\;e\partial_{e}\hat{\Gamma}_{cl}\;,
\begin{array}{ll}{\mathrm{Sector}\\}&{\mathrm{Field~content}}\\{\mathrm{even}}&{[1,1],\,[3,3],\,[5,5],\,[7,7],\,[9,9],\,[11,11],\,[13,13]}\\{\mathrm{odd}}&{[2,2],\,[4,4],\,[6,6],\,[8,8],\,[10,10],\,[12,12]}\\{\mathrm{twisted}}&{[1,13],\,[3,11],\,[5,9],\,[7,7],\,[9,5],\,[11,3],\,[13,1]}\\\end{array}
\begin{array}{ll}{\{{\calA}(x),{\calB}(y)\}_{D}=}&{\{{\calA}(x),{\calB}(y)\}}\\{}&{-\,\displaystyle{\int}\{{\calA}(x),\Phi(z)\}J(z,w)\{\Phi(w),{\calB}(y)\}\,dz\,dw}\\\end{array}
\frac{dr_{(1)}}{d\sigma_{c}}=\pm\left(\frac{b^{2}-2r_{(0)}^{2}}{2r_{(0)}^{2}\sqrt{r_{(0)}^{2}-b^{2}}}r_{g}+\frac{b^{2}}{r_{(0)}^{2}\sqrt{r_{(0)}^{2}-b^{2}}}r_{(1)}\right),
|V\rangle=(\Pi_{Z}^{*})_{-\eta}|l\rangle\Psi^{(l-\frac{1}{2})}(\boldsymbolp,\boldsymbol\theta)+(d_{++})_{-\eta}|l\rangle\Phi^{(l)}(\boldsymbolp,\boldsymbol\theta)+\theta_{-\eta}^{--}|l\rangle\Xi^{(l)}(\boldsymbolp,\boldsymbol\theta).
\overline{{S}}_{F}(x-y)=\langle0|T^{+}(\psi^{(+)}(x)\psi^{(+)^{\dagger}}(y))|0\rangle=\langle0|P^{(+)}T^{+}(\psi(x)\overline{{\psi}}(y))\gamma^{0}P^{(+)}|0\rangle=\int\frac{d^{n}p}{(2\pi)^{n}}\;e^{-ip\cdot(x-y)}\,\overline{{S}}_{F}(p)
\bar{q}_{-}^{(0)\mu\nu}k_{\mu}k_{\nu}\bar{f}_{-(T)}=\frac{\kappa_{5}^{2}e^{-2A_{--}\alpha_{-}^{(0)}}}{\int_{w_{-}}^{w_{+}}dw^{\prime}e^{-3A(w^{\prime})}}\bar{\tau}_{-(T)}
dh=\sum_{n=0}^{\left[\frac{p+2}{2}\right]}\frac{1}{n!}dh^{(p+2-2n)}{\calF}^{n}+\sum_{n=1}^{\left[\frac{p+2}{2}\right]}\frac{1}{(n-1)!}(-1)^{p}h^{(p+2-2n)}(C\Gamma_{\mu}\Gamma_{11})_{\alpha\beta}\Pi^{\mu}\Pi^{\alpha}\Pi^{\beta}{\calF}^{n-1}\quad,
e^{\textstyle-J\alpha_{-}\Phi(\sigma,\tau)}=c_{J}a(\varpi)\sum_{m=-J}^{J}\,(-1)^{J-m}\,{\frac{\lambda_{m}^{(J)}(\varpi)}{\lfloor\varpi+2m\rfloor}}\psi_{m}^{(J)}(\sigma)\,{\overline{\psi}_{m}^{(J)}}(\sigma)\bigr/a(\varpi),
\Gamma^{\mu}=\left(\begin{matrix}{0}&{\sigma^{\mu}}\\{\bar{\sigma}^{\mu}}&{0}\\\end{matrix}\right),\qquad\Gamma^{4}=-i\left(\begin{matrix}{-1}&{0}\\{0}&{1}\\\end{matrix}\right)=-i\,\gamma^{5},\quad\sigma^{\mu}=(1,\vec{\sigma}),\;\;\bar{\sigma}^{\mu}=(1,-\vec{\sigma})\,.
\omega_{n_{1},n_{2},\dots,n_{D-1}}=\left[\left(\frac{n_{1}\pi}{L_{1}}\right)^{2}+\left(\frac{n_{2}\pi}{L_{2}}\right)^{2}+\dots+\left(\frac{n_{D-1}\pi}{L_{D-1}}\right)^{2}\right]^{1/2}.
[P,X^{n}]=-i\sum_{k=0}^{n-1}X^{k}GX^{n-1-k}=\frac{-i}{q^{1/2}+q^{-1/2}}\frac{q^{n}-1}{q-1}\left(U+q^{-(n-1)}U^{\dagger}\right)X^{n-1},
L_{rel}=\frac{1}{2}\sum_{k=1}^{N-1}({\dot{\rho}}_{k}^{2}+{\rho}_{k}^{2}{\dot{\phi}}_{k}^{2})-\frac{1}{2}\sum_{k=1}^{N-1}{\omega}^{2}{\rho}_{k}^{2}+\alpha\hbar\sum_{i<k}{\dot{\phi}}_{ik}.
