harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V\Big[\begin{array}{c}{\alpha+nb_{3}+{m}b_{4}}\\{\beta+\tilde{n}b_{3}+\tilde{m}b_{4}}\\\end{array}\Big]=V\Big[\begin{array}{c}{\alpha+nb_{3}+{m}b_{4}+n_{1}b_{1}+m_{1}b_{2}}\\{\beta+\tilde{n}b_{3}+\tilde{m}b_{4}+n_{2}b_{1}+m_{2}b_{2}}\\\end{array}\Big],
u_{ij}\equiv{\bfe}_{i}{\bfe}_{j}-\delta_{ij},\quad\phi_{a.a+1}\equiv{\widetilde\phi}_{a.a+1}-F_{,a},\quad{\phi}_{0.1}\equiv{\widetilde\phi}_{0.1}+\sum_{i}{\tilde{k}}_{i}F_{,i}-F,
T_{\mu\nu}^{matter}=\frac{1}{4}\left(H_{\mu\nu}^{2}-\frac{1}{6}\tilde{G}_{\mu\nu}H^{2}\right)e^{-\frac{8}{\sqrt{2|D-2|}}\tilde{\phi}}+\left(\partial_{\mu}\tilde{\phi}\partial_{\nu}\tilde{\phi}-\frac{1}{2}(\partial\tilde{\phi})^{2}\tilde{G}_{\mu\nu}\right)~.
I_{2}=\int_{0}^{1}dy\intd^{4}k\,\Bigg(\frac{1}{\Big(k^{2}+2kpy+yp^{2}\Big)^{2}}-\sum_{i}c_{i}\frac{1}{\Big(k^{2}+2kpy+yp^{2}+M_{i}^{2}\Big)^{2}}\Bigg).
S_{\mathrm{IIB}}=\intd^{10}x\sqrt{-\hat{g}_{10}}\left\{\hat{R}_{10}+{\frac{1}{4}}\mathrm{Tr}\left(\hat{\nabla}M\hat{\nabla}M^{-1}\right)-\frac{1}{12}\hat{H}^{T}M\hat{H}\right\},
\langlej^{0}({\bfr})\rangle=-\displaystyle\frac{e}{2}\displaystyle\int_{0}^{\infty}dw\displaystyle\frac{m}{\sqrt{w+m^{2}}}\sum_{l}(|u_{w,l}({\bfr})|^{2}-|v_{w,l}({\bfr})|^{2}).
f_{kl}^{(1)}(x,y)=\left(\begin{array}{cccc}{0}&{0}&{0}&{m^{2}}\\{0}&{0}&{-\delta_{ij}}&{0}\\{0}&{\delta_{ij}}&{0}&{-\partial_{x}^{1}}\\{-m^{2}}&{0}&{\partial_{y}^{1}}&{0}\\\end{array}\right)\delta^{3}(x-y),
\begin{array}{rcl}{\Phi_{0}^{(2)}(u_{1},u_{2},g)}&{=}&{2u_{1}^{2}\Phi_{0}^{(1)}(u_{1},g)\Phi_{0}^{(2)}(u_{1},u_{2},g)+gu_{1}\partial_{u_{1}}^{2}\Phi_{0}^{(2)}(u_{1},u_{2},g)+}\\{}&{}&{\displaystyle\quadu_{2}{\frac{d}{du_{2}}}\left({\frac{u_{1}^{2}u_{2}\Phi_{0}^{(1)}(u_{1},g)-u_{2}^{2}u_{1}\Phi_{0}^{(1)}(u_{2},g)}{u_{1}-u_{2}}}\right)\.}\\\end{array}
\nabla\pi^{(\underline{\alpha})}=d\pi^{(\underline{\alpha})}-\frac14\pi^{(\underline{\beta})}\Omega_{(\underline{\beta})}{}^{(\underline{\alpha})}=0,\nabla\pi_{(\underline{\alpha})}=d\pi^{(\underline{\alpha})}-\frac14\pi_{(\underline{\beta})}\Omega^{(\underline{\beta})}{}_{(\underline{\alpha})}=0.
^{\omega}F_{ij}=-\frac{1}{r^{2}}\left(\gamma_{ij}+\frac{1}{1+\hat{x}_{4}}\hat{x}_{[i}\gamma_{j]k}\hat{x}_{k}\right),\qquad\qquad^{\omega}F_{i4}=\frac{1}{r^{2}}\gamma_{ij}\hat{x}_{j},
{\calH}=\frac{1}{4}\,\pi_{ij}\pi^{ij}+\partial_{i}A_{0j}\pi^{ij}-\frac{1}{4}\,A_{ij}\nabla^{2}A^{ij}+\frac{1}{2}\,A_{ij}\partial^{j}\partial_{k}A^{ik}+\lambda_{0}\,T_{0}
(\Gamma^{0})_{~~\underline{\beta}}^{\underline{\alpha}}=\left(\begin{array}{cc}{0}&{-1}\\{1}&{0}\\\end{array}\right)\,,\quad(\Gamma^{1})_{~~\underline{\beta}}^{\underline{\alpha}}=\left(\begin{array}{cc}{-1}&{0}\\{0}&{1}\\\end{array}\right)\,,\quad\Gamma^{2}=\Gamma^{0}\Gamma^{1}=\left(\begin{array}{cc}{0}&{-1}\\{-1}&{0}\\\end{array}\right)\,.
