harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R_{A}^{B}=\kappa\tau_{A}^{B}\,-\,\epsilon\,\biggl[\delta_{A}^{B}\frac{1}{d}\,{\calR}_{\mathrm{EGB}}^{2}\,-2\,R\,R_{A}^{B}+4\,R_{A\,C}\,R^{C\,B}+4\,R_{C\,D}R_{A}^{~~C\,B\,D}-2\,R_{A\,C\,D\,E}\,R^{B\,C\,D\,E}\biggr],
I=-\frac{1}{4{\pi}^{2}g}\intd^{3}\xi\bigg\{\sqrt{-det\bigg(g_{\alpha\beta}^{ind}+2\pi{\calF}_{\alpha\beta}\bigg)}+\frac{1}{3!}\epsilon^{\alpha\beta\gamma}A_{\mu\nu\rho}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\partial_{\gamma}X^{\rho}\bigg\}.
\intd^{10}\hat{x}\sqrt{|\hat{\jmath}|}\,\left[{\textstyle\frac{1}{2\cdot5!}}\left(\hat{G}_{(5)}^{+}\right)^{2}-{\textstyle\frac{1}{4\cdot5!}}\frac{\hat{\epsilon}}{\sqrt{|\hat{\jmath}|}}\hat{G}_{(5)}^{+}\partial\hat{C}_{(2)}^{-}\mathcal{B}\right]\,,
\begin{array}{rcl}{N=\sum_{a=1}^{3}{\alpha^{a}}^{\dagger}\alpha^{a}\\\}&{,}&{\\\N^{\dagger}=N\\\,}\\{B^{\dagger}=\sum_{a=1}^{3}{\alpha^{a}}^{\dagger}{\alpha^{a}}^{\dagger}\\\}&{,}&{\\\B=\sum_{a=1}^{3}\alpha^{a}\alpha^{a}\\\,}\\\end{array}
E\;=\;\sum_{a=1}^{N}(n_{a}\!-\!m_{a})(n_{a}\!-\!m_{a})-\frac{1}{N}\sum_{a,b=1}^{N}(n_{a}\!-\!m_{a})(n_{b}\!-\!m_{b})=\frac{1}{N}\sum_{a,b=1}^{N}(n_{a}\!-\!m_{a})R_{ab}(n_{b}\!-\!m_{b})\;.
\tilde{K}_{\alpha\dot{\alpha}}^{q}=\frac{1}{3}\sum_{f}\tilde{\psi}_{\dot{\alpha}}^{f\dagger}\tilde{\psi}_{\alpha}^{f}+\frac{1}{3}\sum_{f}\tilde{\bar{\psi}}_{\dot{\alpha}}^{f\dagger}\tilde{\bar{\psi}}_{\alpha}^{f}\;\;,\;\;\;\;\;\;\tilde{K}_{\alpha\dot{\alpha}}^{M}=\frac{1}{3}\mathrm{Tr}\,(\chi_{\dot{\alpha}}^{\dagger}\chi_{\alpha})
\widetilde{M}_{\lambda}(j_{q},\vec{k},\lambda;j_{p})=-i\,\langlej_{q},\vec{k},\lambda|S^{(1)}|j_{p}\rangle=-e\int{d^{4}x}\bar{\psi}_{e}(j_{q},x)\;{A^{\ast}}_{\mu}^{\lambda}(\vec{k},x)\gamma^{\mu}\;\psi_{e}(j_{p},x).
Z_{99}^{(F)}=\eta^{-4}(t)\!\!\!\!\!\!\sum_{a,b=0,1/2}\!\!\!\!\!\!\eta_{ab}\;\vartheta\left[\begin{array}{c}{a}\\{b}\\\end{array}\right](t)\prod_{r=3}^{5}\vartheta\left[\begin{array}{c}{a}\\{b+kv_{r}}\\\end{array}\right](t),
N=-\frac{8\pi}{m_{\mathrm{Pl}}^{2}}\int_{\varphi_{\mathrm{i}}}^{\varphi_{\mathrm{e}}}\mathrm{d}\varphiV(\varphi)\biggl(\frac{\mathrm{d}V}{\mathrm{d}\varphi}\biggr)^{-1}\,,
K_{r}(u)=\left(\begin{array}{ll}{2y_{0}(qtu-q^{-1}t^{-1}u^{-1})\;}&{y_{+}(qu^{2}-q^{-1}u^{-2})}\\{y_{-}(qu^{2}-q^{-1}u^{-2})\;}&{-2y_{0}(t^{-1}u-tu^{-1})}\\\end{array}\right)\,,
e^{-\beta\lambda_{j}\cdot\phi}=\frac{1+\cdots+b_{i}^{m_{j}}e^{-2\pii\lambda_{i}^{\mathrm{v}}\cdot\lambda_{j}}\left(A_{i}\right)^{m_{j}p_{i}}}{\left[1+\cdots+b_{i}\left(A_{i}\right)^{p_{i}}\right]^{m_{j}}}\,,
\left(\begin{array}{c}{\tilde{x}^{1}}\\{\tilde{x}^{0}}\\\end{array}\right)=\frac{1}{\sqrt{1-\beta^{2}}}\left(\begin{array}{cc}{1}&{-\beta}\\{-\beta}&{1}\\\end{array}\right)\left(\begin{array}{c}{x^{1}}\\{x^{0}}\\\end{array}\right).
L=\mathrm{Tr}\left(\frac{1}{2}\nabla_{t}X^{i}\nabla_{t}X_{i}-\frac{1}{4}[X_{i},X_{j}]^{2}+\frac{1}{2}\psi\sigma^{0}\nabla_{t}\psi+\frac{1}{2}\psi^{\dagger}\sigma^{i}[\psi,X_{i}]\right)
f_{1}\ast_{\hbar}f_{2}\equiv\mathrm{e}^{\hbar\frac{\partial}{\partial\bar{z}}\frac{\partial}{\partialz^{\prime}}}f_{1}(z,\bar{z})f_{2}(z^{\prime},\bar{z}^{\prime})|_{z=z^{\prime},\bar{z}=\bar{z}^{\prime}}.
