harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\{~F~,~G~\}^{\mathrm{DB}}=\{~F~,~G~\}^{\mathrm{PB}}-\sum_{s,s^{\prime}}\{~F~,~\theta_{s}~\}^{\mathrm{PB}}(M^{-1})_{ss^{\prime}}\{~\theta_{s^{\prime}}~,~G~\}^{\mathrm{PB}}~.
\zeta_{2}^{r}(s)=\frac{\rho^{2-2s}(\kappa\lambda_{1})^{1-2s}\Gamma(1-s)\Gamma(2s-1)}{4\pi\Gamma(s)}+\frac{\rho^{2s}\Gamma(2s-1)}{2\pi\Gamma(2s)}\sum_{n=1}^{\infty}c_{-}^{1-2s}F(s,2s-1;2s;1-c_{+}/c_{-}).
({\calT}_{\alpha},{\calT}_{\beta})_{\Omega}=0,\quad(\bar{\calP}_{\alpha},\bar{\calP}_{\beta})_{\Omega}=(i\hbar)^{2}U_{\alpha\beta}^{\gamma}\bar{\calP}_{\gamma}(-1)^{\varepsilon_{\alpha}+\varepsilon_{\beta}+\varepsilon_{\gamma}},
{}~~~~~~=\Bigg\{\sum_{\sigma=0}^{N-1}{\frac{w(x_{3},x_{13},x_{1}|\sigma+e-h)w(x_{4},x_{24},x_{2}|\sigma+f-b)s(\sigma,b-g)}{w(x_{4},x_{14},x_{1}|\sigma+e-h)w(x_{3},x_{23},\omegax_{2}|\sigma+f-b)}}\Bigg\}_{0},
S_{L1}=\sum_{n=1}^{\infty}f_{Lq}(n)\,\left({\frac{v^{2}}{R^{2}M_{p}^{9}V_{T}~b^{7-p}}}\right)^{2n}~\left(\frac{v^{2}}{R^{2}M_{p}^{6}b^{4}}\right)^{-2n}=\sum_{n=1}^{\infty}f_{Lq}(n)\,\left({\frac{1}{M_{P}^{3}V_{T}~b^{3-p}}}\right)^{2n}
{\calU}=\left(\begin{matrix}{2\delta_{x}{}^{y}}&{-\frac1{W^{2}}(\partial_{x}K^{Z}){\calJ}_{Z}{}^{Y}}\\{\frac1{W^{2}}{\calJ}_{XZ}\partial^{y}K^{Z}}&{\frac32\delta_{X}{}^{Y}-\frac{1}{W^{2}}{\calJ}_{X}{}^{Z}{\calL}_{Z}{}^{Y}}\\\end{matrix}\right)\,,
{\calE}=\int_{0}^{\infty}du\int_{-\pi}^{\pi}d\phi\,u\left\{\left[b-b_{\mathrm{ext}}\right]^{2}+\left(\partial_{u}f\right)^{2}+\frac{1}{u^{2}}\left(\partial_{\phi}f\right)^{2}+f^{2}\left(j^{2}+\frac{1}{u^{2}}g^{2}\right)+\frac{1}{2}\kappa^{2}\left(1-f^{2}\right)^{2}\right\}.
\omegaZ_{i(2)}\rightarrowZ_{i(2)},\\,omega\tilde{B}_{i(12)}^{1}\rightarrow\tilde{B}_{i(12)}^{1},\\\omega\tilde{\psi}_{i(12)}^{1}\rightarrow\tilde{\psi}_{i(12)}^{1}(i=1,2,3),
\tilde{\calR}\equiv\left({\frac{\tilde{f}_{13}^{T}\tilde{f}_{24}^{T}}{\tilde{f}_{12}\tilde{f}_{34}}}\right)^{s/M_{s}^{2}}\left({\frac{\tilde{f}_{13}^{T}\tilde{f}_{24}^{T}}{\tilde{f}_{14}^{T}\tilde{f}_{23}^{T}}}\right)^{t/M_{s}^{2}}
{\calA}_{0}^{\theta}(t,x)=-\int_{0}^{1}{d\alpha\frac{{2em\Sigma}}{{\lambda^{2}}}}x^{1}\theta=-\frac{e}{{2\pi}}\left({1-\frac{{\frac{{e^{2}}}{\pi}}}{{\lambda^{2}}}}\right)x^{1}\theta.