\Gamma^{0}=\left[\begin{array}{cccc}{0}&{0}&{0}&{-i}\\{0}&{0}&{i}&{0}\\{0}&{-i}&{0}&{0}\\{i}&{0}&{0}&{0}\\\end{array}\right]\qquad\Gamma^{1}=\left[\begin{array}{cccc}{i}&{0}&{0}&{0}\\{0}&{-i}&{0}&{0}\\{0}&{0}&{i}&{0}\\{0}&{0}&{0}&{-i}\\\end{array}\right]\qquad\Gamma^{2}=\left[\begin{array}{cccc}{0}&{0}&{0}&{i}\\{0}&{0}&{-i}&{0}\\{0}&{-i}&{0}&{0}\\{i}&{0}&{0}&{0}\\\end{array}\right]
\!\!\!\!I\!=\!\int_{\Sigma}\!d^{3}\!x\!\sqrt{\gamma}\left(\!R(\gamma)\!-\!{\frac{1}{2}}\!\left(\left(\!DU\right)^{2}\!-\!e^{-2U}\!\!\left(\!DV\right)^{2}\right)\!-\!{\frac{1}{2}}\!\left(\left(\!D\Phi\!\right)^{2}\!-\!e^{-2\Phi}\!\left(\!D\Psi\!\right)^{2}\right)\!-\!\left(\!DT\right)\!^{2}\!+\!e^{-(U\!+\Phi)}V(T)\!\right)
iS_{ij}^{F}({\vec{x}}-{\vec{x^{\prime}}})={\frac{1}{2}}\int{\frac{\mathrm{d}^{3}{\vec{k}}}{(2\pi)^{3}}}{{\frac{e^{i{\vec{k}}.({\vec{x}}-{\vec{x^{\prime}})}}}{\omega_{k}}}}\left(\delta_{ij}-{\frac{k_{i}k_{j}}{\vec{k}}}\right).
\tilde{U}(\phi_{\mathrm{kink}}(x))=\frac{\tilde{m}^{2}}{2\phi_{0}(T)^{2}}(3\phi_{\mathrm{kink}}(x)^{2}-\phi_{0}(T)^{2})=\tilde{m}^{2}\left(-\frac{3}{2}\mathrm{sech}^{2}\frac{\tilde{m}x}{2}+1\right)\,.
S_{eff}=\intdx^{0}\sqrt{-g}\left[{\frac{c^{4}}{16\piG_{eff}}}{\calR}+V_{r}(R)+{\calL}_{1}+\sqrt{\Omega_{R}^{2}-\dot{R}^{2}/c^{2}}\Omega_{R}^{3}{\calL}_{2}\right],
\Psi_{q,j,m_{j}}^{+}=\frac{1}{\sqrt{2j}}\left(\begin{array}{l}{\sqrt{j+m_{j}}\,Y_{j-\frac{1}{2},m_{j}-\frac{1}{2}}^{q}}\\{\sqrt{j-m_{j}}\,Y_{j-\frac{1}{2},m_{j}+\frac{1}{2}}^{q}}\\\end{array}\right)
\sum_{n}\frac{e^{in(\phi-\phi^{\prime})}}{2\pi}\frac{\Gamma(\frac{|n|}{\alpha}+1-\frac{ict}{l_{P}})}{\Gamma(\frac{|n|}{\alpha}+1)}\rho^{\frac{|n|}{\alpha}+1}{_2F_{1}}(\frac{|n|}{\alpha}+1-\frac{ict}{l_{P}};\frac{|n|}{\alpha}+\frac{1}{2};2\frac{|n|}{\alpha}+1;4\rho)
\int\frac{d^{d}u}{\left(u^{2}+a^{2}\right)^{s}}=\frac{\pi^{\frac{d}{2}}}{\Gamma(s)}\Gamma\left(s-\frac{d}{2}\right)\frac{1}{\left(a^{2}\right)^{s-\frac{d}{2}}},
F_{(m)}^{\mu\nu}=\left(\begin{array}{cccc}{0}&{-B_{x}}&{-B_{y}}&{-B_{z}}\\{B_{x}}&{0}&{E_{z}}&{-E_{y}}\\{B_{y}}&{-E_{z}}&{0}&{E_{x}}\\{B_{z}}&{E_{y}}&{-E_{x}}&{0}\\\end{array}\right).
a(\xi)={\frac{-\left({{\gamma}^{2}}\,{\sqrt{M\,\Lambda}}\right)}{2\,M\,\Lambda\,{{\xi}^{2}}}}+{\frac{{{\gamma}^{2}}\,{\sqrt{M\,\Lambda}}}{2\,M\,\Lambda\,{{\left({\frac{2\,{\sqrt{M\,\Lambda}}}{\Lambda}}+\xi\right)}^{2}}}}
\sum_{(n_{1},n_{2},n_{3},n_{4})\neq(0,0,0,0)}=\sum_{n_{1}=0,\vec{N}\neq\vec{0}}+\sum_{n_{1}\neq0,\vec{N}\neq\vec{0}}+\sum_{n_{1}\neq0,\vec{N}=\vec{0}}\,.\,[6pt]
\begin{array}{ccl}{(0~0~0~-\frac{7}{2})}&{\rightarrow}&{(0~0~-\frac{7}{2})(+\frac{7}{2})\oplus(0~0~-\frac{5}{2})(+\frac{1}{2}),}\\{(0~0~0~+\frac{1}{2})}&{\rightarrow}&{(0~0~+\frac{1}{2})(-\frac{1}{2})\oplus(0~0~+\frac{3}{2})(-\frac{7}{2}).}\\\end{array}
C_{J}(\nu_{1},\nu_{2})=(2J+\nu_{1}+\nu_{2}+1)\frac{{\mit\Gamma}(J+1){\mit\Gamma}(J+\nu_{1}+\nu_{2}+1)}{{\mit\Gamma}(J+\nu_{1}+1){\mit\Gamma}(J+\nu_{2}+1)}\,.