S=\fraci{4\pi\alpha^{\prime}}\int\left\{g_{i\bar{\jmath}}(\partialx^{i}\bar{\partial}x^{\bar{\jmath}}+\bar{\partial}x^{i}\partialx^{\bar{\jmath}})-iB_{i\bar{\jmath}}(\partialx^{i}\bar{\partial}x^{\bar{\jmath}}-\bar{\partial}x^{i}\partialx^{\bar{\jmath}})\right\}\,d^{2}z.
0=t-\tau^{*}-R/c=\frac1cu_{t}\tau^{*}+\frac1cs_{t}-\tau^{*}-\frac1c|\vec{u}\tau^{*}+\vec{s}|=\frac1cs_{t}-\frac1c|\vec{u}\tau^{*}+\vec{s}|+o(v/c)\,
{\frac{v_{s}^{2}}{T}}(\eta^{R})={\frac{\left[f(\eta^{R})-\eta^{R}f^{\prime}(\eta^{R})\right]^{2}}{3\left[f(\eta^{R})-\eta^{R}f^{\prime}(\eta^{R})-\frac12\right]}}+f(\eta^{R})+\frac13\eta^{R}f^{\prime}(\eta^{R})\;.
b_{\mu}^{a}=\left(\pi^{(1)a},b_{i}^{a}\right),\;\bar{\eta}_{\mu}^{a}=\left({\calP}^{(1)a},-\bar{\eta}_{i}^{a}\right),\;\eta_{a}^{\mu}=\left(\eta^{(2)a},\eta_{i}^{(2)a}\right).
\begin{array}{rcl}{\widehat{xy}}&{=}&{S^{-1}\left((xy)^{(2)^{\prime}}\right)(xy)^{(1)}}\\{}&{=}&{S^{-1}y^{(2)^{\prime}}S^{-1}x^{(2)^{\prime}}x^{(1)}y^{(1)}}\\{}&{=}&{S^{-1}y^{(2)^{\prime}}\underbrace{\hat{x}y^{(1)}}_{\mathrm{commute}}}\\{}&{=}&{S^{-1}y^{(2)^{\prime}}y^{(1)}\hat{x}}\\{}&{=}&{\hat{y}\hat{x}.}\\\end{array}
L^{+}=\left(\begin{array}{ll}{q^{S^{3}}}&{\OmegaS^{-}}\\{0}&{q^{-S^{3}}}\\\end{array}\right)\\\,\\\L^{-}=\left(\begin{array}{rr}{q^{-S^{3}}}&{0}\\{-\OmegaS^{+}}&{q^{S^{3}}}\\\end{array}\right)
\begin{aligned}{}&{{}F_{0,1,0}(d_{1},0;d_{1},0)={\frac{1}{d_{1}^{2}}}}\\{}&{{}F_{0,0,1}(0,d_{2};0,d_{2})={\frac{1}{d_{2}^{2}}}}\\{}&{{}F_{0,2,0}(d_{1},0;m_{1},m_{2},0,0)=F_{0,0,2}(0,d_{2};0,0,n_{1},n_{2})=0.}\\\end{aligned}
A(x_{1},x_{2})\starB(x_{1},x_{2})\,=\,A(x_{1},x_{2})\,e^{i\frac{\theta}{2}({\overleftarrow\partial_{x_{1}}}{\overrightarrow\partial_{x_{2}}}-{\overleftarrow\partial_{x_{2}}}{\overrightarrow\partial_{x_{1}}})}B(x_{1},x_{2})\;.
\sum_{m=0}^{\infty}(m^{2}-m)\left({\frac{1}{2x}}{\frac{d}{dx}}\right)^{3}\sum_{i=1}^{\infty}{\widetildeS}_{i}(m,\alpha_{m}(x))\sim\sum_{i=1}^{\infty}W_{\infty}^{i},
\begin{array}{lcl}{U^{(J)}}&{=}&{p\tau}\\{}&{}&{}\\{X^{(J)}}&{=}&{{\displaystyle\sum_{n=-\infty}^{\infty}X_{n}^{(J)}(\tau)\;e^{i\frac{2\pi}{\lambda_{J}}n\sigma}}}\\{}&{}&{}\\{Y^{(J)}}&{=}&{0}\\{}&{}&{}\\{X^{j(J)}}&{=}&{0~~,~~j=4,\cdots,D-1}\\\end{array}
\lbrack(\gamma_{1}^{0}\otimes\mathrm{I}+\mathrm{I}\otimes\gamma_{2}^{0})\pi_{0}-(\overrightarrow{\alpha}_{1}\otimes\gamma_{2}^{0}+\gamma_{2}^{0}\otimes\overrightarrow{\alpha}_{2})\cdot\overrightarrow{\pi}
\left<{\chi}_{\mu}^{I}(x){\chi}_{\nu}^{J}(y)\right>\equiv\hbarG_{\mu\nu}^{(\chi)}(x,y)=\frac{1}{2}\hbar{\varepsilon}^{IJ}h^{-1/2}h_{\mu\lambda}{\varepsilon}^{\lambda\kappa\rho}{\partial}_{\rho}G_{\kappa\nu}^{\prime(1)}(x,y),
i\psi^{(q)\dagger}(x)\vec{\alpha}\cdot[\vec{\partial}+{\vec{G}}_{A}(x)T_{s}^{A}]\psi^{(q)}(x){\rightarrow}_{\eta^{(G)}\rightarrow0}\,\,i{\check{\psi}}^{(q)\dagger}(x)\vec{\alpha}\cdot[\vec{\partial}+{\check{\vec{G}}}_{A\perp}(x)T_{s}^{A}]{\check{\psi}}^{(q)}(x),
\tilde{S^{a}}_{L}=[\tilde{S^{a}}_{0}cos\omega_{0}\tau-\bar{\tilde{S^{a}}_{0}}sin\omega_{0}\tau]+\sum_{n\neq0}{\frac{c_{n}}{\sqrt2}}[\tilde{\phi}{_n}\tilde{S^{a}}{_n}+{\frac{i}{m}}(\omega_{n}-n)\Lambda_{b}^{a}{S^{b}}_{n}\phi_{n}],
\int{\calD}A_{\mu}^{\omega}\delta(D_{\mu}(\frac{\deltaf(A_{\nu}^{\omega})}{\deltaA_{\mu}^{\omega}}))det\{\frac{\delta}{\delta\omega}[D_{\mu}(\frac{\deltaf(A_{\nu}^{\omega})}{\deltaA_{\mu}^{\omega}})]\}e^{-S_{0}(A_{\mu}^{\omega})}=0.