S_{\mathrm{KKh}}=-T_{\mathrm{KKh}}\intd^{6}\xi\e^{-2{\hat{\phi}}}{\hat{k}}^{2}\sqrt{|\mathrm{det}(\\partial_{i}{\hat{X}}^{\hat{\mu}}\partial_{j}{\hat{X}}^{\hat{\nu}}{\hat{\Pi}}_{\hat{\mu}\hat{\nu}}-{\hat{k}}^{-2}{\hat{F}}_{i}{\hat{F}_{j}})|}+\mathrm{WZ}\,,
i{\calM}_{fi}=(ig)^{2}{\frac{i}{P^{2}-M^{2}+i\varepsilon}}-(ig)^{2}{\frac{i}{(P^{2}-M^{2}+i\varepsilon)}}i\Pi(P^{2}){\frac{i}{(P^{2}-M^{2}+i\varepsilon)}}
\tilde{E}_{i}[\eta|t]=-\frac{2}{\bar{N}}\epsilon_{ij\rho\sigma}\dot{\eta}^{j}(t)\int\delta\xids\,\omega(\xi(s))E^{\rho}[\xi|s]\omega^{-1}(\xi(s))\dot{\xi}^{\sigma}(s)\dot{\xi}^{-2}(s)\delta(\xi(s)-\eta(t)),
C\;=\;C_{b}^{a}\:\delta_{\beta}^{\alpha}\;,\;\;\;\;\;\;E\;=\;E_{b}^{a}\:\delta_{\beta}^{\alpha}\;,\;\;\;\;\;\;\phi\;=\;\phi_{(b\beta)}^{(a\alpha)}\;,\;\;\;\;\;\;\Xi\;=\;\Xi_{(b\beta)}^{(a\alpha)}\;.
W_{L}(x+\varepsilon\xi,y)\;=\;W_{L}(x,y)\:+\:i\varepsilon\left(A_{j}^{L}(x)\:W_{L}(x,y)-W_{L}(x,y)\:A_{j}^{R}(y)\right)\xi^{j}\:+\:O(\varepsilon^{2})\;.
d{\tilde{s}}^{2}=-\left(M-\frac{r^{2}}{l^{2}}\right)dt^{2}+Jdtd\varphi+r^{2}d\varphi^{2}+\left(\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}}\right)^{-1}dr^{2}
W_{2}(n):=\sum_{k=1}^{N(n)}\frac{W(t_{k})}{t_{k}^{!}}=\frac{1}{n!}\sum_{k=1}^{N(n)}W(t_{k})\Pi(t_{k})W(t_{k})\stackrel{?}{=}\frac{(n-1)!}{2^{n-1}}
\left[\begin{array}{c}{\alpha}\\{m}\\\end{array}\right]_{q^{a}}\equiv\frac{\left[\left[\alpha\right]\right]_{q^{a}}\left[\left[\alpha-1\right]\right]_{q^{a}}\ldots\left[\left[\alpha-m+1\right]\right]_{q^{a}}}{\left[\left[m\right]\right]_{q^{a}}!}
P^{\mu}({\calA})=\frac{4}{3}\left[{\calA}_{\nu}\partial_{\rho}{\calA}_{\sigma}+\partial_{\rho}{\calA}_{\sigma}{\calA}_{\nu}+\frac{3}{2}{\calA}_{\nu}{\calA}_{\rho}{\calA}_{\sigma}\right]\epsilon^{\mu\nu\rho\sigma}
\sum_{j}\gamma_{j}^{L_{0}}=c^{L_{0}L_{1}L_{2}}\left[\sum_{i,sym}\lambda_{pp,i}^{L_{0}}+\sum_{i,sym}\lambda_{pa,i}^{L_{0}}+\sum_{i,sym}\lambda_{ap,i}^{L_{0}}+\sum_{i,sym}\lambda_{aa,i}^{L_{0}}\right],
\int\prod_{i=1}^{2n}{\frac{dt_{i}}{2\pi}}d\theta_{i}~\Theta(t_{1}-t_{2})\Theta(t_{2}-t_{3})\cdots\Theta(t_{2n-1}-t_{2n})\prod_{i<j}|e^{it_{i}}-e^{it_{j}}-ie^{\frac{i}{2}(t_{i}+t_{j})}\theta_{i}\theta_{j}|={\frac{1}{2^{n}}}~,
\begin{array}{ccc}{E_{a}^{\mu}(x)}&{=}&{\left(\begin{array}{cc}{e_{+a}^{\mu}+ie_{-a}^{\mu}}&{0}\\{0}&{e_{+a}^{\mu}-ie_{-a}^{\mu}}\\\end{array}\right)}\\{A_{\mu}(x)}&{=}&{\left(\begin{array}{cc}{a_{+\mu}+ia_{-\mu}}&{0}\\{0}&{a_{+\mu}-ia_{-\mu}}\\\end{array}\right)}\\{\Phi(x)}&{=}&{\left(\begin{array}{cc}{\phi_{+}+iphi_{-}}&{0}\\{0}&{\phi_{+}-i\phi_{-}}\\\end{array}\right),}\\\end{array}
\begin{array}{l}{\lbrackq^{-},\alpha_{0}^{+}]=i\,,}\\{\left[\alpha_{n}^{-}\,,\alpha_{m}^{+}\right]=n\,\delta_{n+m,0}\,\,\,,}\\{\left[s_{n}\,,s_{m}\right]=\left(\frack2-1\right)\,n\,\delta_{n+m,0}\,\,\,,}\\{\left[L_{n}^{\prime}\,,L_{m}^{\prime}\right]=(n-m)L_{n+m}^{\prime}}\\\end{array}
\left.\begin{array}{cl}{\mathrm{2-tensor}}&{b_{\mu\nu}^{TT}}\\{\mathrm{vector}}&{A_{\mu}^{T};B_{\mu}^{T},b_{\mu}^{TL}}\\{\mathrm{scalar}}&{A^{L};\phi,B^{L},B,b^{LL},\beta_{1},}\\\end{array}\right.