L_{eff}^{(1)}=2\pi\kappa\frac{\sigma_{1}\sigma_{2}}{q_{1}q_{2}}\sum_{p_{1}=1}^{n_{1}}\sum_{p_{2}=1}^{n_{2}}\frac{d}{dt}Arg(\vec{R}_{p_{1}}^{(1)}-\vec{R}_{p_{2}}^{(2)})\;\;.
\begin{array}{l}{2(n-n_{1})-2(n_{2}+\cdots+n_{j})+\beta_{i-1j-1}^{(k-1)}\geqa_{1}^{(j)}\geq\cdots\geqa_{n_{j}-n_{j+1}}^{(j)}\geq0,}\\{n-n_{1}-(n_{2}+\cdots+n_{k-1})\geqb_{1}\geq\cdots\geqb_{n_{k-1}}\geq0,}\\\end{array}
V_{\alpha(-1/2)}^{f}(z)=e^{-c/2}S_{\alpha}e^{i(\alpha-1/2)S_{12}}e^{i(\beta-1/2)S_{34}}e^{i(\gamma-1/2)S_{56}}Ge^{i\frac{1}{2}KX}e^{i\frac{1}{2}K\bar{X}}
\tilde{Z}_{AB}=i\sigma_{2}\left(\begin{matrix}{z_{1}}&{0}\\{0}&{z_{2}}\\\end{matrix}\right),\qquad\tilde{Z}_{AB}=i\sigma_{2}\left(\begin{matrix}{z_{1}}&{0}&{0}&{0}\\{0}&{z_{2}}&{0}&{0}\\{0}&{0}&{z_{3}}&{0}\\{0}&{0}&{0}&{z_{4}}\\\end{matrix}\right),
\gamma_{0}=\left(\begin{matrix}{1}&{0}\\{0}&{-1}\\\end{matrix}\right)\qquad\gamma_{1}=\left(\begin{matrix}{0}&{1}\\{-1}&{0}\\\end{matrix}\right)\qquad\gamma_{5}=\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)
{\calA}(p_{1},p_{2},p_{3},p_{4})~=~V_{CKG}^{-1}\int\prod_{i=1}^{4}dz_{i}~<{V_{F}(1)~V_{B}(2)~[V_{B}(3)~V_{F}(4)~+~V_{B}(4)~V_{F}(3)]}>~,
i\frac{\sqrt{8m}}{\hbar}\delta_{\mu}^{2}{\calF}_{\alpha\psi}^{\left(\mu\right)}\gamma^{\mu}+\left[\overline{{V}}_{\alpha}^{\left(\mu\right)}-m\right]\delta_{\mu}{\calF}_{\alpha\psi}^{\left(\mu\right)}\left(\delta_{\mu}^{2}{\calF}_{\alpha\psi}^{\left(\mu\right)}\psi_{\alpha}^{\left(\mu\right)}-\delta_{\mu}{\calF}_{\alpha\psi}^{\left(\mu\right)}\right)=0
w:=\left(\begin{array}{c}{\tilde{x}_{1}}\\{.}\\{.}\\{.}\\{\tilde{x}_{n}}\\{\hat{y}_{1}}\\{.}\\{.}\\{.}\\{\hat{y}_{n}}\\\end{array}\right)\;,\;z:=\left(\begin{array}{c}{\tilde{y}_{1}}\\{.}\\{.}\\{.}\\{\tilde{y}_{n}}\\{\hat{x}_{1}}\\{.}\\{.}\\{.}\\{\hat{x}_{n}}\\\end{array}\right)\;\;.
I_{W}=(I_{CSG})_{|M\approxR\times\Sigma}=\frac{1}{\sqrt{\Lambda}}\intdt\int_{\Sigma}d^{2}x(-2\epsilon^{ij}E_{ia}\dot{\omega}_{j}^{a}+E_{t}^{a}\Psi_{a}+\omega_{ta}{\calG}^{a}),
C_{ab}^{b}=E_{i}^{b}(\xi_{b}^{j}\xi_{a,j}^{i}-\xi_{a}^{j}\xi_{b,j}^{i})=A_{~c}^{b}(A_{a,j}^{~c}\xi_{b}^{j}-A_{b,j}^{~c}\xi_{a}^{j}).
E_{ABCD}^{E}a_{E}=0\,\Longleftrightarrow\,\frac{\partial}{\partialx^{D}}C^{ABC}a_{C}=0\,\Longleftrightarrow\,\frac{\partial}{\partialx^{F}}R^{E}{}_{\!AB}{}^{\!D}a_{D}=0\.
J^{(2)}(4-2\varepsilon;1,1)=\mathrm{i}\;\pi^{2-\varepsilon}\;\Gamma(\varepsilon)\;\frac{m_{0}^{1-2\varepsilon}}{\sqrt{k_{12}^{2}}}\;\left\{\Omega_{1}^{(2;4-2\varepsilon)}+\Omega_{2}^{(2;4-2\varepsilon)}\right\},
\frac{d^{2}\phi}{d\rho^{2}}+\frac{5\rho^{4}-\rho_{0}^{4}}{\rho(\rho^{4}-\rho_{0}^{4})}\frac{d\phi}{d\rho}-\frac{\rho^{2}l(l+4)}{\rho^{4}-\rho_{0}^{4}}\phi+\frac{\rho^{4}(\rho^{4}+(\omegaR)^{4})}{(\rho^{4}-\rho_{0}^{4})^{2}}\phi=0\,.
\delta_{\alpha}\Psi_{i_{1}i_{2}...i_{l}}(x)\equiv\frac{\delta\Psi_{i_{1}i_{2}...i_{l}}(x)}{\delta\phi}\delta_{\alpha}\phi,\mathrm{\mathrm{~or}}\delta_{\alpha}\Psi(x,u)\equiv\frac{\delta\Psi(x,u)}{\delta\phi}\delta_{\alpha}\phi.
M_{n}=4\pi\int_{0}^{\infty}dr~r^{2}\{\frac{F_{\pi}^{2}}{8}[F^{2}+2\frac{sin^{2}nF}{n^{2}r^{2}}]+\frac{1}{2e^{2}}~\frac{sin^{2}nF}{r^{2}}~[\frac{sin^{2}nF}{n^{2}r^{2}}+2F^{2}]\}
\Pi_{2,1}^{0,\beta}(m_{0})=\frac{\lambda^{2}}{4(4\pi)^{2}}~\frac{1}{\tilde{\epsilon}}\left[I_{\beta}^{\epsilon}(m_{0})+\frac{m_{0}}{2}\frac{\partial}{\partialm_{0}}\left(I_{\beta}^{\epsilon}(m_{0})\right)\right].