\begin{aligned}{P_{i}}&{{}=AF_{i}-2AA^{*}F_{iA^{*}}-\frac{g_{i}}{6mM_{P}},}\\{X}&{{}=\left(\begin{matrix}{F_{AA^{*}}}&{F_{Aj^{*}}}\\{F_{iA^{*}}}&{F_{ij^{*}}}\\\end{matrix}\right).}\\\end{aligned}
F_{ij}^{a}=\partial_{j}A_{i}^{a}-\partial_{i}A_{j}^{a}-g{\epsilon}^{abc}A_{i}^{b}A_{j}^{c}\,,\;\;\,F_{i0}^{a}=\partial_{0}A_{i}^{a}+\partial_{i}A_{0}^{a}+g{\epsilon}^{abc}A_{i}^{b}A_{0}^{c}\,,
S_{C}^{\prime}=\sum_{C}\sum_{x_{0}(\inC)}\sum_{x(\inC)}\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}\phi^{ab}[C;x_{0}]B_{\mu\nu}^{a}[C;x,x_{0}]B_{\lambda\rho}^{b}[C;x,x_{0}].
\delta\left(Z_{\;\;\beta_{2}}^{\alpha_{1}}\eta_{\alpha_{1}}^{*}+\frac12C_{\beta_{2}}^{\alpha_{0}\beta_{0}}\Phi_{\alpha_{0}}^{*}\Phi_{\beta_{0}}^{*}\right)=D_{\;\;\beta_{2}}^{\alpha_{2}}\Phi_{\alpha_{2}}^{*},
U\left({\bfy^{\prime}},\bfy\right)\left|E\right\rangle\equiv\overline{\psi}\left({\bfy}\right)\left({1+ig\int_{\bfy^{\prime}}^{\bfy}{dz^{i}}A_{i}^{c}\left(z\right)T^{c}}\right)\psi\left({\bfy^{\prime}}\right)\left|E\right\rangle.
{\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})=\prod_{m=1}^{\eta}\overline{{{\calY}^{\alpha[m]}}}({\bf{r}})\;\;\;\mathrm{with}\;\;\;\overline{{{\calY}^{\alpha}}}({\bfr})={\textstyle\frac{\partial_{j}}{\partial^{2}}\overline{{{\calA}_{j}^{\alpha}}}({\bfr})}\,.
\delta\chi^{i}=-{\frac{1}{2\sqrt{2}}}{\frac{\partial_{r}H}{H}}\epsilon^{ij}\gamma^{r}\left[\epsilon_{j}+f^{-1/2}(\eta\gamma_{\overline{{0}}}\epsilon_{jk}\epsilon^{k}+\sqrt{2}grH^{1/2}\gamma_{r}\epsilon^{j})\right],
R=\left(\begin{array}{llll}{q}&{0}&{0}&{0}\\{0}&{1}&{\Omega}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{q}\\\end{array}\right)\\\,\\\R^{\prime}=\left(\begin{array}{llll}{q^{-1}}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{-\Omega}&{1}&{0}\\{0}&{0}&{0}&{q^{-1}}\\\end{array}\right)
\frac{1}{24}(x_{1}^{24}+x_{2}^{24})+\frac{1}{12}x_{3}^{12}+\frac{1}{2}x_{5}^{2}-\psi_{0}x_{1}x_{2}x_{3}x_{4}x_{5}-\frac{1}{6}(x_{1}x_{2}x_{3})^{6}-\frac{1}{12}(x_{1}x_{2})^{12}=0.
{\calL}_{NJL}=\bar{q}[i\gamma^{\mu}\partial_{\mu}-g(\sigma+i\vec{\pi}\cdot\vec{\tau}\gamma_{5})]q-{\frac{\mu^{2}}{2}}(\sigma^{2}+\vec{\pi}^{2})+{\frac{M_{0}\mu^{2}}{g}}\sigma\,\,,
\begin{array}{ll}{{\calE}=}&{2\pi\intrdr\left[\frac{1}{2}\left(\frac{A^{\prime}}{rh}-h\varphi^{2}e^{-2gM}+2mM\right)^{2}+\frac{1}{r}\varphi^{2}e^{-2mM}A^{\prime}-\frac{2}{rh}mMA^{\prime}+\right.}\\{}&{\left.+\left(\varphi^{\prime}-\frac{A\varphi}{r}\right)^{2}e^{-2gM}+\frac{2}{r}\varphi\varphi^{\prime}Ae^{-2gM}+\left(M^{\prime}+\frac{G}{gr}\right)^{2}-\frac{2}{gr}GM^{\prime}\right];}\\\end{array}
\left\langleT_{11}(\vec{x})\right\rangle=-\frac{1}{32\pi^{2}}F_{0}(\vec{x})-\frac{1}{16\pi^{2}}\left(D_{1}(\vec{x})-D_{2}(\vec{x})\right)+\frac{1}{64\pi^{2}}\left(F_{1}(\vec{x})-F_{2}(\vec{x})\right)
\left[{\frac{\partial}{\partialr^{2}}}+{\frac{2}{r}}{\frac{\partial}{\partialr}}-V_{\mathrm{eff}}\right]R_{l\omega}=0~~;~~~V_{\mathrm{eff}}=-D_{1}D_{2}D_{3}D_{4}\omega^{2}+{\frac{l(l+1)}{r^{2}}}:
G_{3}=\sum_{k^{a},k^{b},k^{c}}\delta(k^{c}-k^{a}-k^{b})\tilde{A}_{a}^{\dagger}(k^{a})\tilde{B}_{b}^{\dagger}(k^{b})\tilde{C}_{c}(k^{c})V_{abc}(k^{a},k^{b},k^{c})+h.c.