{\frac{\partial(\deltaa_{j})}{\partiala_{i}^{*}}}=\mp{\frac{\partial(\deltaa_{i})}{\partiala_{j}^{*}}}\,,\quad{\frac{\partial(\deltaa_{j}^{*})}{\partiala_{i}}}=\mp{\frac{\partial(\deltaa_{i}^{*})}{\partiala_{j}}}\,,\quad{\frac{\partial(\deltaa_{i})}{\partiala_{j}}}+{\frac{\partial(\deltaa_{j}^{*})}{\partiala_{i}^{*}}}=0.
d_{{\bf{q}}/2}({\bf{q}})=(\frac{\omega_{{\bf{q}}}+\epsilon_{{\bf{q}}}+\rho_{0}v_{{\bf{q}}}}{2\mathrm{~}\omega_{{\bf{q}}}})^{\frac{1}{2}}f_{{\bf{q}}}-(\frac{-\omega_{{\bf{q}}}+\epsilon_{{\bf{q}}}+\rho_{0}v_{{\bf{q}}}}{2\mathrm{~}\omega_{{\bf{q}}}})^{\frac{1}{2}}f_{-{\bf{q}}}^{\dagger}
{\calS}_{WZW}\,=\,\frac{k}{4\pi}\int_{\Sigma}d^{2}\sigmaL_{a}^{i}L^{ja}\Omega_{ij}\,\+\frac{ik}{12\pi}\int_{B}d^{3}\sigma\,\epsilon^{abc}L_{a}^{i}L_{b}^{j}L_{c}^{k}{f_{ij}}^{\ell}\Omega_{k\ell}\,.
\left\{|p_{x}^{\prime},p_{t}^{\prime},\pi^{\prime},{\calC}^{\prime},{\bar{\calC}}^{\prime},\tau^{\prime}\rangle\equiv|\tau^{\prime}\rangle\right\},\quad\quad\left\{\langlex^{\prime\prime},t^{\prime\prime},\pi^{\prime\prime},{\calC}^{\prime\prime},{\bar{\calC}}^{\prime\prime},\tau^{\prime\prime}|\equiv\langle\tau^{\prime\prime}|\right\},
M=\langleS(\alpha)\rangle=\frac{\sum_{m}\sum_{n}x^{2kn^{2}+2mn+m^{2}/2k+k/8-l/2}(-1)^{(m-l)/2+nk}C_{m}^{l}}{\sum_{m}\sum_{n}x^{2kn^{2}+(2m-k)n+(m-k/2)^{2}/2k}C_{m}^{l}}\;,
A=\left(\begin{array}{cccc}{0}&{-1}&{0}&{0}\\{-1}&{\frac{b}{r_{0}}}&{0}&{0}\\{0}&{0}&{-\frac{b}{r_{0}}}&{-1}\\{0}&{0}&{-1}&{0}\\\end{array}\right),\quadB=\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{-r_{0}}&{b}&{0}&{0}\\{0}&{0}&{0}&{-r_{0}}\\{0}&{0}&{0}&{-b}\\\end{array}\right)
\frac{1}{2}\int\prod_{i=1}^{4}dg_{i}~~~\phi^{2}(g_{1},\ldots,g_{4})=\frac{1}{2}\sum_{N_{1}\ldotsN_{4}}\phi_{\mathcal{B}}{}_{\alpha_{1}\ldots\alpha_{4}}^{N_{1}\ldotsN_{4}}\\phi_{\mathcal{B}}{}_{\alpha_{1}\ldots\alpha_{4}}^{N_{1}\ldotsN_{4}}.
{{}^{\ast}F}_{\mu\nu}^{a}F_{\mu\nu}^{a}=\partial_{\mu}\mathrm{tr}(\textstyle{\frac{4}{3}}\epsilon_{\mu\nu\lambda\rho}{}A_{\nu}A_{\lambda}A_{\rho}-4A_{\nu}{}{{}^{\ast}F}_{\mu\nu})
\int_{0}^{s}ds_{1}\int_{0}^{s_{1}}ds_{2}Tr(K_{0}(s-s_{1})(i{\cal{W}}_{1}+{\cal{W}}_{2})K_{0}(s_{1}-s_{2})(i{\cal{W}}_{1}+{\cal{W}}_{2})K_{0}(s_{2})+\ldots
\mathrm{d}s_{q}^{2}=\left(\begin{matrix}{\mathrm{d}x_{+},}&{\mathrm{d}x_{-}}\\\end{matrix}\right)\left(\begin{matrix}{0}&{q^{\frac{1}{2}}\Gamma}\\{q^{-\frac{1}{2}}\Gamma}&{0}\\\end{matrix}\right)\left(\begin{matrix}{\mathrm{d}x_{+}}\\{\mathrm{d}x_{-}}\\\end{matrix}\right),
{\cal{F}}={\mathrm{\bf{1}}}\bigg(g^{2}{\cal{F}}^{(0)}-\frac{i}{2}g{\cal{F}}_{\mu\nu}^{(2)}dx^{\mu}\wedgedx^{\nu}\bigg)+{\mathrm{\bf{i}}}ig{\cal{F}}_{\mu}^{(1)}dx^{\mu}.