\tilde{Z}_{ij}=i\sigma_{2}\left(\begin{matrix}{|Z|}&{0}\\{0}&{|Z|}\\\end{matrix}\right)\,\qquad\tilde{Z}_{ij}=i\sigma_{2}\left(\begin{matrix}{|Z|}&{0}&{0}&{0}\\{0}&{|Z|}&{0}&{0}\\{0}&{0}&{|Z|}&{0}\\{0}&{0}&{0}&{|Z|}\\\end{matrix}\right)\,
C_{\mu\nu}=\frac{\displaystyle1}{\displaystylea^{2}+b^{2}}\left(aF_{\mu\nu}+bF_{\mu\nu}^{*}\right)\,;B_{\mu\nu}=\frac{\displaystyle1}{\displaystylea^{2}+b^{2}}\left(bF_{\mu\nu}-aF_{\mu\nu}^{*}\right)\,,
Tr\left({\tilde{L}}(\lambda|x_{N},\partial_{x_{N}})\ldots{\tilde{L}}(\lambda|x_{1},\partial_{x_{1}})\right)=\prod_{i=1}^{N}-(\lambda+\beta_{i})\frac{w_{i}(\lambda-1)}{w_{i}(\lambda)}+\prod_{i=1}^{N}\frac{w_{i}(\lambda+1)}{w_{i}(\lambda)}~.
\frac{\partial\varphi}{\partialZ_{a}}=\sum_{i=1}^{m}\frac{d}{d\mu_{i}}\sum_{\mathrm{\scriptsize{part}}\atop\alpha_{1}\inI_{a}}\prod_{b=1}^{N}\sum_{\mathrm{\scriptsize{perm}}}\frac{E_{-\alpha_{1_{b}}}^{b}\ldotsE_{-\alpha_{q_{b}}}^{b}}{(\mu_{1_{b}}-\mu_{2_{b}})\ldots(\mu_{q_{b}}-Z_{b})}|\lambda^{b}>.
e^{iS_{\Lambda-\delta\Lambda}(\phi_{+},\phi_{-})}\equiv\int\prod_{\Lambda_{0}>|{\vec{q}}|>\Lambda-\delta\Lambda}{\calD}[\phi_{+}({\vec{q}},t)]{\calD}[\phi_{-}({\vec{q}},t)]\;e^{iS_{cl}[\phi_{+},\phi_{-}]}
\mathscr{L}_{GF+FP}={\frac{\alpha}{2}}\left(\mathscr{B}+{\frac{1}{\alpha}}\partial_{\mu}\mathscr{A}^{\mu}\right)^{2}-{\frac{1}{2\alpha}}(\partial_{\mu}\mathscr{A}^{\mu})^{2}+i\bar{\mathscr{C}}\cdot\partial_{\mu}\mathscr{D}^{\mu}[\mathscr{A}]\mathscr{C}.
\Pi_{1}(m_{0})=\frac{\lambda}{2}~T\sum_{n}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{\omega_{n}^{2}+\omega_{k}^{2}}=\frac{\lambda}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{1}{\omega_{k}(m_{0})}\left[n_{B}(\omega)+\frac{1}{2}\right],
\chi_{\nu}^{\mu}=-\frac{\kappa^{2}}{2}\frac{1-\Psi_{0}}{\Psi_{0}}\left(\stackrel{A}{T_{\\nu}^{\mu}}+\stackrel{B}{T_{\\nu}^{\mu}}\right)-\frac{l}{2\Psi_{0}}\left(\delta\Psi_{\|\nu}^{|\mu}-\delta_{\nu}^{\mu}\Box\delta\Psi\right)\.
\eta_{i}\cdot\delta_{\perp}X^{\prime\prime}\to\eta_{i}\cdot\delta_{\perp}X^{\prime\prime}|_{flat}+R_{\mu\alpha\nu\beta}X^{\mu}{}^{\prime}X^{\nu}{}^{\prime}\eta^{\alpha}{}_{j}\eta^{\beta}{}_{i}\Psi^{j}\,,
\beta_{1}=\frac{1}{2}\left\langle\sum_{X_{i}\in({\itG}_{1})_{X_{i^{\prime}}^{\prime}}}\sum_{\stackrel{i_{\alpha_{1}}<j_{\alpha_{1}}<i_{\alpha_{2}}<j_{\alpha_{2}}}{\left\{i_{\alpha_{1}},j_{\alpha_{1}},i_{\alpha_{2}},j_{\alpha_{2}}\right\}\inJ_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}{\bf1}_{(w(t_{i_{\alpha_{1}}})=w(t_{j_{\alpha_{1}}}))}\right.\times
S={\frac{T_{(1,0)}}{2}}\intdt\,\left[3(3\ell_{1}-\ell_{2})\dot{\ell_{1}}^{2}-2(3\ell_{1}-\ell_{2})\dot{\ell_{1}}\dot{\ell_{2}}+(\ell_{1}+8\ell_{2})\dot{\ell_{2}}^{2}\right]\,.
H=\sum_{i=1\atopi\mathrm{odd}}^{2N-3}{h_{(i,i+1,i+2)}^{(A)}}\sum_{i=1\atopi\mathrm{odd}}^{2N-1}{h_{(i,i+1)}^{(B)}}+h_{(1)}^{C}+h_{(2N-1,2N)}^{D}.
S_{\mathrm{H}}=\frac{1}{16\pi\alpha^{\prime4}}\intd^{10}x\sqrt{|g_{10}|}e^{-\Phi}\left[R_{10}+\left(\nabla\Phi\right)^{2}-\frac{1}{12}H_{3}^{2}-\frac{1}{4}F_{2}^{2}\right]
E_{D}=-\frac{1}{2(4\pi)^{\frac{p+1}{2}}}\left(\prod_{i=1}^{p}L_{i}\right)\Gamma\left(-\frac{1+p}{2}\right)\sum_{n_{p+1},\dots,n_{D-1}}\left[\nu_{n_{p+1}}^{2}(a_{p+1})+\dots+\nu_{n_{D-1}}^{2}(a_{D-1})\right]^{\frac{1+p}{2}}.