{{\tilde{H}}^{\alpha\beta}}(\tilde{x},{\tilde{\tau}},{\tilde{\omega}})=(i{\tilde{\omega}}^{\rho}T_{\rho}{}^{\mu\nu}-\frac{1}{2}{\tilde{\omega}}^{2}{\calD}^{\mu}\psi^{\nu})T_{\mu\nu}^{-1}{}^{\alpha\beta},
0=\delta(y)\mathrm{i}(\bar{\Psi}_{+})_{\mu}^{A}\gamma^{\mu}\left[2g\alpha_{0}+\frac{1}{4\sqrt{2}}\lambda_{0}(1+3\alpha_{0}^{2}\epsilon(y)^{2})\right](\delta-\gamma_{5}\sigma_{3})_{A}^{\;B}(\eta_{+})_{B}\,
\int_{0}^{\infty}dy\,y^{1-s}\frac{d}{dy}t^{2(\rho-1)}=(1-\rho)\,\frac{\Gamma\left(\frac{\displaystyle3-s}{\displaystyle2}\right)\Gamma\left(\rho-\frac{\displaystyle3-s}{\displaystyle2}\right)}{\Gamma(\rho)},\quad3-2\mathrm{~Re~}\rho<\mathrm{~Re~}s<3.
\mathrm{\Psi(\tilde{R},\tilde{\varphi})=\tilde{R}^{\frac{1-p}{2}}Z_{\nu}\left(\frac{\sqrt{2}\tilde{a}^{2}}{3}\tilde{R}^{3}\right)\,e^{\pmim\tilde{\varphi}},\qquad\nu=\frac{1}{6}\sqrt{(1-p)^{2}-(2m)^{2}}\,,}
\psi(x)=\langlex^{ml}\vert\psi\rangle=N^{1/2}(n)(\beta/2\pi)^{n/4}\int_{I_{n}}d^{n}\rho~(1-\beta\rho^{2}/2)^{1/2+\sqrt{1/4+n/8}}e^{ix\cdot\rho/\hbar}\psi(\rho)
S^{\prime}={\frac{1}{16\piG_{10}}}\intd^{10}x\sqrt{-G^{str}}[e^{-2\phi}({\calR}^{str}+2\partial_{M}\phi\partial^{M}\phi)-{\frac{1}{2\cdot8!}}F_{M_{1}\cdotsM_{8}}F^{M_{1}\cdotsM_{8}}],
k_{1}^{\mu}T_{\rho\sigma\mu\nu}^{(3)abcd}(p_{1},p_{2};k_{1},k_{2})=C_{3}k_{2\nu}S_{\rho\sigma}^{(3)}(p_{1},p_{2},k_{1})-C_{3}\Gamma_{\sigma\nu\rho}^{(3)}(p_{1},p_{2},q_{3})
\widetilde{G}_{A_{1}\dotsA_{N}}^{\,amp}\,\overset{(s_{1})}{e}{\!\!}_{A_{1}}\,\dots\,\overset{(s_{N})}{e}{\!\!}_{A_{N}}\;=\;\widetilde{G}_{A_{1}\dotsB_{\ell}\dotsA_{N}}^{\,amp}\,\overset{(r)}{e}{\!}_{B_{\ell}}\,\overset{(r)}{e}{\!}_{A_{\ell}}\,\dots\,\overset{(s_{N})}{e}{\!\!}_{A_{N}}\;,
b_{1}(y,y^{\prime},p_{0})=a_{1}(p_{0})+{\frac{1}{12}}\left(R_{\mu\nu}(p_{0})(y-y^{\prime})^{\mu}(y-y^{\prime})^{\nu}+R_{\mu\lambda\nu\rho}(p_{0})(y-y^{\prime})^{\mu}(y-y^{\prime})^{\nu}y^{\lambda}y^{\rho}\right)
L_{q}^{new}=\left[e\hbarc\intd^{3}x\,\bar{q}(x)\gamma^{\mu}q(x)A_{\mu}(x)\right]_{em}\cdot\left[g\hbarc^{\prime}\intd^{3}x^{\prime}\,\bar{q}_{i}(x^{\prime})\gamma^{\mu^{\prime}}\frac{\lambda_{ij}^{a}}{2}q_{j}(x^{\prime})A_{\mu^{\prime}}^{a}\right]_{color}.
\left(Q_{1}^{\prime}\right)^{2}=\left(Q_{2}^{\prime}\right)^{2}=H^{\prime},\\left[Q_{1}^{\prime},H^{\prime}\right]=\left[Q_{2}^{\prime},H^{\prime}\right]=0,\\left\{Q_{1}^{\prime},Q_{2}^{\prime}\right\}=2p_{3}^{2}\neq0,
V(ker(S))=\prod_{k=0}^{N}\biggl(\!det(\widetilde{T}_{k+1}^{*}\widetilde{T}_{k+1})^{1/2}det(\Phi_{k}^{*}\Phi_{k})^{-1/2}\,V(\Gamma_{\!k+1})V(H^{k}(R(S)))\!\biggr)^{(-1)^{k}}
\deltaB_{i}\vert_{\Sigma_{2}^{i}}=-2\epsilon_{i}{\frac{T_{3}}{T_{6}^{(i)}}}\chi_{2}^{1}(N_{i})-2\kappa_{11}^{2}T_{6}^{(i)}\chi_{2}^{1}(N_{i}^{\prime})\vert_{\Sigma_{2}^{1}}.