\deltaL=\frac{d}{dt}(p_{\mu}\circ\deltaq^{\mu})-\dot{p}^{\mu}\circ\deltaq^{\mu}+\frac{1}{2}\dot{q}^{\mu}(\deltaq^{\alpha}\partial_{\alpha}g_{\mu\nu})\dot{q}^{\nu}-\deltaq^{\mu}\partial_{\mu}U_{q}+\frac{1}{2}[\deltaq^{\mu},\dot{g}_{\mu}]
{\calL}_{\mathrm{Bose}}=g_{i\bar{j}}\left(-\partial_{\mu}\phi^{i}\partial^{\mu}\bar{\phi}^{j}+F^{i}\bar{F}^{j}\right)+\left(F^{i}\frac{\partial{W}}{\partial\phi^{i}}+\bar{F}^{i}\frac{\partial\bar{W}}{\partial\bar{\phi}^{i}}\right)
[\delta_{L_{1}},\delta_{L_{2}}]e{^a}_{\mu}=\beta{^a}_{b}e{^b}_{\mu}-ia\varepsilon^{abcd}\{\bar{\psi}_{b}\gamma_{c}\gamma_{5}\psi_{e}(\partial_{\mu}\beta{_d}^{e})-{\frac{i}{4}}\varepsilon{_c}^{efg}\bar{\psi}_{b}\gamma_{g}\psi_{d}(\partial_{\mu}\beta_{ef})\}
E_{\underline{{\mu}}}^{\\nu}=\left(\begin{matrix}{\begin{matrix}{1,}\\\end{matrix}}&{\begin{matrix}{0,}&{\ldots}&{0}\\\end{matrix}}\\{\begin{matrix}{{e}^{{1}}}\\{\vdots}\\{{e}^{{d}}}\\\end{matrix}}&{\begin{matrix}{{h}^{i}}\\\end{matrix}_{\underline{{a}}}}\\\end{matrix}\right),
\begin{array}{llll}{[E_{1},E_{2}]=E_{3}}&{[F_{1},F_{2}]=-E_{3}}&{[E_{1},F_{2}]=F_{3}}&{[F_{1},E_{2}]=F_{3}}\\{\mathrm{}[E_{2},E_{3}]=E_{1}}&{[F_{2},F_{3}]=-E_{1}}&{[E_{2},F_{3}]=F_{1}}&{[F_{2},E_{3}]=F_{1}}\\{\mathrm{}[E_{3},E_{1}]=-E_{2}}&{[F_{3},F_{1}]=E_{2}}&{[E_{3},F_{1}]=-F_{2}}&{[F_{3},E_{1}]=-F_{2}}\\\end{array}
\Big\langle\psi_{\alpha_{1}}^{(1)}...\psi_{\alpha_{N}}^{(N)}\psi_{\alpha_{1}^{\prime}}^{(1)}...\psi_{\alpha_{n}^{\prime}}^{(n)}(-L,0)U({\calC}^{(L)})^{N+n}\overline{{\psi}}_{\beta_{1}}^{(1)}...\overline{{\psi}}_{\beta_{N}}^{(N)}\overline{{\psi}}_{\beta_{1}^{\prime}}^{(1)}...\overline{{\psi}}_{\beta_{n}^{\prime}}^{(n)}(L,0)\Big\rangle\;.
\oint\mathrm{d}u_{1}\oint\mathrm{d}u_{2}\oint\mathrm{d}u_{3}\frac{(u_{1}-u_{2})^{2}}{(u_{1}-u_{3})^{2}(u_{2}-u_{3})^{4}}\prod_{i=1}^{4}\frac{(z_{i}-u_{1})^{1/2}(z_{i}-u_{2})^{1}}{(z_{i}-u_{3})^{1}}\,.
(K^{(l)}\:V\:K^{(r)})(x,y)\;=\;\frac{1}{2^{l+r}}\left(\frac{\partial}{\partial_{u}}+\frac{\partial}{\partialv}\right)^{l}\left(\frac{\partial}{\partial_{u}}-\frac{\partial}{\partialv}\right)^{r}\frac{\partial}{\partialu}\left.A_{uv}\right|_{u=0=v}\;.
\tilde{g}_{\mu\nu}={\frac{g_{\mu\nu}}{g_{\tau\tau}}}~~~,~~~\tilde{\phi}_{s}=(g_{\tau\tau})^{{\frac{D-2}{4}}}\phi_{s}~~~,~~~\tilde{\psi}_{d}=(g_{\tau\tau})^{{\frac{D-1}{4}}}\psi_{d}~~~.
{\frac{\partial^{2}\psi}{\partialk_{x}^{2}}}\;+\;{\frac{\partial^{2}\psi}{\partial\kappa_{x}^{2}}}\;=\;0\qquad\qquad\left({\frac{\partial\;}{\partialp_{x}}}{\frac{\partial\psi}{\partial\overline{{p_{x}}}}}\;=\;0\right)
\varrho_{*}\Big(\sum_{k=1}^{n}\,f_{k}\,d^{\prime}f_{k}^{\prime}\Big)\:=\:\sum_{k=1}^{n}\,\sum_{\iota\inI}\,\big(r_{U_{\iota}\capU}^{U}(f_{k}^{\prime})\,d^{\prime}r_{U_{\iota}\capU}^{U}(f_{k})\big)\otimes1\,+\,0,
\{\hat{\beta}_{e}^{b},\hat{\beta}_{e^{\prime}}^{c}\}=\delta_{e+e^{\prime}}\,,\quad\{\hat{\beta}_{o}^{b},\hat{\beta}_{o^{\prime}}^{c}\}=\delta_{o+o^{\prime}}\,,\quad\{\bar{\beta}_{o}^{b},\bar{\beta}_{o^{\prime}}^{c}\}=\delta_{o+o^{\prime}}\,.