{\frac{M^{2}\pi}{g^{2}N}}=\left(\begin{array}{cccc}{3x+6y}&{0}&{0}&{0}\\{0}&{{\frac{81}{4}}+9y}&{-3i\sqrt{{\frac{3y}{2}}}}&{0}\\{0}&{3i\sqrt{{\frac{3y}{2}}}}&{{\frac{27}{2}}+{\frac{3}{2}}x+3y}&{-{\frac{27}{2\sqrt{2}}}}\\{0}&{0}&{-{\frac{27}{2\sqrt{2}}}}&{{\frac{27}{4}}+3x+{\frac{3}{2}}y}\\\end{array}\right)
{\frac{d^{2}x^{i}}{d\tilde{\lambda}^{2}}}=-{\frac{\kappa^{2}}{6}}\varrho\left[{\frac{\kappa^{4}}{36}}(2+3\omega)\varrho^{2}y_{0}^{2}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}+y_{0}^{2}\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}-2\right]{\frac{dx^{i}}{d\tilde{\lambda}}}.
\Gamma(\phi_{0})=\intd^{3}{\bfx}\left\{\frac{1}{2}(\partial_{i}\phi_{0})^{2}+\frac{1}{2}\mu_{R}^{2}\phi_{0}^{2}+\frac{\lambda_{R}T}{4}\phi_{0}^{4}+\frac{(N+26)}{9}\frac{\lambda_{R}^{3}\zeta(3)}{2^{9}\pi^{4}}\phi_{0}^{6},\right\}.
\phi(g_{1},\ldots,g_{4})=\sum_{N_{1}\ldotsN_{4}}\phi_{\mathcal{B}}{}_{\alpha_{1}\ldots\alpha_{4}}^{N_{1}\ldotsN_{4}}\D_{\alpha_{1}\beta_{1}}^{(N_{1})}(g_{1})\ldotsD_{\alpha_{4}\beta_{4}}^{(N_{4})}(g_{4})\S_{\beta_{1}\ldots\beta_{4}}^{N_{1}\ldotsN_{4}}.
\begin{array}{rl}{<0|T\left(\hat{\phi}(x_{f})\hat{\phi}(x_{i})\right)|0>=}&{\theta(x_{f}^{0}-x_{i}^{0})\,<0|\hat{\phi}(x_{f})\hat{\phi}(x_{i})|0>}\\{}&{}\\{+}&{\theta(x_{i}^{0}-x_{f}^{0})\,<0|\hat{\phi}(x_{i})\hat{\phi}(x_{f})|0>\,}\\\end{array}
{\calT}(x-y)=i\left(\begin{array}{cc}{\langleTO(x)O(y)\rangle}&{\langleO(y-i\beta/2)O(x)\rangle}\\{\langleO(x-i\beta/2)O(y)\rangle}&{\langle\overline{{T}}O(x-i\beta/2)O(y-i\beta/2)\rangle}\\\end{array}\right)~,
g_{00}^{2}=4(h_{0}+{\frac{q_{0}}{r}})\left((h_{1}+{\frac{p_{1}}{r}})(h_{2}+{\frac{p_{2}}{r}})(h_{3}+{\frac{p_{3}}{r}})+a(h_{3}+{\frac{p_{3}}{r}})^{3}\right),
L_{\mathrm{int}}=2g^{2}{\psi}_{-}^{\dagger2}\psi_{-}^{1}\psi_{+}^{\dagger1}\psi_{+}^{2}e^{-2i\rho}+2g^{2}{\psi}_{+}^{\dagger2}\psi_{+}^{1}\psi_{-}^{\dagger1}\psi_{-}^{2}e^{2i\rho}\quad.
S_{fi}=e^{-i[(u+w)/2]^{2}T/2}\!\int\!d^{9}b\,e^{-iq_{\mu}b_{\mu}}\,\langle{\calH}^{3}|\langle{\calH}^{4}|e^{i\Gamma_{\mathrm{(1)}}(u_{\mu}+w_{\mu}/2,b_{\mu},\theta^{3})}|{\calH}^{1}\rangle|{\calH}^{2}\rangle
\frac{d^{m-s}}{dz^{m-s}}\left.\left(\frac{\prod_{j=2}^{k}(z-u_{j})}{z^{k-2}}\right)^{h_{1}}\right|_{z=\zeta}=\left(\frac{\prod_{j=2}^{k}(\zeta-u_{j})}{\zeta^{k-2}}\right)^{h_{1}-m+s}\left[\binom{h_{1}}{m-s}+O(\zeta^{-1})\right]\,,
\mathrm{Tr}[A]=\int\mathrm{d}^{6}\bar{z}\,\left.{A(z,z^{\prime})}\right|_{\bar{z}=\bar{z}^{\prime}},\quad\quad\mathrm{\overline{T}r}[C]=\int\mathrm{d}^{6}z\,\left.{C(z,z^{\prime})}\right|_{z=z^{\prime}}.
{\calS}_{3}=4i({^+\tau}-{^-\tau})\int_{M}eF^{\mu\nui}\,{^\starF}_{\mu\nu}^{j}g_{ij}-4({^+\tau}+{^-\tau})\int_{M}eF^{\mu\nui}F_{\mu\nu}^{j}g_{ij}\,,
\frac{{\delta}^{2}H}{{\delta}{R^{\prime}}^{2}}=\frac{1}{2\pig}\frac{R(1+D^{2}{R^{\prime}}^{2})}{P(1+{R^{\prime}}^{2})^{3/2}},\;\;\frac{{\delta}^{2}H}{{\deltaR}{\delta{R}^{\prime}}}=\frac{1}{2\pig}\frac{R^{\prime}(D^{2}+1)}{P\sqrt{1+{R^{\prime}}^{2}}}=\frac{{\delta}^{2}H}{{\delta{R}^{\prime}}{\deltaR}}
\Omega_{2}^{(1)}(x)=\intdy\\lambda_{i_{1}}^{(1)}(x,y)L_{i_{1}}^{(1)}(y)=-\partial_{1}\alpha_{2}^{(1)}+e\alpha_{3}^{(1)}+e\alpha_{4}^{(1)}+\alpha_{5}^{(1)}=0.