\dot{\alpha}_{0}\dot{\Psi}_{0}+\dot{\alpha}_{0}^{2}\Phi_{0}=-\alpha_{1}\left(\dot{\alpha}_{0}\dot{\xi}_{0}^{y}-\dot{\alpha}_{0}^{2}\xi_{0}^{y}-\dot{\alpha}_{0}^{2}\alpha_{1}l^{2}\phi_{0}+\dot{\alpha}_{0}\ddot{\alpha}_{0}e^{\alpha_{0}}lb_{0}-\frac{\dot{\alpha}_{0}\ddot{\alpha}_{0}}{\alpha_{1}}e^{2\alpha_{0}}\dot{\hat{E}}_{0}\right)+\dot{\alpha}_{0}\dot{\hat{\Psi}}_{0}.
\Phi_{0}=a_{0}^{-1}\sum_{M\geq0}\frac{x^{M}}{\Gamma^{2}(M+1)}\;\frac1{2\pii}\int_{C}\frac{\Gamma(4s+1)\Gamma(2M-s)}{\Gamma^{3}(s+1)}(e^{-\pii}y)^{s}\,ds,
e^{\hat{\mu}}=\tilde{E}_{\alpha}^{\hat{\mu}}d\theta^{\alpha}+\tilde{E}_{\nu}^{\hat{\mu}}dx^{\nu}+\tilde{E}_{i}^{\hat{\mu}}d\Phi(x,\theta)^{i}+\tilde{E}_{\alpha^{\prime}}^{\hat{\mu}}d\Psi^{\alpha^{\prime}}(x,\theta),
{\calU}(f,h,u)=\frac{(u^{2}-1)^{2}}{2e^{2}r^{4}}+\frac{\eta^{2}u^{2}h^{2}}{r^{2}}+\frac{\eta^{2}f^{2}}{r^{2}}+\frac{\lambda\eta^{4}}4(h^{2}-1)^{2}+\frac{\lambda\eta^{4}}4(f^{2}-1)^{2}\,
H^{(1D)}=m_{D}\;\intd^{3}\vec{x}\;\bar{\psi}\psi=\sum_{h=\pm1}\sum_{\vec{p}}H_{\vec{p},h}^{(1D)},\;\;\;H_{\vec{p},h}^{(1D)}=m_{D}\;(\;b_{\vec{p},h}^{(0)+}d_{\vec{p},h}^{(0)}+d_{\vec{p},h}^{(0)+}b_{\vec{p},h}^{(0)}\;).
\nabla\times\vec{v}=\left(\frac{\partialv_{3}}{\partialx_{2}}-\frac{\partialv_{2}}{\partialx_{3}},\frac{\partialv_{1}}{\partialx_{3}}-\frac{\partialv_{3}}{\partialx_{1}},\frac{\partialv_{2}}{\partialx_{1}}-\frac{\partialv_{2}}{\partialx_{2}}\right).
\begin{array}{lll}{T(z)}&{=}&{\frac{1}{(k+\check{g})}\{\frac{1}{2}(U_{a}U^{a}+U^{a}U_{a})+\frac{1}{2\check{g}}S_{a}^{b}S_{b}^{a}\}(z)}\\{\}&{\}&{\}\\{\}&{\}&{-\frac{1}{(k+\check{g})}\frac{1}{(2\check{g})}\{(S_{a}^{b}+\bar{S}_{a}^{b})(S_{b}^{a}+\bar{S}_{b}^{a})\}(z)}\\{\}&{\}&{\}\\{\}&{\}&{-\frac{1}{2}\{\psi_{a}\partial\psi^{a}+\psi^{a}\partial\psi_{a}\}(z)}\\\end{array}
\bar{q}_{\pm}^{(0)\mu\nu}k_{\mu}k_{\nu}\left[1+l_{S\pm}^{2}\bar{q}_{\pm}^{(0)\mu\nu}k_{\mu}k_{\nu}+O(\mu^{2})\right]\bar{\sigma}_{\pm(Y)}=-16\piG_{N}\bar{\tau}_{\pm(Y)}.
{\left.\frac{\delta\Omega^{\dot{a}\,a}}{\deltaC^{\dot{b}\,\alpha}}\right\vert}_{C={\calP}=\lambda=0}=T_{\alpha}\delta^{\dot{a}}{}_{\dot{b}}\,,\quad{\left.\frac{\delta\Omega^{\dot{a}\,a}}{\delta\lambda^{b\,\alpha}}\right\vert}_{C=\lambda=0}=-{\calP}_{\alpha}^{\dot{a}}\delta^{a}{}_{b}=-{\calP}_{\dot{b}\,\alpha}\epsilon^{\dot{b}\dot{a}}\,\delta^{a}{}_{b}\,,
q^{-C_{R_{1}}-C_{R_{2}}}~\sum_{\ell}~dim_{q}~\ell~~q^{C_{\ell}}~=~q^{C_{R_{1}}+C_{R_{2}}}~\sum_{\ell}~dim_{q}~\ell~~q^{-C_{\ell}}~=~{\frac{S_{\Lambda_{R_{1}}\Lambda_{R_{2}}}}{S_{00}}}~.
v\rightarrow{\calM}_{CY}^{\epsilon}v=\left(\begin{matrix}{}&{}&{}&{0}&{0}&{0}\\{}&{{\calM}_{-1}^{+\prime}}&{}&{0}&{2}&{0}\\{}&{}&{}&{0}&{0}&{2}\\{}&{}&{}&{}&{}&{}\\{}&{0}&{}&{}&{{\calM}_{-1}^{+\prime}}&{}\\{}&{}&{}&{}&{}&{}\\\end{matrix}\right)v\.