{\calG}=\frac{1}{2}M_{ab}d{\bfx}_{a}\cdotd{\bfx}_{b}+\frac{g^{4}}{2(4\pi)^{2}}(M^{-1})_{ab}(d\xi_{a}+{\bfW}_{ac}\cdotd{\bfx}_{c})(d\xi_{b}+{\bfW}_{bd}\cdotd{\bfx}_{d}),
\tilde{\alpha}_{2,1}=\frac{r_{+}^{2}}{4}+\frac{3r_{+}^{4}}{2l^{2}}\pm\frac{r_{+}^{2}}{2}\sqrt{\frac{9r_{+}^{4}}{l^{4}}+\frac{11r_{+}^{2}}{3l^{2}}-\frac{5}{12}}.
\bar{\rho}_{\alpha}=-\rho_{\alpha+1}+2\rho_{\alpha}-\rho_{\alpha-1},\quad(\rho_{\alpha})_{x}+\sum_{s=1}^{min(m_{1},m_{2})}\sum_{k=0}^{s-1}f_{\alpha-k}^{s}\bar{f}_{\alpha-k}^{s}=0
\int_{-\infty}^{\infty}d\tau\left[A\frac{v^{6}}{r^{11}}+B\frac{v^{4}(v\cdotr)^{2}}{r^{13}}+C\frac{v^{2}(v\cdotr)^{4}}{r^{15}}+D\frac{(v\cdotr)^{6}}{r^{17}}\right].
\prod_{a=1}^{\tilde{n}_{c}}(\Lambda\omega^{2\ell-1}+\delta_{\ell}-\phi_{a})^{2}=(\Lambda\omega^{2\ell-1})^{\tilde{n}_{c}}\left[1+2\sum_{a=1}^{\tilde{n}_{c}}\frac{\delta_{\ell}-\phi_{a}}{\Lambda\omega^{2\ell-1}}\right].
ds^{2}=-2dx^{+}dx^{-}-\beta^{2}\left(4\Big[(\tilde{x}^{1})^{2}+(\tilde{x}^{2})^{2}\Big]+\sum_{i=3}^{6}(\tilde{x}^{i})^{2}\right)+\sum_{i=1}^{6}(d\tilde{x}^{i})^{2}+\sum_{i=7}^{9}(dx^{i})^{2},
\int_{0}^{T}d\tau_{a}\int_{0}^{T}d\tau_{b}\,G_{Bab}^{k-{\frac{D}{2}}}=B\Bigl(k+1-{\frac{D}{2}},k+1-{\frac{D}{2}}\Bigr)T^{2+k-{\frac{D}{2}}}
\left.+\frac{1}{2}\left(\{\psi_{+}(x),\psi_{+}(y)\}+\{\psi_{-}(x),\psi_{-}(y)\}\right)\left(\frac{1}{2}\phi^{\prime}(x)U(y)+\frac{1}{2}U(x)\phi^{\prime}(y)\right)\right]\\.
\langle-D_{a}y_{a}\cdotD_{b}y_{b}\rangle=\mathrm{tr}\biggl[D_{a}D_{b}\hat{\calG}_{ab}+{\frac{1}{2}}{\frac{D_{a}(\hat{\calG}_{aa}-\hat{\calG}_{ab})D_{b}(\hat{\calG}_{ab}-\hat{\calG}_{bb})}{\bar{T}-{\frac{1}{2}}\hat{\calC}_{ab}}}\biggr]\;.\nonumber\,
u=\frac{\bar{y}_{1}y_{1}^{\prime}+\bar{y}_{2}y_{2}^{\prime}}{y_{1}y_{2}^{\prime}-y_{1}^{\prime}y_{2}},\qquad\bar{u}=\frac{y_{1}\bar{y}_{1}^{\prime}+y_{2}\bar{y}_{2}^{\prime}}{\bar{y}_{1}\bar{y}_{2}^{\prime}-\bar{y}_{1}^{\prime}\bar{y}_{2}}.
{\frac{1}{R_{11}^{2}}}\left(\ell_{2}^{2}+{\frac{1}{\tau_{2}^{2}}}(\ell_{2}\tau_{1}-\ell_{1})^{2}\right)={\frac{|\ell_{1}-\ell_{2}\tau|^{2}}{(\tau_{2}R_{11})^{2}}}.
{\calP}_{n}^{\pm}(\beta)=N_{\pm}^{2}(\beta)\left[|T_{n}(\beta)|^{2}+|T_{n}(-\beta)|^{2}{\pm}\left(T_{n}^{*}(\beta)T_{n}(-\beta)+T_{n}(\beta)T_{n}^{*}(-\beta)\right)\right].
\varphi_{k}^{*}\dot{\varphi}_{k^{\prime}}^{*}\dot{\varphi}_{k}\varphi_{k^{\prime}}+\varphi_{k}\dot{\varphi}_{k^{\prime}}\dot{\varphi}_{k}^{*}\varphi_{k^{\prime}}^{*}=2+\frac12(\dot{\varphi}_{k}\varphi_{k}^{*}+\varphi_{k}\dot{\varphi}_{k}^{*})(\dot{\varphi}_{k^{\prime}}\varphi_{k^{\prime}}^{*}+\varphi_{k^{\prime}}\dot{\varphi}_{k^{\prime}}^{*})
{\vert{\calM}\vert}^{2}={\frac{4g_{A}^{2}G_{F}^{2}}{e^{2}}}\epsilon^{\alpha}(a)\epsilon^{*\alpha^{\prime}}(a)\left(p_{1}^{\beta}p_{2}^{\beta^{\prime}}+p_{1}^{\beta^{\prime}}p_{2}^{\beta}\right)\Pi_{\alpha\beta}^{5}\Pi_{\alpha^{\prime}\beta^{\prime}}^{5},
S_{\sigma}={\frac{1}{4\pi\alpha^{\prime}}}\int_{{\calW}_{s}}d^{2}\!\sigma\,g^{1/2}\,\left\{(g^{ab}G_{\mu\nu}(X)+\epsilon^{ab}B_{\mu\nu}(X))\partial_{a}X^{\mu}\partial_{b}X^{\nu}+\alpha^{\prime}\phiR\right\}\,
{\calL}[v_{\mu},\theta]=-\,\frac{1}{4}\,\left(\partial_{[\mu}v_{\nu]}\right)^{2}-\frac{\alpha_{j}\,e^{2}}{2}\,v_{\mu}v^{\mu}-\frac{\alpha_{j}\,e}{2}\,\epsilon^{\mu\nu}\,v_{\mu}(\partial_{\nu}\theta)+\frac{\alpha_{j}}{8}\,(\partial_{\mu}\theta)(\partial^{\mu}\theta).