\varepsilon_{0}^{\beta\rightarrow0}=-\frac{\verte\vertH}{(2\pi)^{2}}\sum_{n\sigma}\int_{-\infty}^{\infty}dp\;\epsilon_{n\sigma}\;\left\{\frac{2\beta(\alpha_{n\sigma}+p^{2})}{4+\beta^{2}(\alpha_{n\sigma}+p^{2})}\right\}
M_{1}(u)=\frac{1}{2\sqrt{\pi}g}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\frac{\Gamma(\frac{1}{2}+n(1-1/g))}{\Gamma(1-n/g)}u^{-2n(1-1/g)}+{\calO}(x^{n}).
G(x)=e^{-ix}x^{\frac{\tilde{\beta}+1}{2}}\left(P\,\,F\left[1+\frac{1}{2}(1+i)\tilde{\beta}),2+\tilde{\beta};2ix\right]+Q\,\,U\left[1+\frac{1}{2}(1+i)\tilde{\beta},2+\tilde{\beta};2ix\right]\right),
\hat{T}_{4}=\frac{i}{2}\hat{F}_{ij}\ast(\hat{F}_{jk}\ast\hat{F}_{ki}-\hat{F}_{ki}\ast\hat{F}_{jk})=\frac{1}{2}\hat{F}_{ij}\ast[\hat{D}_{k},\hat{D}_{i}]\hat{F}_{jk},
\begin{array}{l}{\epsilon(k_{i})=1~,~~~~\epsilon(X_{i}^{-})=0=\epsilon(X_{i}^{+})~,}\\{S(k_{i})=k_{i}^{-1}~,~~~S(X_{i}^{-})=-X_{i}^{-}k_{i}^{-1}~,~~~S(X_{i}^{+})=-k_{i}X_{i}^{+}~.}\\\end{array}
(g)^{\mu\nu}=\left(\begin{array}{ccc}{0}&{e^{-\psi/2}}&{0}\\{e^{-\psi/2}}&{(f^{2}x_{I}^{2}-h_{++})e^{-\psi}+g_{+I}\gamma^{IJ}g_{J+}}&{-e^{-\frac{3}{2}\psi}g_{+I}\gamma^{IJ}}\\{0}&{-e^{-\frac{3}{2}\psi}g_{+J}\gamma^{JI}}&{e^{-\psi}\gamma^{IJ}}\\\end{array}\right)
\sum_{m=0}^{\infty}\sum_{j=1}^{23}{\frac{(\Gamma({\frac{j}{24}}+m))^{3}\pi\Gamma({\frac{1}{2}}+{\frac{j}{24}}+m)(-)^{m}\prod_{j\neqk=1}^{23}\Gamma({\frac{k-j}{24}}-m)\prod_{n=3}^{10}\Gamma({\frac{n}{8}}+{\frac{j}{24}}+m)\Gamma({\frac{11}{12}}+{\frac{j}{24}}+m)}{m!(m+{\frac{j}{24}})\prod_{l=10}^{1}\Gamma({\frac{l}{12}}-m-{\frac{j}{24}})}}z^{-m-{\frac{j}{24}}}.
F_{C}=-T\int_{0}^{T}(E_{C1}(t)-E_{C1}(0)){\frac{dt}{t^{2}}}-T\int_{0}^{T}(E_{C2}(t)-E_{C2}(0)){\frac{dt}{t^{2}}}+\frac{5\,\xi^{2}}{32a\pi}
\gamma^{1}=\left(\begin{array}{cc}{0\,\,}&{\,\,{\mathbb1}}\\{{\mathbb1}\,\,}&{\,\,0}\\\end{array}\right)\quad\gamma^{2}=\left(\begin{array}{cc}{0}&{-i\sigma_{x}}\\{i\sigma_{x}}&{0}\\\end{array}\right)\quad\gamma^{3}=\left(\begin{array}{cc}{0}&{-i\sigma_{y}}\\{i\sigma_{y}}&{0}\\\end{array}\right)\quad\gamma^{4}=\left(\begin{array}{cc}{0}&{-i\sigma_{z}}\\{i\sigma_{z}}&{0}\\\end{array}\right)\quad
B=D_{1}\,\frac{\Gamma(1-2\mu)\Gamma(2\nu)}{\Gamma(-\mu+\nu-\lambda)\Gamma(1-\mu+\nu+\lambda)}e^{-i\pi\mu}+D_{2}\,\frac{\Gamma(1+2\mu)\Gamma(2\nu)}{\Gamma(\mu+\nu-\lambda)\Gamma(1+\mu+\nu+\lambda)}e^{i\pi\mu}
\tilde{Z}_{3}=1-\frac{\tilde{e}^{2}}{8\pi}\int_{2m}^{\infty}da\,\frac{(1/a^{2})(a^{2}-2m\tilde{\mu})^{2}+(2m-\tilde{\mu})^{2}}{(a^{2}-\tilde{\mu}^{2})^{2}}+O(\tilde{e}^{4}).