\langle\alpha,\,\beta\rangle\=\\alpha_{0}\,\beta_{0}^{*}\,+\,\frac12\,alpha_{1,A}\,\beta_{1,A}^{*}\,+\,\frac14\,alpha_{2,A}\,\beta_{2,A}^{*}\,+\,\frac18\\alpha_{3}\,\beta_{3}^{*}\.
D=\sum_{r,s}(\phi_{r}^{\dagger}T_{A}^{r}\phi_{r})(\phi_{s}^{\dagger}T_{A}^{s}\phi_{s})=\sum_{r,s}(\phi_{r}\otimes\phi_{s})^{\dagger}(T_{A}^{r}\otimesT_{A}^{s})(\phi_{r}\otimes\phi_{s}).
\omega_{\mathrm{phys}}^{2}(k_{\mathrm{phys}})=k_{\mathrm{phys}}^{2}\left[\frac{\epsilon_{1}}{1+\mathrm{e}^{x}}+\frac{\epsilon_{3}\mathrm{e}^{x}}{(1+\mathrm{e}^{x})^{2}}\right],
{\frac{1}{N}}\left(\begin{array}{ccccc}{N-1}&{-1}&{\cdots}&{\cdots}&{-1}\\{-1}&{N-1}&{}&{}&{\vdots}\\{\vdots}&{}&{\ddots}&{}&{\vdots}\\{\vdots}&{}&{}&{\ddots}&{\vdots}\\{-1}&{\cdots}&{\cdots}&{\cdots}&{N-1}\\\end{array}\right),
H=-\sum_{i}\sum_{\sigma=\uparrow,\downarrow}(c_{i,\sigma}^{\dagger}c_{i+1,\sigma}+c_{i+1,\sigma}^{\dagger}c_{i,\sigma})+U\sum_{i}(n_{i,\uparrow}-\frac{1}{2})(n_{i,\downarrow}-\frac{1}{2}).
is_{z}e^{3\psi/8-\phi/2}\{{\bfF}_{\alpha}^{2}+F_{\alpha}^{2}\}\epsilon_{6}=\left\{\partial_{\beta}(\phi-\psi/2)+is_{z}e^{\phi}\gamma_{7}\partial_{\beta}\kappa\right\}\left(\delta_{\alpha}^{\beta}+\gamma_{\alpha}{}^{\beta}\right)\epsilon_{6},
T_{\mathrm{RG}}=\frac{2\pi}{\omega-\frac{1}{\omega}\frac{\epsilon^{2}A^{2}}{24b^{2}}(\omega^{2}+b^{2})}=\frac{2\pi}{\sqrt{ab}}\{1+\frac{\epsilon^{2}A^{2}}{24b}(\frac1a+\frac1b)\}+o(\epsilon^{2}),
W_{\mathrm{T}}^{(\xi)}=\frac{1}{4\xi}\intd^{4}x\,\mathrm{tr}\,F_{\mu\nu}\tilde{F}^{\mu\nu}-\frac{1}{24\xi}\epsilon^{\mu\nu\rho\sigma}\bar{Q}_{\mu}^{(\xi)}\bar{Q}_{\nu}^{(\xi)}\bar{Q}_{\rho}^{(\xi)}\bar{Q}_{\sigma}^{(\xi)}\intd^{4}x\,\mathrm{tr}\,\phi^{2},
F[f](x)-c_{2}^{\prime}\delta((x^{0})^{2}-|{\bfx}|^{2})=\sum_{k=0}^{N}a_{k}\left(\sum_{\nu=0}^{3}\eta^{\nu\nu}\left(\frac{\partial}{\partialx^{\nu}}\right)^{2}\right)^{k}\delta(x).
\sum_{j=1}^{\infty}{\calH}_{\alpha=-1}^{j}\vertn_{1},n_{2},n_{3},\cdots;\bar{n}_{1},\bar{n}_{2},\bar{n}_{3},\cdots\rangle=\sum_{j=1}^{\infty}\{\epsilon_{j}^{\prime}+{\frac{j}{2}}\}\vertn_{1},n_{2},n_{3},\cdots;\bar{n}_{1},\bar{n}_{2},\bar{n}_{3},\cdots\rangle
g_{2\ell}={\frac{1}{2\pii}}\oint{\frac{dz}{z^{2\ell+1}}}{\frac{1}{2}}\Biggl[\biggl({\frac{z+1}{z-1}}\biggr)^{1/6}-\biggl({\frac{z+1}{z-1}}\biggr)^{1/6}\Biggr],
4\left(-\nabla_{\mu}\nabla^{\mu}\Phi+\nabla_{\mu}\Phi\nabla^{\mu}\Phi\right)=R-2\Lambda-\frac{1}{12}H_{\alpha\beta\gamma}H^{\alpha\beta\gamma}-\kappa^{2}T_{3}\;e^{\Phi}\;V(T)\sqrt{g^{-1}X},
(f^{(2)})^{-1}=\left(\begin{array}{cccc}{0}&{\delta_{ij}-a_{i}a_{j}}&{a_{i}}&{0}\\{-\delta_{ij}+a_{i}a_{j}}&{a_{j}\pi_{i}-a_{i}\pi_{j}}&{-\pi_{i}}&{a_{i}}\\{-2a_{i}}&{-2\pi_{i}}&{0}&{-\delta_{ij}}\\{0}&{2a_{i}}&{\delta{ij}}&{0}\\\end{array}\right)
\int_{0}^{\infty}\langleT_{\mu\nu}(\tau)\rangle_{\mathrm{\tiny~Ren.}}u^{\mu}u^{\nu}d\tau={\frac{1}{128\pi}}{\frac{\gamma}{v_{x}}}\left(2-v_{y}^{2}-v_{z}^{2}\right)\int_{0}^{\infty}\omega^{2}\,\left[n^{2}(\omega)-1\right]\,d\omega\geq0.