S_{OSp}={\frac{1}{2}}\int\left[E^{b}(x,\theta)u_{b}^{~a}(y)\bar{\lambda}\gamma_{a}\lambda+{\frac{1}{2}}\left(\Omega^{cd}(x,\theta)u_{c}^{~a}u_{d}^{~a}+(u^{-1}du)^{ab}\right)\bar{\lambda}\gamma_{ab}\lambda\right].
\langle\sum_{i=1}^{N}(x_{i}+iy_{i})^{J_{1}}\sum_{j=1}^{N}(x_{j}+iy_{j})^{J_{2}}\sum_{k=1}^{N}(x_{k}-iy_{k})^{J_{1}+J_{2}}\rangle=N{\frac{(J_{1}+J_{2})!}{\omega^{J_{1}+J_{2}}}}.
\ast(dx^{i_{1}}\wedgedx^{i_{2}}\wedge\cdots\wedgedx^{i_{p}})={\frac{1}{(n-p)!}}{\epsilon}^{i_{1}i_{2}\cdotsi_{p}}{}_{i_{p+1}\cdotsi_{n}}dx^{i_{p+1}}\wedge\cdots\wedgedx^{i_{n}}
\psi^{(+)}\stackrel{r\sim\infty}{\rightarrow}{\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(b)\Gamma(c-a)}}e^{(-Qr)}+{\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(a)\Gamma(c-b)}}e^{(+Qr)}
{\bfF}(\Box)\simeq\left[\begin{array}{cr}{\,\,\displaystyle{\frac{l^{2}\Box}{2}}\,\,}&{\,\,\displaystyle{\frac{l^{2}\Box}{2J_{2}^{-}}}\,\,\,\,\,}\\{}&{}\\{\,\,\,\displaystyle{\frac{l^{2}\Box}{2J_{2}^{-}}}\,\,}&{\,\,-\displaystyle{\frac{J_{1}^{-}}{J_{2}^{-}}}\,\frac{l}{a}\sqrt{\Box}}\\\end{array}\right].
ds^{2}=\left(\frac{l}{z}\right)^{2}\left(dz^{2}-(1+2\phi)d\tau^{2}+2b_{,i}dx^{i}d\tau+\left((1-2\hat{\Psi})\delta_{ij}+2\hat{E}_{,ij}\right)dx^{i}dx^{j}\right)\.
L_{D}^{**}=\int_{\calM}d^{3}x{\frac{4\pi}{g}}\varepsilon^{lmn}\bigg(B_{l}^{AB}\partial_{m}B_{nAB}-{\frac{4\pi}{g}}f_{ABCDEF}B_{l}^{AB}B_{m}^{CD}B_{n}^{EF}\bigg).
\stackrel{\cdot\cdot}{\chi}+\left\{C\left[m_{r}^{2}+(\xi_{r}-\frac16)R-12\kappa_{r}\phi_{c}^{2}+15\lambda_{r}^{2}\phi_{c}^{4}+\sum_{i}\frac{k_{i}^{2}}{a_{i}^{2}}\right]+Q\right\}\chi_{k}=0
\varphi_{\alpha}(\lambda,\omega)\sim{\frac{1}{(4\pi)^{d/2}}}\sum_{n=0}^{\infty}\left[a_{n}^{\prime}(\omega)~{\frac{\lambda^{d/2-n+\alpha-1}}{\Gamma\left(\fracd2-n+\alpha\right)}}+b_{n}^{\prime}(\omega)~{\frac{\lambda^{(d-1)/2-n+\alpha-1}}{\Gamma\left({\frac{d-1}{2}}-n+\alpha\right)}}\right],
\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{-1}&{0}\\{0}&{0}&{0}\\\end{array}\right)\quad\mathrm{~and~}\quad\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{1}&{0}\\{0}&{0}&{-1}\\\end{array}\right).
\stackrel{\wedge}{\Sigma}_{(2)}(p,m,\lambda,\Lambda_{0})=\frac{i\lambda^{2}}{192\pi^{6}}\int_{0}^{1}dx\int_{0}^{1}dy\fracy{\left[yC\right]^{3/2}}\intd^{3}kd^{3}l\frac1{\left[k^{2}+l^{2}+Dp^{2}-m^{2}+i\epsilon\right]^{3}}
K_{MN}^{(0)}\equiv\left(\begin{array}{ccc}{g_{\mu}^{\;\rho}g_{\nu}^{\;\sigma}}&{0}&{-g_{\mu\nu}}\\{0}&{-g_{\mu}^{\;\rho}g_{\nu}^{\;\sigma}}&{0}\\{0}&{0}&{8}\\\end{array}\right)
\ddot{\eta}_{i}({\bfk}_{n},t)+3H{\dot{\eta}}_{i}({\bfk}_{n},t)+[e^{-2Ht}{\bfk}_{n}^{2}+\mu_{i}^{2}]\eta_{i}({\bfk}_{n},t)+\alpha_{i}\phi({\bfk}_{n},t)=0.
