harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\chi_{1}^{a}=\frac{\partial_{j}A_{j}^{a}}{\nabla^{2}}\,;\quad\chi_{2}^{a}=f^{abc}\frac{\partial_{j}}{\nabla^{2}}\left(\chi_{1}^{b}A_{j}^{c}+\frac12(\partial_{j}\chi_{1}^{b})\chi_{1}^{c}\right)\,;\quad\dots
\begin{array}{ll}{ds^{2}=}&{-\left(1-\frac{r_{H}^{7-p}+\alpha}{r^{7-p}}\right)dt^{2}+\left(1+\frac{\alpha}{r^{7-p}}\right)dx^{\scriptscriptstyle{10}^{\scriptstyle{2}}}\mp2\frac{\sqrt{\alpha\left(r_{H}^{7-p}+\alpha\right)}}{r^{7-p}}dtdx^{10}}\\{}&{+\frac{dr^{2}}{1-\left(\frac{r_{H}}{r}\right)^{7-p}}+r^{2}d\Omega_{8-p}^{2}+dy^{s}dy_{s},}\\\end{array}
\begin{array}{rclcrcl}{\displaystyle[t_{a},\,t_{b}]}&{=}&{iF_{abc}t_{c}}&{,}&{F_{ab0}}&{=}&{0}\\{\displaystyle\{t_{a},\,t_{b}\}}&{=}&{D_{abc}t_{c}}&{,}&{D_{ab0}}&{=}&{\displaystyle\sqrt{\frac{2}{N}}\delta_{ab}}\\\end{array}
\begin{array}{l}{P^{-ab}P^{-cd}=P^{-cb}P^{-ad},}\\{(\partial_{a}\bar{\theta}\Gamma^{\mu}\partial_{b}\theta)(\Lambda\Pi_{c})=-\displaystyle\frac12\partial_{a}\bar{\theta}\Gamma^{\mu}\{\tilde{\Lambda},\tilde{\Pi}_{c}\}\partial_{b}\theta,}\\\end{array}
E_{i}^{\Lambda_{i}+1}(z)\Phi_{L.W.}^{(\lambda,\mu)}(w)=\frac{-[\Lambda_{i}+\nu(\vec{\alpha}_{i},\vec{\beta})]}{(z-w)}a_{ii}^{*^{\Lambda_{i}}}\Phi_{L.W.}^{(\lambda,\mu)}(w)+R.T..
h_{ab}=\left(\begin{array}{cc}{-\alpha^{2}+\frac{\beta^{2}}{\gamma}}&{\beta}\\{\beta}&{\gamma}\\\end{array}\right)~,~h^{ab}=\left(\begin{array}{cc}{-\frac{1}{\alpha^{2}}}&{\frac{\beta}{\alpha^{2}\gamma}}\\{\frac{\beta}{\alpha^{2}\gamma}}&{\frac{1}{\alpha}-\frac{\beta^{2}}{\alpha^{2}\gamma^{2}}}\\\end{array}\right),
H:=\{Q,Q^{\dagger}\}=\frac{\hbar^{2}}{m}\left(\begin{array}{cc}{bb^{\dagger}}&{0}\\{0}&{b^{\dagger}b}\\\end{array}\right)=\left(\begin{array}{cc}{H_{+}}&{0}\\{0}&{H_{-}}\\\end{array}\right)\quad,
S=\int_{M}d^{3}x\;\epsilon^{\sigma\mu\nu}\left[{B_{\sigma}}\!^{a}({F_{\mu\nu}}\!^{a}+\epsilon^{abc}{\phi_{\nu}}\!^{b}{\phi_{\mu}}\!^{c})+{\eta_{\sigma}}\!^{a}D_{[\mu}{\phi_{\nu]}}^{a}\right]\;.
\begin{array}{rl}{\Phi_{TT}^{(l=1)}\left[\Gamma^{[r,0]}\right]=}&{\Phi_{TT}^{(A)}\left[D^{[r,0]},\Gamma_{VTV}^{[r,0]}\right]+\Phi_{TT}^{(B)}\left[{\tilde{D}}^{[r,0]},\Gamma_{GT\overline{{G}}}^{[r,0]}\right]}\\{\mathrm{}}&{+\Phi_{TT}^{(C)}\left[D^{[r,0]}\right]+\sum_{F}\,\,\Phi_{TT}^{(D)}\left[S_{F}^{[r,0]},\,\Gamma_{FT\overline{{F}}}^{[r,0]}\right]\,.}\\\end{array}
\left(V_{\zeta_{1}}^{cl}\otimes{\calV}_{j_{1}}\right)\bigotimes\left(V_{\zeta_{2}}^{cl}\otimes{\calV}_{j_{2}}\right)=\left(\bigoplus_{\zeta_{1}+\zeta_{2}\le\zeta_{3}}V_{\zeta_{3}}^{cl}\right)\bigotimes\left(\bigoplus_{j_{3}=\vertj_{1}-j_{2}\vert}^{\mathrm{min}\,\{j_{1}+j_{2},p-2-J_{1}-j_{2}\}}{\calV}_{j_{3}}\right).
u^{\prime\prime}(z)+\left[\frac{1}{4}\sum_{\nu=1}^{4p}\frac{1-\alpha_{\nu}^{2}}{(z-\alpha_{\nu})^{2}}+\frac{1}{2}\sum_{\nu=1}^{4p}\frac{\beta_{\nu}}{z-\alpha_{\nu}}\right]u(z)=0~~.