{\bar{\gamma}}^{0}=i\gamma^{0},\\\{\bar{\gamma}}^{i}=\gamma^{i},\\\,gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}={\bar{\gamma}}^{0}{\bar{\gamma}}^{1}{\bar{\gamma}}^{2}{\bar{\gamma}}^{3}
\frac{2\pi}{\tau_{2}}\left\{1-\frac{\beta^{2}}{8\pi^{2}\alpha^{\prime}}-\frac{\beta^{2}({A}^{2}+{B}^{2})}{16\pi^{2}\alpha^{\prime}}+\sqrt{\left(1-\frac{\beta^{2}({A}+{B})^{2}}{16\pi^{2}\alpha^{\prime}}\right)\left(1-\frac{\beta^{2}({A}-{B})^{2}}{16\pi^{2}\alpha^{\prime}}\right)}\right\}
0=\left(\begin{matrix}{0}&{-\omega_{01}}&{\ldots}&{-\omega_{0d}}\\{\omega_{10}}&{0}&{\ldots}&{\omega_{1d}}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{\omega_{d0}}&{\omega_{d1}}&{\ldots}&{0}\\\end{matrix}\right)\left(\begin{matrix}{rk^{0}}\\{0}\\{\vdots}\\{k^{0}}\\\end{matrix}\right)=\left(\begin{matrix}{\omega_{0d}}\\{r\omega_{10}+\omega_{1d}}\\{\vdots}\\{\omega_{d0}}\\\end{matrix}\right)k^{0}
F_{\sigma}^{(n)}(\zeta_{1},\cdots,\zeta_{2n-1},x^{2}\zeta_{2n})^{k\cdotsk_{2n-1}k}=\sigma\frac{\{k\}}{\{k_{2n-1}\}}F_{\sigma}^{(n)}(\zeta_{2n},\zeta_{1},\cdots,\zeta_{2n-1})^{k_{2n-1}k\cdotsk_{2n-1}}.
g(q)=g_{ij}(q)\,dq^{i}\,dq^{j}=2\int\,\frac{dx\,dy}{(1+r^{2})^{2}}\frac{\partialW}{\partialq^{i}}\frac{\partial\bar{W}}{\partialq^{j}}\,dq^{i}\,dq^{j}
\bar{G}_{n}^{(N)}(L,r,N-1)=q^{\frac{L}{2}+\frac{r}{N}}\widetilde{\sum_{\tilde{m}\in{\bfZ}^{N+1}}}q^{mC^{-1}m}(q)_{L-1}\prod_{i=0}^{N}\frac{1}{(q)_{m_{i}}}\left\{q^{-e_{n-1}C^{-1}m}-q^{\frac{L}{2}-\frac{r}{N}-e_{n}C^{-1}m}\right\}
\Pi_{\lambda\lambda^{\prime}}^{(1)ab}(q)=-g^{2}f^{acd}f^{bcd}[g_{\lambda\lambda^{\prime}}g_{\mu\nu}-g_{\lambda\mu}g_{\lambda^{\prime}\nu}]\int\frac{d^{4}k}{(2\pi)^{4}}\frac{g^{\mu\nu}}{(k^{2}-M^{2}+i\varepsilon)}
\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}=-u_{1}+u_{3}\rightarrowB_{3}^{1}=\left[B_{\bar{1}}^{0},B_{3}^{0}\right]=-D_{11}^{\bar{8}}+D_{12}^{\bar{6}}-D_{13}^{\bar{5}}
\begin{aligned}{M_{3(a)}(q)}&{{}=\int{\frac{d^{4}l}{(2\pi)^{4}}}\tau_{1}^{\mu\nu}(k_{1},l+k_{1},m_{1})\tau_{1}^{\alpha\beta}(l+k_{1},k_{2},m_{1})\tau_{2}^{\sigma\rho\gamma\delta}(k_{3},k_{4},m_{2})}\\{}&{{}\times\Big[\frac{i{\calP}_{\alpha\beta\gamma\delta}}{(l+q)^{2}}\Big]\Big[\frac{i{\calP}_{\mu\nu\sigma\rho}}{l^{2}}\Big]\Big[\frac{i}{(l+k_{1})^{2}-m_{1}^{2}}\Big]}\\\end{aligned}
ds^{2}=-\left(1-\frac{r_{0}^{2}}{r^{2}}\right)\left(1+\frac{r_{1}^{2}}{r^{2}}\right)^{-1}\left(1+\frac{r_{2}^{2}}{r^{2}}\right)^{-1}\beta^{2}dt^{2}+\frac{\Lambda^{-1}}{r^{2}}\left(1-\frac{r_{0}^{2}}{r^{2}}\right)^{-1}dr^{2}
{h_{a}^{(I)}}^{2}={v_{b}^{(I)}}^{2}=\sigma^{(I)},\qquad\mathrm{where}\qquad\sigma^{(I)}=\frac{1}{t^{(I)}d_{I}+\frac{1}{4}\left.\vec{\alpha}^{(I)}\right.^{2}},\qquadt^{(I)}=\frac{D-2-d_{I}}{2(D-2)}.