{\calN}_{Q}[\tilde{\calH}_{x}]={\frac{1}{2}}\tilde{\Pi}_{x}^{2}+{\frac{1}{2}}(\nabla\tilde{\phi}_{x})^{2}-{\frac{1}{2}}I_{0}[Q^{2}]+{\frac{1}{4}}Q^{2}I_{1}[Q^{2}]+{\calN}_{Q}[V(\tilde{\phi}_{x})]
V(\tilde{\varphi},\tilde{f})=8m^{2}e^{{\frac{10}{\sqrt{15}}}\tilde{\varphi}}+e^{{\frac{4}{\sqrt{15}}}\tilde{\varphi}}(\mu^{2}e^{{\frac{6}{\sqrt{10}}}\tilde{f}}-R_{4}e^{{\frac{1}{\sqrt{10}}}\tilde{f}}),
\bar{j}_{{}_{\mathrm{REG}}}^{\varphi}(r)=-\frac{s}{2\pi}\int_{0}^{\infty}dk\,\frac{k^{2}}{|E|}\sum_{l=1}^{\infty}[J_{l-F}(kr)J_{l+1-F}(kr)-J_{l+F}(kr)J_{l-1+F}(kr)].
\zeta_{D}^{c}(2,x^{2})=-\frac{\alpha\,R}{8\,x^{3}}+\frac{3\,\alpha}{512\,R\,x^{5}}+\frac{555}{2^{17}}\,\frac{\alpha}{R^{3}\,x^{7}}+{\calO}(x^{-9})\,{.}
V(f_{1},f_{2})=\left|\frac{\mu}{2}\right|^{2}\frac{1}{f_{1}^{i}{\bar{f}}_{1i}+f_{2}^{i}{\bar{f}}_{2i}}\left\{-|f_{1}^{i}{\bar{f}}_{2i}-f_{2}^{i}{\bar{f}}_{1i}|^{2}+(f_{1}^{i}{\bar{f}}_{1i}+f_{2}^{i}{\bar{f}}_{2i})^{2}\right\}.
b_{-r_{m}}^{i_{m}}\cdotsb_{-r_{2}}^{i_{2}}b_{-r_{1}}^{i_{1}}|\Omega_{\mathrm{NS}}\rangle,\qquadr_{l}\in{\mathbbN}_{0}+\frac{1}{2},\qquadi_{l}=1,2,\ldots,N,
\{\psi_{\alpha}^{a},\psi_{\beta}^{b}\}=0=\{\tilde{\psi}_{\alpha}^{a},\tilde{\psi}_{\beta}^{b}\},\qquad\{\tilde{\psi}_{\alpha}^{a},\psi_{\beta}^{b}\}=2\delta^{ab}\delta_{\alpha\beta}.
\alpha^{ijk}\Pi_{i}\Pi_{k}\Pi_{j}\longrightarrow-\left({\frac{3}{2}}{\frac{\alpha^{ijk}\alpha^{ijk}}{\alpha}}+{\frac{9}{4}}{\frac{\alpha^{iik}\alpha^{jjk}}{\alpha}}\right)J^{2}-{\frac{5}{8}}{\frac{\alpha^{ijk}\alpha^{ijk}}{\alpha}}+{\frac{3}{16}}{\frac{\alpha^{iik}\alpha^{jjk}}{\alpha}}
\left(\int_{0}^{1}ds\,\sqrt{x^{\,\prime\,2}}\right)^{-1}\int_{0}^{1}{\frac{ds}{\sqrt{x^{\,\prime\,2}}}}\,{\frac{\deltaS_{\mathrm{cl.}}}{\deltax^{\mu}(s)}}{\frac{\deltaS_{\mathrm{cl.}}}{\deltax_{\mu}(s)}}=-2m^{2}{\frac{\partialS_{\mathrm{cl.}}}{\partialA}}\.
\left[(dr^{3}\partial_{r})^{2}+\omega^{2}r^{6}f-{\frac{8dr^{4}r_{\pm}^{4}}{(r^{2}+r_{\pm}^{2})^{2}}}\left(1+{\frac{r_{0}^{2}}{r_{\pm}^{2}}}\right)\right]\tilde{\phi}_{\pm}=0,
\dot{G}_{Bi_{1}i_{2}}\dot{G}_{Bi_{2}i_{3}}\cdots\dot{G}_{Bi_{n}i_{1}}\rightarrow\dot{G}_{Bi_{1}i_{2}}\dot{G}_{Bi_{2}i_{3}}\cdots\dot{G}_{Bi_{n}i_{1}}\nonumber\,-2^{n}{G}_{p\,i_{1}i_{2}}^{\chi}{G}_{p\,i_{2}i_{3}}^{\chi}\cdots{G}_{p\,i_{n}i_{1}}^{\chi}\nonumber\,
|\Psi_{(+)a}>=|0>{\bfM}_{a}+\frac{1}{2}b_{i}^{\dagger}b_{j}^{\dagger}|0>{\bfM}_{a}^{ij}+\frac{1}{24}\epsilon^{ijklm}b_{j}^{\dagger}b_{k}^{\dagger}b_{l}^{\dagger}b_{m}^{\dagger}|0>{\bfM}_{ai}
S(\sigma)=\frac{1}{4}\int_{0}^{2\pi}d\sigma^{\prime}d\sigma^{\prime\prime}e^{\frac{1}{4}\gamma\etap\left(\epsilon(\sigma-\sigma^{\prime})-\epsilon(\sigma-\sigma^{\prime\prime})\right)}e^{\psi^{+}(\sigma^{\prime})}e^{\psi^{-}(\sigma^{\prime\prime})}
\tilde{E}_{M}^{~A}=\left(\begin{matrix}{1+2\zeta\theta}&{2\bar{\zeta}\bar{\theta}}&{\zeta}&{-\bar{\zeta}}\\{-2\zeta\theta}&{1-2\bar{\zeta}\bar{\theta}}&{-\zeta}&{\bar{\zeta}}\\{\theta}&{0}&{1}&{0}\\{0}&{-\bar{\theta}}&{0}&{1}\\\end{matrix}\right).