\begin{array}{ll}{sZ}&{=tr_{(1)}(Z_{1}D_{1})Z_{2}=tr_{(1)}(Z_{1}(VZ_{2}D_{1}V^{-1}+{\calP}))}\\{\,}&{=tr_{(1)}(Z_{2}VZ_{1}D_{1}V^{-1}+Z_{1}{\calP})=Z_{2}tr_{(1)}(VZ_{1}D_{1}V^{-1})+Z_{2}=Z(s+1)}\\\end{array}
H^{n}({\calH})=\Biggl(\bigoplus_{\stackrel{\scriptstylen=N^{c}-N^{b}}{N^{c},N^{b}\geq0}}H^{-N^{b}}\left({\calF}(\alpha_{-m}^{0},b_{-m};k^{0})\right)\otimesH^{N^{c}}\left({\calF}(c_{-m}^{})\right)\Biggr)\otimes{\calH}_{K}.
\int\frac{d^{2}\rho}{\pi}\e^{-|\rho|^{2}}e^{\alpha^{\prime}k_{\mu}^{2}/4n}(\rho|e^{\sqrt{\alpha^{\prime}/2}k_{\mu}\alpha_{-n}^{\mu}z^{n}/n}e^{-\sqrt{\alpha^{\prime}/2}k_{\mu}\alpha_{n}^{\mu}z^{-n}/n}|q^{n}\rho)
\delta_{\perp}(\deltaS)=\epsilon_{\mu}\left\{\int_{m}\sqrt{-\gamma}\left[K^{j}\Phi_{j}{\calE}^{i}n^{\mu}{}_{i}+(\tilde{\delta}_{\perp}{\calE}^{i})n^{\mu}{}_{i}-{\calE}^{i}(\widetilde\nabla_{a}\Phi_{i})e^{\mu\,a}\right]+\int_{m}\nabla_{a}\delta_{\perp}{\calP}^{a\mu}\right\}\,.
Q_{f}=f_{\mu_{1}\dots\mu_{r}}\Pi^{\mu_{1}}\psi^{\mu_{2}}\dots\psi^{\mu_{r}}+\frac{i}{r+1}(-1)^{r+1}f_{[\mu_{1}\dots\mu_{r};\mu_{r+1}]}\psi^{\mu_{1}}\dots\psi^{\mu_{r+1}}.
\begin{array}{lll}{}&{\Psi(z,\bar{z})=S_{-1/2}\phi(z,\bar{z});}&{}\\{}&{\bar{\Psi}(z,\bar{z})=\bar{S}_{-1/2}\phi(z,\bar{z});}&{}\\{}&{F(z,\bar{z})=S_{-1/2}\bar{S}_{-1/2}\phi(z,\bar{z})}&{}\\\end{array}
{\frac{d^{2}y}{d\tilde{\lambda}^{2}}}-{\frac{1}{2}}\left[1-\left({\frac{dy}{d\tilde{\lambda}}}\right)^{2}\right]\partial_{y}\left({\calW}g_{\mu\nu}\right){\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}=0.
\langle\Phi_{f}|H_{I}^{\prime}|\Phi_{i}\rangle=-{\frac{\hbar^{2}}{4m}}\int_{-\infty}^{\infty}dx\,\partial_{x}\theta(x)\left[-\varphi_{2}^{\prime}(x)\varphi_{1}(x)+\varphi_{2}(x)\varphi_{1}^{\prime}(x)\right].
|Dp,f,x^{i},x^{a},\eta\rangle_{U}={\calN}_{1,f}\int\Bigl(\prod_{i}dk_{i}e^{ik_{j}x^{j}}\Bigr)\Bigl(\prod_{a}\sum_{m_{a}}e^{im_{a}x^{a}/R^{a}}\Bigr)|Dp,f,k,m\rangle_{X,U}|Dp,f,\eta\rangle_{\psi,U}
\underline{{A}}^{\mu}=\frac1\ell\underline{{\Theta}}^{\mu}+{\frac{1}{2}}\epsilon^{\mu}\_{\nu\rho}\underline{{\Omega}}^{\nu\rho}\qquad,\qquad\underline{{\tilde{A}}}^{\mu}=-\frac1\ell\underline{{\Theta}}^{\mu}+{\frac{1}{2}}\epsilon^{\mu}\_{\nu\rho}\underline{{\Omega}}^{\nu\rho}\qquad.