{\calW}\,(\phi,\widetilde\phi)=\frac{m^{2}}{4\lambda}\,\phi-\frac{\lambda}{3}\,\phi^{3}+\lambda\,\phi\,\widetilde\phi^{\,2}+\frac{p\,m}{2}\,\widetilde\phi^{\,2}+\frac{q\lambda}{3}\,\widetilde\phi^{\,3}\,,
\mu_{0}J_{\mathrm{em}\,\mu}=\frac{i\hbar}{2q\lambda^{2}}\left[\psi^{*}\left(\nabla_{\mu}\psi+i\frac{q}{\hbar}A_{\mu}\psi\right)-\left(\nabla_{\mu}\psi^{*}-i\frac{q}{\hbar}A_{\mu}\psi^{*}\right)\psi\right].
\textstyle\hat{\tau}^{k}=\tau^{k}+\Upsilon\circ\tau^{k-1}+\frac{1}{2}\Upsilon\circ\Upsilon\circ\tau^{k-2}+\cdots+\frac{1}{k!}\underbrace{\Upsilon\circ\cdots\circ\Upsilon}_{k}\circ\tau^{0}.
C_{K3}^{full}\equive_{A}\,\cap\,e_{B}=\left(\begin{matrix}{\begin{matrix}{\sigma_{1}}&{0}&{0}\\{0}&{\sigma_{1}}&{0}\\{0}&{0}&{\sigma_{1}}\\\end{matrix}}&{{\bf0}}\\{{\bf0}}&{\begin{matrix}{-C(E_{8})}&{{\bf0}}\\{{\bf0}}&{-C(E_{8})}\\\end{matrix}}\\\end{matrix}\right)\,
{\calK}_{\mu\nu}=(g_{\mu\rho}\,g_{\nu\lambda}+g_{\mu\lambda}\,g_{\nu\rho})\,\eta^{\kappa\lambda\alpha\beta}D_{\gamma}\,[{\tilde{R}}_{\,\,\,\,\,\alpha\beta}^{\rho\gamma}\,\partial_{\kappa}f]
\begin{array}{rcl}{\chi_{1,r(k+1)}^{(p,kp+p-1)}(q)}&{=}&{\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k}^{2}+(r-1)(n_{1}+\cdots+n_{k})}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-1}-n_{k}}(q)_{2n_{k}+r-1}}F_{p-1;r}^{(2n_{k}+r-1)}(q).}\\\end{array}
P_{\mu\nu\rho\tau}^{(\mu)}\left(p\right)P^{(\mu)\rho\tau}{}_{\alpha\beta}\left(p\right)=P_{\mu\nu\alpha\beta}^{(\mu)}\left(p\right)\;\,,\;\,P_{\mu\nu\alpha\beta}^{(\mu)}\left(p\right)=P_{\alpha\beta\mu\nu}^{*(\mu)}\left(p\right)\;\,.
\left(1-\frac{w\kappa}{4}e^{2\phi}\right)\left(\dot{\phi}^{\prime}-\dot{\rho}\phi^{\prime}-\rho^{\prime}\dot{\phi}\right)-\dot{\phi}\phi^{\prime}=\frac{\kappa}{8}e^{2\phi}\left[\dot{Z}Z^{\prime}+2\dot{Z}^{\prime}-2\left(\dot{\rho}Z^{\prime}+\rho^{\prime}\dot{Z}\right)\right].
\left(i\hbarc\gamma^{\mu}\partial_{\mu}-mc^{2}\right)_{\alpha\beta}\left(i\hbar{c}^{\prime}\gamma^{\mu^{\prime}}\partial_{\mu^{\prime}}-m{c^{\prime}}^{2}\right)_{\alpha^{\prime}\beta^{\prime}}q_{\beta\beta^{\prime}}(x,x^{\prime})=0.
S=\int\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}\left(B_{\mu\nu}^{a}(x)R_{\lambda\rho}^{a}(x)+\underbrace{\phi^{ab}(x)B_{\mu\nu}^{a}(x)B_{\lambda\rho}^{b}(x)}_{\mathrm{constraint\:term}}\right).
{\cal{L}}_{\theta}=-{\frac{1}{2}}e^{2\tilde{U}}e^{2\tilde{\Phi}}\theta^{2}+{\frac{1}{6}}\Lambda\epsilon_{ijk}\left(\theta\epsilon^{ijk}-\partial^{[i}b^{jk]}-b^{[i}V^{jk]}\right)\,
\partial_{i}G_{jk}=\hat{D}_{i}G_{jk}=\hat{D}_{i}(e_{j}^{a}e_{k}^{a})=M_{ij}^{l}e_{l}^{a}e_{k}^{a}+M_{ik}^{l}e_{l}^{a}e_{j}^{a}=M_{ij}^{l}G_{kl}+M_{ik}^{l}G_{jl}.
\begin{array}{ll}{a=}&{\frac{1}{2}e^{2}\rho^{2}}\\{b=}&{\frac{1}{4\beta}\left[q^{2}v^{2}+2\alpha^{2}e^{2}\rho^{2}\right]}\\{\delta=}&{-\frac{e^{2}\alpha\rho^{2}}{2\sqrt{\beta}}}\\\end{array}
H_{1}=1+\sum_{i}{\frac{Q_{i}}{[|\vec{x}-\vec{x}_{0\,i}|^{2}+{\frac{4Q}{(n-2)^{2}}}|\vec{z}-\vec{z}_{0}|^{n-2}]^{\frac{2}{n-2}}}},\\\\H_{2}={\frac{Q}{|\vec{z}-\vec{z}_{0}|^{4-n}}}.