P_{\calR}^{1/2}=\sqrt{\frac{k^{3}}{2\pi^{2}}}\left|{\frac{u_{k}}{z}}\right|={\frac{2^{\nu-3/2}}{2\pi}}{\frac{\Gamma\left(\nu\right)}{\Gamma\left(3/2\right)}}\left({\frac{H}{m_{\mathrm{Pl}}\sqrt{\epsilon}}}\right)\left|C_{\mathrm{+}}\left(k\right)+C_{-}\left(k\right)\right|\left({\frac{y}{1-\epsilon}}\right)^{3/2-\nu},
{\calT}_{11}(q)=-\frac{1}{2\pi}\int_{-\infty}^{\infty}dq_{0}^{\prime}\left(\frac{M^{+}(q_{0}^{\prime},\vec{q})}{q_{0}-q_{0}^{\prime}+i\epsilon}-\frac{M^{-}(q_{0}^{\prime},\vec{q})}{q_{0}-q_{0}^{\prime}-i\epsilon}\right)~,
{\frac{M^{2}\pi}{g^{2}N}}=\left(\begin{array}{cccc}{9x}&{0}&{0}&{0}\\{0}&{{\frac{81}{4}}+3x+6y}&{-3i\sqrt{{\frac{y}{2}}}}&{-{\frac{3i}{2}}\sqrt{y}}\\{0}&{3i\sqrt{{\frac{y}{2}}}}&{{\frac{9}{2}}x}&{0}\\{0}&{{\frac{3i}{2}}\sqrt{y}}&{0}&{{\frac{81}{4}}+{\frac{9}{2}}y}\\\end{array}\right)
D(x,x^{\prime})=\sum_{n=-\infty}^{\infty}\int{\frac{d^{4}k}{(2\pi)^{4}}}\int_{0}^{\infty}{\frac{qdq}{2\pi\alpha}}{\frac{e^{ik_{\mu}(x^{\mu}-x^{\mu})}e^{in(\theta-\theta^{\prime})/\alpha}}{k_{4}^{2}+q^{2}}}J_{\left|n/\alpha\right|}(qr)J_{\left|n/\alpha\right|}(qr^{\prime}),
L_{0}=-\frac{1}{2}\left(\begin{array}{rr}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,,\qquadL_{+}=\frac{1}{\sqrt{2}}\left(\begin{array}{rr}{0}&{0}\\{1}&{0}\\\end{array}\right)\,,\qquadL_{-}=\frac{1}{\sqrt{2}}\left(\begin{array}{rr}{0}&{1}\\{0}&{0}\\\end{array}\right)
M(q)\ddot{q}+\frac{1}{2}M^{\prime}(q)\dot{q}^{2}-\frac{\alpha\Gamma(n+1)}{\pi}\int_{-\infty}^{\infty}d\tau^{\prime}\frac{q(\tau)-q(\tau^{\prime})}{\mid\tau-\tau^{\prime}\mid^{n+1}}=V^{\prime}(q)+\deltaV^{\prime}(q),
\intd^{N}{\mathbf{k}}\int_{k}^{\infty}\frac{k^{s}g(z)dz}{\sqrt{z^{2}-k^{2}}}=\frac{\pi^{N/2}}{\Gamma(N/2)}B\left(\frac{N+s}{2},\frac{1}{2}\right)\int_{0}^{\infty}dzz^{N+s-1}g(z),
{\calL}_{\calA}=-\frac{1}{2g^{2}}\,\mathrm{Tr}\,\left(F_{\mu\nu}^{2}\right)+\frac{1}{g^{2}}\,\mathrm{Tr}\,\left\{{\calA}_{\mu}\left(D^{2}g_{\mu\nu}{\calA}_{\nu}-2i\left[F_{\mu\nu},{\calA}_{\nu}\right]\right)\right\}+\ldots\,
\begin{array}{c}{R_{ij}=R_{i}{}^{k}{}_{jk},R_{ia}=P_{i}{}^{k}{}_{ak}=-P_{ia},}\\{R_{ai}=P_{a}{}^{b}{}_{ib}=P_{ai},R_{ab}=S_{a}{}^{c}{}_{bc}=S_{ab}}\\\end{array}
T_{\nu}^{\mu}=\delta_{\nu}^{\mu}f_{0}\,,\quadT_{\rho}^{\rho}=f_{\rho}\,,\quadT_{\theta}^{\theta}=f_{\theta}\,,\quadT_{\varphi}^{\varphi}=f_{\varphi}\,,\quad\mathcal{L}=Lm_{W}\,,\quadJ=\frac{H}{\eta\,e\,\rho}.
V^{i^{\prime}}=\frac{n^{\prime}}{n^{\prime}+\frac{n^{\prime}}{(e{\calP}^{\prime})}(eQ)}\left[1+\frac{n^{\prime}}{(e{\calP}^{\prime})}(eQ)\right]\left[Q^{i^{\prime}}-{\calP}_{L}^{i^{\prime}}\frac{(eQ)}{(e{\calP}^{\prime})}\right]
\left(\begin{array}{ccccc}{M_{00}}&{M_{02}}&{0}&{0}&{\cdots}\\{0}&{M_{22}}&{M_{24}}&{0}&{\cdots}\\{0}&{0}&{M_{44}}&{M_{46}}&{\cdots}\\{0}&{0}&{0}&{M_{66}}&{\cdots}\\{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}\\\end{array}\right)
\langle\underline{{O}}|\underline{{S}}^{(\pm)}(x)|\underline{{O}}\rangle=\intd\Gamma_{m}(\vec{p})\langle\underline{{O}}|\underline{{I}}_{\vec{p}}|\underline{{O}}\rangle(\gamma\cdotp\pmm)e^{\mpip\cdotx}=Z\intd\Gamma_{m}(\vec{p})\chi(\vec{p})(\gamma\cdotp\pmm)e^{\mpip\cdotx}.\nonumber
A_{\mu}=\mp{\frac{2i}{g}}\,(H_{1}\,{\frac{v_{\mu}^{1}}{\rho_{1}}}+g\,\kappa\,E_{13}^{\pm}\,R_{\mu}^{1})\mp\,{\frac{2i}{g}}\,(H_{2}\,{\frac{v_{\mu}^{2}}{\rho_{2}}}+g\,\kappa\,E_{23}^{\pm}\,R_{\mu}^{2}).