\phi_{int}=-\frac{2m_{1}m_{2}}{\epsilonP\cdotr}\alphaf(\nu)-\frac{\alpha^{2}h(\nu)}{\epsilonP\cdotr}\left(\frac{m_{1}^{2}}{b_{1}}+\frac{m_{2}^{2}}{b_{2}}\right)+O(\alpha^{3}),
\tau^{2}=~~\left(\begin{matrix}{1~}&{~-i~}&{~0~}&{~0}\\{i~}&{~0~}&{~0~}&{~0}\\{0~}&{~0~}&{~0~}&{~-i}\\{0~}&{~0~}&{~i~}&{~0}\\\end{matrix}\right)\quad~;\qquad\tau^{3}=\left(\begin{matrix}{1~}&{~0~}&{~0~}&{~0}\\{0~}&{~-1~}&{~0~}&{~0}\\{0~}&{~0~}&{~1~}&{~0}\\{0~}&{~0~}&{~0~}&{-1}\\\end{matrix}\right)
{\frac{d}{dt}}E={\frac{1}{n!}}\left.\intd\Omega\left[\dot{E_{1}}\left|Y_{a_{1}\cdotsa_{n}}^{(l)}\right|^{2}+\dot{E_{2}}\left|Y_{a_{2}\cdotsa_{n}}^{(l)}\right|^{2}+\dot{E_{3}}\left|Y_{a_{3}\cdotsa_{n}}^{(l)}\right|^{2}\right]\right|_{\rho={\frac{\pi}{2}}},
S(g)=-\frac{1}{8\pi}\int_{z}\mathrm{d}^{2}\sigma\mathrm{tr}(g^{-1}\partial_{\sigma}g)(g^{-1}\partial_{\bar{\sigma}}g)+\frac{1}{24\pi}\int_{{\calB}}\mathrm{d}^{3}\hat{\sigma}i\epsilon^{\hat{i}\hat{j}\hat{k}}\mathrm{tr}(g^{\prime-1}\partial_{\hat{k}}g^{\prime})(g^{\prime-1}\partial_{\hat{j}}g^{\prime})(g^{\prime-1}\partial_{\hat{i}}g^{\prime})
\Phi_{c}(p,J)=\tilde{C}_{\alpha}^{\,\alpha_{0}}(p)\,\Bigl\{\,(2\pi)^{\,4}\,\delta_{\Phi(-p)}\,L^{\,\alpha,\,\alpha_{0}}(\Phi)\,\vert_{\,\Phi=\tilde{C}_{\alpha}^{\,\alpha_{0}}\!J}\,-J(p)\,\Bigr\}
\tau(x_{1},x_{2})={\frac{1}{1+c}}\Bigl[\phi(x_{1})\phi(x_{2})+\hat{g}x_{1}^{-1}(\tau(x_{1},x_{2})-\phi_{yy}(x_{2}))+\hat{g}x_{2}^{-1}(\tau(x_{1},x_{2})-\phi_{yy}(x_{1}))\Bigr]\.
\begin{array}{rclrcl}{F^{(1)m}}&{=}&{2\partialA^{(1)m}\,,}&{H}&{=}&{\partialB-{\textstyle\frac{1}{2}}A^{(1)m}F^{(2)}{}_{m}-{\textstyle\frac{1}{2}}A^{(2)}{}_{m}F^{(1)m}\,,}\\{}&{}&{}&{}&{}&{}\\{F^{(2)}{}_{m}}&{=}&{2\partialA^{(2)}{}_{m}\,,}&{{\calF}_{m}}&{=}&{F^{(2)}{}_{m}+F^{(1)n}B_{nm}\,,}\\\end{array}
c_{I,L}(N)g_{m}^{2L-2}{\frac{F^{I}}{X^{3L+2(I-2)}}}=c_{I,L}(N){\frac{F^{2}}{g_{m}^{2}}}\left({\frac{g_{m}^{2}F^{2}}{X^{7}}}\right)^{L}\left({\frac{F}{X^{2}}}\right)^{I-2L-2}.
\begin{array}{rcl}{[2M]\vertj_{k-1},M\rangle}&{=}&{\displaystyle\frac{t-t^{-1}}{q-q^{-1}}\vertj_{k-1},M\rangle}\\{}&{=}&{\left(ef-fe\right)\vertj_{k-1},M\rangle}\\{}&{=}&{[j_{k-1}-M+1]e\vertj_{k-1},M-1\rangle-[j_{k-1}+M+1][j_{k-1}-M]\vertj_{k-1},M\rangle}\\{}&{}&{-\left[\begin{array}{c}{j_{k}-M-1}\\{j_{k-1}-M}\\\end{array}\right][j_{k}-M]\vertj_{k},M\rangle~.}\\\end{array}
\langlep^{\prime}|J^{\mu}|p\rangle=4N\int\frac{d^{2}k}{(2\pi)^{2}}\frac{(m^{2}-k^{2}+p\cdotp^{\prime})k^{\mu}+(k^{2}-m^{2}-k\cdotp^{\prime})p^{\mu}+(k^{2}-m^{2}-k\cdotp)p^{\prime\mu}}{(k^{2}-m^{2}+i\epsilon)((k-p)^{2}-m^{2}+i\epsilon)((k-p^{\prime})^{2}-m^{2}+i\epsilon)},
\int_{\calM}dx^{2}\epsilon_{ab}\epsilon^{\mu\nu}(E_{\mu}^{a}(q)\Pi^{b}\partial_{\nu}\phi-E_{\mu}^{a}(q)\phi\partial_{\nu}\Pi^{b}+E_{\mu}^{a}(q)E_{\nu}^{b}(q)\Pi_{l}\Pi^{l}).
I=\int_{t_{0}}^{t_{1}}\int_{0}^{1}\Big[x(y^{\prime}\dot{z}-\dot{y}z^{\prime})-H_{1}\Big(\frac{\partialH_{2}}{\partialx}x^{\prime}+\frac{\partialH_{2}}{\partialy}y^{\prime}+\frac{\partialH_{2}}{\partialz}z^{\prime}\Big)\Big]ds\,dt.