V[\sigma]=\frac{N}{2\lambda_{B}}\sigma^{2}+\frac{N}{2\pi^{3/2}}\int_{1/\Lambda^{2}}^{\infty}\frac{dt}{t^{5/2}}\,\left(1+\frac{1}{2t}\right)e^{-t\sigma^{2}}e^{\not\nabla^{2}t}.
D^{--}=\partial^{--}-2i\theta^{-\alpha}\bar{\theta}^{-\dot{\alpha}}\partial_{\alpha\dot{\alpha}}+\theta^{-\alpha}{\frac{\partial}{\partial\theta^{+\alpha}}}+\bar{\theta}^{-\dot{\alpha}}{\frac{\partial}{\partial\bar{\theta}^{+\dot{\alpha}}}}\;.
d_{\beta}\,\sim\,\gamma_{n_{1}}\,(-1)^{k_{1}+\cdots+k_{N_{f}}}\,(2k_{0})^{4n_{1}-3}\,\prod_{i=1}^{N_{f}}\,{_{n_{1}}C_{k_{i}}}\qquad(0\leq{^\forall}k_{i}\leqn_{1}),
[\Lambda\frac{\partial\phantom{a}}{\partial\Lambda}+\mu\frac{\partial\phantom{a}}{\partial\mu}+\delta\,m\frac{\partial\phantom{a}}{\partialm}+\beta\frac{\partial\phantom{a}}{\partialg}-N\gamma]\Gamma^{(N)}(p_{1},\ldotsp_{N})\approx0,
V_{i}\cdotV_{j}=\left(\begin{array}{cc}{-1}&{-{\frac{1}{2}}}\\{-{\frac{1}{2}}}&{-1}\\\end{array}\right)~,~~~k_{ij}=\left(\begin{array}{cc}{k_{00}}&{{k_{10}+{\frac{1}{2}}}}\\{k_{10}}&{{k_{10}+{\frac{1}{2}}}}\\\end{array}\right)~.
ds_{5}^{2}=\frac{1}{z^{2}}\left[\left(\frac{z}{g}\right)^{2}\left(-dt^{2}+\left(g_{,r}^{2}+g_{,z}^{2}\right)\left(dr^{2}+dz^{2}\right)+\left(\frac{f}{r}\right)^{2}r^{2}d\Omega_{2}^{2}\right)\right]
A_{4F}=\int_{z_{4}}^{z_{2}}dz_{3}<O_{-\frac{1}{2}}^{(1)}(z_{1})O_{-\frac{1}{2}}^{(2)}(z_{2})V_{-\frac{1}{2}}^{(3)}(z_{3})O_{-\frac{1}{2}}^{(4)}(z_{4})>
\hat{G}_{MN}=\left(\begin{matrix}{e^{2{\bar{\varphi}}}h_{\bar{\mu}\bar{\nu}}+\bar{G}_{\bar{m}\bar{n}}\bar{A}_{\bar{\mu}}^{\bar{m}}\bar{A}_{\bar{\nu}}^{\bar{n}}}&{\bar{A}_{\bar{\mu}}^{\bar{m}}\bar{G}_{\bar{m}\bar{n}}}\\{\bar{A}_{\bar{\nu}}^{\bar{n}}\bar{G}_{\bar{m}\bar{n}}}&{\bar{G}_{\bar{m}\bar{n}}}\\\end{matrix}\right),
\begin{array}{c}{e_{1}e_{2}=e_{3}\,\e_{2}e_{3}=e_{1}\,\e_{3}e_{1}=e_{2}}\\{e_{3}e_{2}=-e_{1}\,\e_{2}e_{1}=-e_{3}\,\e_{1}e_{3}=-e_{2}\.}\\\end{array}
T_{-p+2}^{+q}=-2i\gamma_{q\dot{q}}^{i}\psi_{+2\dot{q}}^{1-}\gamma_{p\dot{p}}^{i}\psi_{-2\dot{q}}^{2+},\quadT_{+2-2}^{+q}=\frac{i}{16}\nabla_{-p}T_{-p+2}^{+q},
R_{\mu\nu}-2\nabla_{\mu}\nabla_{\nu}\phi+{\textstyle\frac{1}{4}}H_{\mu}{}^{\rho\sigma}H_{\nu\rho\sigma}-{\textstyle\frac{1}{4}}e^{2\phi}\sum_{n}{\textstyle\frac{(-1)^{n}}{(n-1)!}}T^{(n)}{}_{\mu\nu}\,,
\nu_{R}(E)=\mathrm{Tr}_{R}~\delta(\hat{H}-E)={\frac{1}{2\pi}}\int_{-\infty}^{\infty}d\alpha~\mathrm{Tr}_{R}~e^{i\alpha(E-\hat{H})}={\frac{1}{2\pi}}\int_{-\infty}^{\infty}d\alpha~e^{i\alphaE}Z_{R}(i\alpha)~~~.