\mathcal{L}_{LC}=-\frac{h^{\tau\tau}}{4\pi\alpha^{\prime}}\left(2G_{+-}\dot{Z}^{-}+G_{ij}\dot{Z}^{i}\dot{Z}^{j}-\frac{1}{(h^{\tau\tau})^{2}}G_{ij}Z^{i\,\prime}Z^{j\,\prime}\right)
h_{1}=0,h_{3}=h_{\overline{{3}}}=\frac{2}{9},h_{8}=\frac{1}{2},h_{6}=h_{\overline{{6}}}=\frac{5}{9},h_{10}=h_{\overline{{10}}}=1,h_{15}=h_{\overline{{15}}}=\frac{8}{9}\\.
d_{q_{1}q_{2}q_{3}q_{4}}=\delta(\sum_{i}q_{i})t^{-2+(|q_{1}|+|q_{2}|+|q_{3}|+|q_{4}|)/2}\left(1+{\frac{\vertq_{1}+q_{2}\vert}{2}}-{\frac{\vertq_{1}+q_{3}\vert}{2}}-{\frac{\vertq_{1}+q_{4}\vert}{2}}\right)\.
\deltaT_{0}^{0}=\delta\varepsilon=\varepsilon_{,X}\deltaX+\varepsilon_{,\varphi}\delta\varphi=\frac{\varepsilon+p}{c_{S}^{2}}\left[\left(\frac{\delta\varphi}{\dot{\varphi}}\right)^{.}-\Phi\right]-3H\left(\varepsilon+p\right)\frac{\delta\varphi}{\dot{\varphi}}
\begin{array}{l}{\varepsilon=-P^{j}=t_{ren}^{00}\,,\quadj=\overline{{3,d}}\,,}\\{P^{1}=\frac{\left(x^{1}\right)^{2}}{\left(x^{1}\right)^{2}+\left(x^{2}\right)^{2}}t_{ren}^{rr}+\left(x^{2}\right)^{2}t_{ren}^{\varphi\varphi}\,,}\\{P^{2}=\frac{\left(x^{2}\right)^{2}}{\left(x^{1}\right)^{2}+\left(x^{2}\right)^{2}}t_{ren}^{rr}+\left(x^{1}\right)^{2}t_{ren}^{\varphi\varphi}\,.}\\\end{array}
D_{\nu_{1}}\left[\mathrm{e}^{\mathrm{i}r\theta}\{\hat{f}}_{m}^{\alpha^{\prime}}(\theta)\right]={\frac{d}{d\theta}}\left[P_{1-\nu_{1}}\left(\mathrm{e}^{\mathrm{i}(r+m+\alpha^{\prime})\;{\theta}}{\bfP}_{\alpha^{\prime}}(f)\right)\right](\theta).\tag{B.40}
\chi_{1}^{D}=\sqrt{2}\left\lbrack\begin{array}{c}{1}\\{0}\\{0}\\{0}\\\end{array}\right\rbrack,\;\;\;\;\chi_{2}^{D}=\sqrt{2}\left\lbrack\begin{array}{c}{0}\\{1}\\{0}\\{0}\\\end{array}\right\rbrack.
p^{i}L_{i}\,(\equivp^{\alpha}K^{\alpha}{}_{\alpha}+\Phi^{u}R_{u})=-\frac{1}{2}\varphi^{\hat{\imath}}H_{\hat{\imath}}\,(\equiv-\frac{1}{2}\phi^{\hat{a}}H_{\hat{a}}-\frac{1}{2}\Phi^{u}R_{\hat{u}})\,,
e^{-\frac{1}{2}\betaK^{ij}u_{i}u_{j}}<x|e^{\varphi^{i}v_{i}}|y>=\sum_{n=0}^{\infty}(-\frac{1}{2}\betaK^{ij}u_{i}u_{j})^{n}/n!\,<x|e^{\varphi^{i}v_{i}}|y>.
\deltaA^{k_{s},\alpha_{s}}=d\varphi^{k_{s},\alpha_{s}}-C_{i_{p},\beta_{p}\;j_{q},\gamma_{q}}^{k_{s},\alpha_{s}}\;\varphi^{i_{p},\beta_{p}}A^{j_{q},\gamma_{q}}\;,\;\deltaF^{k_{s},\alpha_{s}}=C_{i_{p},\beta_{p}\;j_{q},\gamma_{q}}^{k_{s},\alpha_{s}}\;F^{i_{p},\beta_{p}}\varphi^{j_{q},\gamma_{q}}\;,
{\calL}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}(A_{\mu}+\partial_{\mu}\theta)^{2}+A^{\mu}\partial_{\mu}B+d_{2}\partial_{\mu}\bar{\calC}\partial^{\mu}{\calC}-\frac{1}{2}\alphaB^{2}.
{\calH}^{(1)}=\sum_{\sigma}\int\frac{d^{3}p}{(2\pi)^{3}}E_{p}\,\left[a_{\sigma}^{\dagger}(\vec{p})a_{\sigma}(\vec{p})+b_{\sigma}(\vec{p})b_{\sigma}^{\dagger}(\vec{p})+c_{\sigma}^{\dagger}(\vec{p})c_{\sigma}(\vec{p})+d_{\sigma}(\vec{p})d_{\sigma}^{\dagger}(\vec{p})\right]\,,
X(\beta_{i},\bar{\beta}_{i}|\beta_{i+1},\bar{\beta}_{i+1}|\alpha_{i},\alpha_{i}^{\prime})=\sum_{\gamma_{i}}w(\beta_{i},\alpha_{i},\beta_{i+1},\gamma_{i}|u)w(\bar{\beta}_{i},\gamma_{i},\bar{\beta}_{i+1},\alpha_{i}^{\prime}|u+v)~.