\alpha^{\prime}m_{\mathrm{vec}}^{2}\equiv-\alpha^{\prime}\sum_{\sigma,\rho=0}^{p^{\prime}}G^{\sigma\rho}k_{\sigma}k_{\rho}=0~,\qquad\sum_{\sigma,\rho=0}^{p^{\prime}}G^{\sigma\rho}k_{\sigma}\zeta_{\rho}(k)=0~.
\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(\vec{p})=\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}(0)+\sum_{j=1}^{3}\,\sum_{\mu=0}^{3}\,p_{\mu,j}\int_{0}^{1}d\lambda\,\left(\,\partial_{\mu,j}\,\Gamma_{l,s}^{\,\alpha,\,\alpha_{0}}\right)(\lambda\,\vec{p})\quad,
\Pi_{odd}(p)=-\frac{4mg^{2}}{\beta}\sum_{n=-\infty}^{\infty}~\int_{0}^{1}dx\int\frac{d^{2}l}{(2\pi)^{2}}[\frac{1-\frac{1}{2}\{e^{il\tilde{p}}+e^{-il\tilde{p}}\}}{(M_{1}^{2}+((2n+1)\frac{\pi}{\beta}+i\mu)^{2}}~]~~~,
{\calA}^{(3)}=4\pi\widetilde{g}\int\delta\xids\epsilon_{\mu\nu\sigma\rho}\,\dot{\xi}^{\sigma}(s)\,\bar{\psi}(\xi(s))\Omega_{\xi}(s,0)T^{i}\,\Omega_{\xi}^{-1}(s,0)\,\psi(\xi(s))\gamma^{\rho}L_{i}^{\mu\nu}[\xi|s]\,.
\Psi_{E}^{\ell}\left(r,\theta\right)=\left(\begin{array}{l}{A_{\ell}^{+}J_{-a}\left(kr\right)e^{i\ell\theta}}\\{-i\frac{k}{E-m}A_{\ell}^{+}J_{1-a}\left(kr\right)e^{i\left(\ell+1\right)\theta}}\\{A_{\ell}^{-}J_{-a}\left(kr\right)e^{i\ell\theta}}\\{-i\frac{k}{E+m}A_{\ell}^{-}J_{1-a}\left(kr\right)e^{i\left(\ell+1\right)\theta}}\\\end{array}\right)\qquad\mathrm{for}\,a<\frac12\,.
\delta^{(1)}(F_{\mu\nu}^{(0)})_{\overline{{\mathrm{m}}}}=ig[\delta\omega_{\mathrm{m}},(F_{\mu\nu}^{(0)})_{\overline{{\mathrm{m}}}}]-\delta^{(0)}(F_{\mu\nu}^{(1)})_{\overline{{\mathrm{m}}}},
\begin{array}{rcl}{AB\stackrel{X}{\longrightarrow}A\circ_{X}B}&{=}&{(AX)(X^{\dagger}B)\stackrel{Y}{\longrightarrow}}\\{(A\circ_{X}Y)\circ_{X}(Y^{\dagger}\circ_{X}B)}&{=}&{[((AX)(X^{\dagger}Y))X][X^{\dagger}((Y^{\dagger}X)(X^{\dagger}B))]}\\{}&{=}&{[((A(YX))X^{\dagger})X][X^{\dagger}(X((X^{\dagger}Y^{\dagger})B))]}\\{}&{=}&{[A(YX)][(X^{\dagger}Y^{\dagger})B]}\\{}&{=}&{A\circ_{(YX)}B,}\\\end{array}
P_{-}=\int_{-{\infty}}^{\infty}i{\Psi}_{+}^{\ast}(0,x^{-}){\partial}_{-}{\Psi}_{+}(0,x^{-})dx^{-}=\int_{-{\infty}}^{\infty}({\partial}_{-}\tilde{\phi})^{2}dx^{-}.
T_{\mu\nu}^{(0)}=\frac{1}{2\pi}\{2(g_{\mu\nu}\BoxF-\nabla_{\mu}\nabla_{\nu}F)-Ug_{\mu\nu}+2V\nabla_{\mu}\phi\nabla_{\nu}\phi-g_{\mu\nu}V(\nabla\phi)^{2}\}\mathrm{,}
S_{\mathrm{ct}}=-\frac{1}{8\piG}\int_{\cal\partialM}d^{n}x\sqrt{-h}\left[\frac{n-1}{l}+\frac{l}{2(n-2)}{\calR}+\frac{l^{3}}{2(n-4)(n-2)^{2}}\left({\calR}_{ab}{\calR}^{ab}-\frac{n}{4(n-1)}{\calR}^{2}\right)\right],
\partial_{i}V={\frac{1}{4}}\bar{Z}^{CD}\bar{Z}^{AB}P_{ABCD,i}+{\frac{1}{4}}Z_{AB}Z_{CD}P^{ABCD}{}_{,i}+{\frac{1}{2}}\bar{Z}_{I}P_{AB,i}^{I}\bar{Z}^{AB}+{\frac{1}{2}}Z^{I}P_{I,i}^{AB}Z_{AB}=0\.
A=\frac{C_{l}}{2\pi}\int\frac{dt}{t}e^{-(\frac{b^{2}t}{2\pi\alpha^{\prime}})}(8\pi^{2}\alpha^{\prime}t)^{-(\sharpNN)/2}\Theta_{3}(0,itL^{2}/2\pi^{2}\alpha^{\prime})B\timesJ.
j^{1}=-\gamma\frac{i}{2}(\bar{\phi}D_{0}\phi-\phi\overline{{D_{0}\phi}})-\frac{1}{2}B^{2}-\frac{1}{2}|D_{1}\phi|^{2}+\frac{1}{2}|D_{2}\phi|^{2}+\frac{\lambda}{8}(1-|\phi|^{2})^{2}+\gammaa_{0}
K=\left(\begin{array}{ccccc}{\frac{\pi}{\pi\!+\!gN}\frac{1}{-\triangle\!+\!e^{2}N/(\pi\!+\!gN)}}&{}&{}&{}&{}\\{}&{\frac{1}{-\triangle}}&{}&{}&{}\\{}&{}&{.}&{}&{}\\{}&{}&{}&{.}&{}\\{}&{}&{}&{}&{\frac{1}{-\triangle}}\\\end{array}\right)\;.