S=\frac{1}{2\kappa_{4}^{2}}\int\!d^{4}x\,(-\tilde{g}_{4})^{1/2}\Biggl(\tilde{R}_{4}-2\frac{\partial_{\mu}\tau\partial^{\mu}\bar{\tau}}{|\tau-\bar{\tau}|^{2}}-6\frac{\partial_{\mu}\rho\partial^{\mu}\bar{\rho}}{|\rho-\bar{\rho}|^{2}}\Biggl)\.
S_{I,\theta}=-\frac{\sigma}{2}\int\frac{d\nu}{2\pi}\int_{\bar{R}_{1}}^{\bar{R}_{2}}dr\tilde{\theta}_{I}^{*}\left[-\frac{\partial}{\partialr}\gamma^{-1}r^{2}\frac{\partial}{\partialr}-(\nu^{2}-\omega^{2})\gammar^{2}\right]\tilde{\theta}_{I}\,,
4\hat{\calD}_{c}({\frac{1}{1-{{\frac{2}{3}}k^{2}}}}m_{d}^{c}h^{dab})=-{\frac{1}{4!}}\epsilon^{abcdef}E_{c}^{\underline{D}}E_{d}^{\underline{C}}E_{e}^{\underline{B}}E_{f}^{\underline{A}}F_{\underline{{ABCD}}}.
{\frac{1}{2}}\left\{\left[\begin{array}{c}{Q_{\alpha}}\\{\tilde{Q}_{\alpha}}\\\end{array}\right],\left[Q_{\beta}\\tilde{Q}_{\beta}\right]\right\}=\left[\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}\right]M\delta_{\alpha\beta}+\left[\begin{array}{cc}{p}&{q/g_{s}}\\{q/g_{s}}&{-p}\\\end{array}\right]{\frac{{L(\Gamma^{0}\Gamma^{1})}_{\alpha\beta}}{2\pi{\alpha^{\prime}}}}\.
\sqrt{\gamma_{1}}\,\sqrt{\gamma_{2}}\C_{\mathrm{tot}}(X_{1},X_{2})=32\,G_{N}\,T^{2}\,(\partial_{+}\,X_{1}^{\mu}\\partial_{+}\,X_{2\mu})(\partial_{-}\,X_{1}^{\nu}\\partial_{-}\,X_{2\nu})\,.
d\xi=\frac{2i\pi}{L_{\Phi}}\xi\sqrt{\varphi(z)}dz\leftrightarrows\varphi(z)\left(dz\right)^{2}=-\frac{L_{\Phi}^{2}}{\left(2\pi\right)^{2}}\frac{\left(d\xi\right)^{2}}{\xi^{2}}\tag{3.13}
x(u)={\frac{\sigma(\xi-u)}{\sigma(\xi)\sigma(u)}}\,e^{bu},\quadx_{d}(u)={\frac{\sigma(2\xi-u)}{\sigma(2\xi)\sigma(u)}}\,e^{2bu},\quadx_{t}(u)={\frac{\sigma(3\xi-u)}{\sigma(3\xi)\sigma(u)}}\,e^{3bu}.
\begin{array}{ll}{y_{0}=1}&{y_{4}=ibx_{1}}\\{y_{1}=-(J_{1}^{2}+J_{2}^{2}+\frac{1}{4}+2\alpha/x_{3}^{2})}&{y_{5}=ibx_{2}}\\{y_{2}=-2J_{3}}&{y_{6}=-\frac{1}{2}ib\{x_{3},J_{1}\}}\\{y_{3}=b^{2}x_{3}^{2}}&{y_{7}=-\frac{1}{2}ib\{x_{3},J_{2}\}\quad,}\\\end{array}
Q_{\calM}=\left(\begin{array}{ccccccc}{1}&{1}&{-3}&{1}&{0}&{0}&{2\zeta_{1}+\zeta_{2}+\zeta_{3}+\zeta_{4}}\\{0}&{0}&{1}&{-2}&{1}&{0}&{\zeta_{2}+\zeta_{5}}\\{0}&{0}&{0}&{1}&{-2}&{1}&{\zeta_{3}+\zeta_{6}}\\\end{array}\right).
E_{2}(r,a)=\frac{(\varepsilon-1)^{2}}{16\pi^{2}a^{4}}\,\frac{23}{60}\,\frac{(d^{4}-10d^{2}-15)}{(1+d)^{4}(1-d)^{4}}\,,d=\frac{r}{a}\,,r<a.
{\cal{Z}}_{rs}=P^{\mu}(C\Gamma_{\mu}^{(11)})_{rs}+Z_{\bf{O}}^{\mu\nu}(C\Gamma_{\mu\nu}^{(11)})_{rs}=Z_{\bf{O}}^{[\mu_{1}\ldots\mu_{5}]}(C\Gamma_{\mu_{1}\ldots\mu_{5}}^{(11)})_{rs}\,,
{\calW}_{F}^{(2;3g)}={\frac{C_{F}C_{A}(LT)^{2}(2T)^{4-4\omega}}{\pi^{2\omega}}}e^{-2i\pi\omega}[I_{1}(\beta^{2})+I_{2}(\beta^{2})+I_{3}(\beta^{2})+I_{4}(\beta^{2})]
\Lambdae_{i}(p)=L(\Lambdap)R(\Lambda,p)e_{i}=\sum_{i^{^{\prime}}}L(\Lambdap)e_{i^{^{\prime}}}R_{i^{^{\prime}}i}(\Lambda,p)=\sum_{i^{^{\prime}}}e_{i^{^{\prime}}}(\Lambdap)R_{i^{^{\prime}}i}(\Lambda,p)