\Gamma_{\underline{\alpha}\underline{\beta}}^{i}=\left(\begin{array}{cc}{0}&{\delta_{\beta}^{\alpha}(\gamma^{i})_{q}^{~r}}\\{-\delta_{\alpha}^{\beta}(\gamma^{i})_{~r}^{q}}&{0}\\\end{array}\right)\,,\quadi=1,...,5\,,
{\calW}\,(\varphi_{1},\varphi_{2})=\frac{m^{2}}{4\lambda}\,\varphi_{1}-\frac{\lambda}{3}\,\varphi_{1}^{3}+q\,\lambda\,\varphi_{1}\phi_{2}^{2}+\frac{p\,m}{2}\,\varphi_{2}^{2}\,,
\widehat{d}_{-}(p)=\frac{1}{(4\pi)^{3}}\theta(p^{2}-m^{2})\theta\left(-p^{0}\right)\left(1-\frac{m^{2}}{p^{2}}\right)\left[\frac{m}{2}\beta^{\mu}\beta_{\mu}+p\!\!\!/\left(1+\frac{m^{2}}{p^{2}}\right)\right]\mathrm{~.}
I_{N}(p)\rightarrow-(2\pi)^{\ell}q^{1-\ell}\int_{0}^{\infty}\!dr\,\,\,r^{\ell}J_{\ell-1}(qr)\left[\frac{\Gamma(\ell-1)}{4\pi^{\ell}r^{2\ell-2}}\right]^{N};\qquadq^{2}\equiv-p^{2},
(r^{2}-r_{H}^{2})\partial_{r}^{2}R(r)+2r\partial_{r}R(r)+\frac{1}{(r^{2}-r_{H}^{2})}[\omega^{2}\ell^{4}+m^{2}\ell^{2}(r^{2}-r_{H}^{2})]R(r)=0.
S=\frac{\hat{j}l_{c}t\phi_{0}^{3/2}}{2\pil_{s}^{2}}\int_{1}^{\sqrt{\Lambda/\phi_{0}}}\sqrt{\frac{\chi^{2}(\chi^{2}+\hat{j})(2\chi^{2}+\hat{j})^{2}}{(\chi^{2}-1)(\chi^{2}+\hat{p}^{2})}}d\chi
\Gamma_{ij}^{k}\Gamma_{ij}^{k}-\Gamma_{ij}^{k}\Gamma_{ik}^{j}-\Gamma_{ij}^{j}\Gamma_{ik}^{k}+2\Gamma_{ii}^{j}\Gamma_{jk}^{k}-\Gamma_{ii}^{k}\Gamma_{jj}^{k}=0
ds^{2}=e^{\sigma}(d\rho^{2}+\rho^{2}d\phi^{2}+\sum_{i,j=1}^{d-2}(\gamma_{ij}(\theta)+h_{ij}(\theta)\rho^{2})d\theta^{i}d\theta^{j}+...)\equive^{\sigma}d\tilde{s}^{2}
J({\hat{k}};M_{1},M_{2})\;=\;\int\frac{d^{2}{\hat{p}}}{(2\pi)^{2}}\,\frac{-i\not\!{\hat{p}}}{[({\hat{k}}-{\hat{p}})^{2}+M_{1}^{2}]({\hat{p}}^{2}+M_{2}^{2})}\;.
L_{1/4}=\frac{1}{2}h_{ij}(y)\dot{y}^{i}\dot{y}^{j}+\frac{1}{2}L_{\alpha\beta}(y)(\dot{\chi}^{\alpha}+w_{i}^{\alpha}(y)\dot{y}^{i})(\dot{\chi}^{\beta}+w_{j}^{\beta}(y)\dot{y}^{j})-\frac{1}{2}L_{\alpha\beta}(y)a^{\alpha}a^{\beta}.
\hat{\phi}_{n}^{(i)\mu}=\sqrt{\frac{\alpha^{\prime}}{2n}}i\left(\hat{a}_{n}^{(i)\mu}-\hat{a}_{n}^{\dagger(i)\mu}\right)~,\quad\hat{\pi}_{n\mu}^{(i)}=\sqrt{\frac{n}{2\alpha^{\prime}}}g_{\mu\nu}\left(\hat{a}_{n}^{(i)\nu}+\hat{a}_{n}^{\dagger(i)\nu}\right)~.
\left[J_{a}^{\rho}(x),J_{b}^{\sigma}(y)\right]=if_{abc}J_{c}^{\sigma}(x)\gamma^{0}\gamma^{\rho}\delta(x-y)+i\frac{\delta_{ab}}{4\pi}\fracm{\left|m\right|}\epsilon^{\rho\sigma\mu}\partial_{\mu}\delta^{(3)}(x-y).