\frac{1}{a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}}=\frac{\Gamma[\alpha_{1}+\alpha_{2}]}{\Gamma[\alpha_{1}]\Gamma[\alpha_{2}]}\int_{0}^{1}dx\,\frac{x^{\alpha_{1}-1}(1-x)^{\alpha_{2}-1}}{[xa_{1}+(1-x)a_{2}]^{\alpha_{1}+\alpha_{2}}}
\bar{g}(s)=\frac{gs^{1/4}}{1-\frac{\piA(y)b(y)g}{y}(s^{1/4}-1)}=\frac{g_{IR}^{*}|\tilde{g}|s^{1/4}}{1+|\tilde{g}|s^{1/4}},\\\\\\\tilde{g}=\frac{g}{g-g_{IR}^{*}}\\in\\(-\infty,0).
\frac{1}{16\pi}\int_{B_{e}}\left(2\Lambda-F^{2}\right)\sqrt{g}d^{4}x=\beta_{RN}\Lambda\left[\frac{r_{s}^{3}-r_{h}^{3}}{12}\right]\,+\,\frac{q^{2}}{4}\beta_{RN}\left(\frac{1}{r_{h}}-\frac{1}{r_{s}}\right)
S=\frac{\piu}{\lambda^{2/3}}\left[\frac{|m^{i}+\taun^{i}+\omegap^{i}|^{2}}{\tau_{2}}+\lambda^{2}\left(p^{i}\right)^{2}\right]-\piu^{3}+2\piiC_{ijk}m^{i}n^{j}p^{k}\,,
{\cal{V}}=-\frac{\pi^{2}}{V}\frac{s(N-s)}{N}+\frac{V}{2\pi^{2}}\left(\frac{4\pi}{V}\sum_{i=1}^{s}\sum_{j=s+1}^{N}|A_{+}^{(ji)}|^{2}+\frac{1}{2\pi^{2}}\sum_{k}(B^{k})^{2}+\frac{1}{\pi^{2}}\sum_{i}\sum_{j>i}|X^{(ij)}|^{2}\right).
W=\left(\begin{array}{ccc}{1}&{0}&{0}\\{\phi_{2}}&{1}&{0}\\{f}&{\phi_{1}}&{1}\\\end{array}\right)\left(\begin{array}{ccc}{\lambda_{1}^{-1}\lambda_{2}^{-1}}&{0}&{0}\\{0}&{\lambda_{2}^{-1}}&{0}\\{0}&{0}&{\lambda_{1}}\\\end{array}\right)\left(\begin{array}{ccc}{1}&{\Phi_{1}}&{F}\\{0}&{1}&{\Phi_{2}}\\{0}&{0}&{1}\\\end{array}\right),
{B}_{L}=\frac{\ell}{16\piG}\int_{|r|=\bar{r}}\left[\partial_{a}(\sqrt{-\stackrel{(0)}{\bfg}}\stackrel{(0)}g\strut^{ab}\partial_{b}\phi)+\partial_{\varphi}(\omega_{t}\phi)-\partial_{t}(\omega_{\varphi}\phi)\right]d\varphidt\qquad,
\lambda=2\bigl(z-z_{1}\bigr){\frac{dx}{y}}-{\frac{1}{2}}m^{2}\theta_{2}^{4}\theta_{3}^{4}\bigl(z-z_{1}\bigr){\frac{dx}{y\bigl(x+{\frac{1}{4}}m^{2}\theta_{2}^{4}\theta_{3}^{4}\bigr)}}\cdotp
\zeta_{\overline{{\Theta}}}(s)=\mathrm{Tr}\\overline{{\Theta}}^{-s}=\sum_{a}\sum_{\mu}\intd^{4}x\sqrt{g}\,mbox{tr}\,\langlex|\left(\overline{{\Theta}}_{\mu}^{a}\right)^{-s}|x\rangle
L_{int}(\phi;\Phi)={\frac{\lambda_{\Phi}}{24}}\Phi^{4}+{\frac{\lambda_{\Psi}}{24}}\Psi^{4}+{\frac{\lambda_{\Phi}}{6}}\phi\Phi^{3}+{\frac{\lambda_{\Phi\Psi}}{2}}\phi\Phi^{3}~,
\delta_{B}(\left[v^{3}\theta^{2}\right])=-iv^{2}\left(\left(\frac{d}{\phid\phi}\right)f_{1}^{(2)}v^{i}\phi^{j}\phi^{k}+f_{1}^{(2)}v^{i}\delta^{jk}\right)(\epsilon\gamma^{k}\theta)(\theta\gamma^{ij}\theta).
\left[\delta_{1},\delta_{2}\right]\psi_{\alpha}=\left(\varepsilon_{2}^{\prime}\sigma_{\nu}\bar{\varepsilon}_{1}^{\prime}\,-\varepsilon_{1}^{\prime}\sigma_{\nu}\bar{\varepsilon}_{2}^{\prime}\right)\partial^{\nu}\tilde{\partial}_{+}\psi\,_{\alpha}\,\,\,,
\delta(q_{2}-\tilde{q}_{2}-\Theta)=\int_{-\infty}^{\infty}d\rho\,e^{i\rho(q_{2}-\tilde{q}_{2}-\Theta)}=\int_{-\infty}^{\infty}d\rho\,e^{-i\rho\Theta}\,e^{i\rho(q_{2}-\tilde{q}_{2})}.
H_{n}(z_{2}^{\prime},z_{3}^{\prime},z_{A}^{\prime},\mu,\beta_{A}(z_{2}^{\prime},z_{3}^{\prime},z_{A}^{\prime},\mu))=-\frac{1}{2}\beta_{n}(z_{2}^{\prime},z_{3}^{\prime},z_{A}^{\prime},\mu).