\epsilon^{\mu\nu\rho}(\partial_{\nu}\stackrel{(0)}{B_{\rho}^{a}}-\partial_{\rho}\stackrel{(0)}{B_{\nu}^{a}})=\sum_{i=1}^{n}\ointdz_{(i)}^{\mu}{\delta}^{3}(x-z_{(i)})I_{(i)}^{a}(0)
V_{en}^{PQ}|0>=(-)^{a+1}\sum_{j=0}^{a-1}(-1)^{j}\left(\begin{array}{c}{j}\\{a-1}\\\end{array}\right)\sum_{l=0}^{\frac{P}{a}-1}\mathrm{exp}\left[\frac{2\pii}{k}Qe(Pn-l-\frac{P}{a}j-1/2)\right]|Pe,Pn-l-\frac{P}{a}j>
X^{(2)}=\nu^{2}\sum_{i,l,n}[(a_{i}^{\dagger}-a_{l}^{\dagger})^{-1}(\xi_{1i}-\xi_{1l})K_{il},(a_{i}^{\dagger}-a_{n}^{\dagger})^{-1}(\xi_{2i}-\xi_{2n})K_{in}]\,.
\vec{W}_{1,2}=\frac{2}{\pi^{2}}\mathrm{Re}\int\frac{x-y_{1}}{|x-y_{1}|^{4}}q_{1}^{+}i\vec{\tau}q_{2}\frac{x^{+}-y_{2}^{+}}{|x-y_{2}|^{4}}d^{4}x~.
({}^{(3)}\bigtriangleupA)_{r}={\frac{\partial^{2}A_{r}}{\partialr^{2}}}+{\frac{2}{r}}{\frac{\partialA_{r}}{\partialr}}+{\frac{1}{r^{2}}}({}^{(2)}\bigtriangleupA)_{r}-{\frac{2}{r^{2}}}A_{r}-{\frac{2}{r^{3}}}A_{p}^{\;\;\midp},
S_{\mathrm{W}_{B}}^{\mathrm{BI}}=-T_{0}\intd\tau~\mathrm{Str}\Bigl\{k^{-1}\sqrt{-P[E_{00}+E_{0i}(Q^{-1}-\delta)_{k}^{i}E^{kj}E_{j0}]\\mathrm{det}(Q_{j}^{i})}\\Bigr\},
\{{\calJ}_{\mu},x_{\nu}\}^{*}=-\epsilon_{\mu\nu\lambda}x^{\lambda}+\gamma_{i}e_{\nu}^{(i)}\left(-\frac{e_{\mu}^{(0)}}{\sqrt{-p^{2}}}+e_{\mu}^{(j)}\partial^{\sigma}e_{\sigma}^{(j)}\right),
\begin{array}{cc}{Q^{i}=\left(\begin{array}{c}{{\calQ}_{+}^{a}}\\{(\bar{{\calQ}}_{a}^{+})^{t}}\\\end{array}\right)~~~~~}&{~~~~~\tilde{Q}^{i}=\left(\begin{array}{c}{{\calQ}_{-}^{a}}\\{(\bar{{\calQ}}_{a}^{-})^{t}}\\\end{array}\right)}\\\end{array}
\begin{array}{ccccccccc}{}&{}&{}&{}&{\frac{1}{l_{M}^{3}}}&{}&{}&{}&{}\\{}&{}&{}&{}&{\mid}&{}&{}&{}&{}\\{\frac{R_{1}}{l_{M}^{3}}}&{-}&{\frac{R_{1}R_{2}}{l_{M}^{6}}}&{-}&{\frac{R_{1}R_{2}R_{3}}{l_{M}^{9}}}&{-}&{\frac{R_{1}R_{2}R_{3}R_{4}}{l_{M}^{9}}}&{-\dots-}&{\frac{1}{R_{d+1}}}\\\end{array}
(2\piS\,T)^{2}=r_{0}^{2}={\frac{det^{1/2}\{Q_{\alpha}^{c},Q_{\gamma}^{b}\}}{[{\frac{1}{4N}}\Omega_{ac}\epsilon^{\beta\gamma}\{Q_{\alpha}^{c},Q_{\gamma}^{b}\}]^{N-2}}}\.
\left.+\left(\frac{1}{2}\mathrm{tr}p_{r}^{2}-\frac{1}{4}p_{12}p_{21}+\frac{85}{72}\right)R^{2}+\left(\frac{9}{2}\Lambda+\mathrm{tr}p_{r}p_{l}\right)R+\left(\frac{9}{2}\lambda^{2}+\frac{1}{2}\mathrm{tr}p_{l}^{2}\right)\right]\;.
C_{abcd}=C_{abcd}^{(+)}+C_{abcd}^{(-)}\Leftrightarrow\Psi_{ABCD\mathrm{~}}\epsilon_{\stackrel{.}{A}\stackrel{.}{B}}\epsilon_{\stackrel{.}{C}\stackrel{.}{D}}+\overline{{\Psi}}_{\stackrel{.}{A}\stackrel{.}{B}\stackrel{.}{C}\stackrel{.}{D}\mathrm{~}}\epsilon_{AB}\epsilon_{CD}\,.