\frac{1}{\sqrt{2}}\left(\begin{array}{cccccc}{}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{-i}&{}\\{}&{}&{}&{}&{}&{}\\{}&{i}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}\\\end{array}\right)\;.
{\calH}={\frac{1}{2}}(\pi_{\phi}-eA_{1})^{2}+{\frac{1}{2}}\pi_{1}^{2}+{\frac{1}{2}}\phi^{2}-{\frac{ag^{2}}{2}}A_{1}^{2}-{\frac{1}{2ae^{2}}}(\pi_{1}^{\prime}+e\phi^{\prime})^{2}.
S=\intd^{4}x\sqrt{g}\left({\frac{m_{s}^{8}v_{1}v_{2}v_{3}}{(2\pi)^{7}\lambda^{2}}}R+{\frac{1}{4}}{\frac{m_{s}^{6}v_{1}v_{2}v_{3}}{(2\pi)^{7}\lambda}}F^{2}+{\frac{1}{4}}\sum_{i=1}^{3}{\frac{m_{s}^{2}v_{i}}{(2\pi)^{3}\lambda}}\tilde{F}_{i}^{2}+\dots\right)
\int\frac{d\omega^{\prime\prime}}{2\pi}\int\frac{dk^{\prime\prime}}{2\pi}\frac{i}{\omega^{2}-k^{2}-m^{2}+i\epsilon}\left[1+K_{b}^{+}(k^{\prime\prime})e^{-2ik^{\prime\prime}x^{\prime\prime}}\right].
K_{\nu}(z)=\sqrt{\frac{\pi}{2z}}\frac{e^{-z}}{\Gamma(\nu+\frac{1}{2})}\int_{0}^{\infty}ds\,e^{-s}s^{\nu-\frac{1}{2}}\left[1+\frac{s}{2z}\right]^{\nu-\frac{1}{2}},
S_{\mathrm{loc};\,0}[e^{\sigma(x)}g(x),\phi(x)]-S_{\mathrm{loc};\,0}[g(x),\phi(x)]=\intd^{d}x\,\sigma(x)\,g_{ij}\,{\frac{\deltaS_{\mathrm{loc};0}}{\deltag_{ij}}}.
n_{\mathrm{eff}}^{2}(n,\eta)\simeqn^{2}-n^{2}\biggl(\frac{n}{n_{\mathrm{inf}}}\biggr)^{2}(1-n_{\mathrm{b}}^{2}\eta^{2}),\quad\frac{a^{\prime\prime}}{a}\simeqn_{\mathrm{b}}^{2}\biggl[1-6\biggl(\frac{\eta}{\eta_{0}}\biggr)^{2}\biggr].
S=\intd^{4}x\sqrt{|g|}\left[e^{-\varphi}\left(R+\left(\nabla\varphi\right)^{2}-\frac{1}{2}\left(\nabla\beta\right)^{2}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}\right)-\frac{1}{48}e^{\sqrt{3}\beta}F_{\mu\nu\lambda\kappa}F^{\mu\nu\lambda\kappa}\right]
G(x)=\psi_{0}(x)I_{4}-\psi_{\mu}(x)\gamma_{\mu}+\frac12\psi_{[\mu\nu]}(x)\gamma_{[\mu}\gamma_{\nu]}+i\widetilde{\psi}_{\mu}(x)\gamma_{\mu}\gamma_{5}-i\widetilde{\psi}_{0}(x)\gamma_{5}.
{\mathcal{D}}{\mit\Pi}^{(\alpha)}{\mathcal{D}}A^{(\alpha)}{\mathcal{D}}\varepsilon{\mathcal{D}}\beta\prod_{x}\delta(\varepsilon)\delta(\beta)={\mathcal{D}}{E}^{i}{\mathcal{D}}B^{i}\prod_{x}\delta({\nabla}^{i}{E}^{i}-iJ_{4})\delta({\nabla}^{i}B^{i}).
\dot{\xi}^{i}=\partial_{\mu}\xi^{i}\dot{x}^{\mu}=\partial_{\mu}\xi^{i}\frac{p^{\mu}}{N\cdotp}=\frac{p^{\mu}e_{\mu}^{\;\;i}}{N\cdotp}\equiv\frac{\pi^{i}}{N\cdotp}\;,
_2F_{1}[a,b;c;z]=\sum_{l=0}^{\infty}{\frac{(a)_{l}(b)_{l}}{(c)_{l}}}{\frac{z^{l}}{l!}}=1+{\frac{ab}{c}}{\frac{z}{1!}}+{\frac{a(a+1)b(b+1)}{c(c+1)}}{\frac{z^{2}}{2!}}+\cdots.
{\bfi.}~~~A_{1}=\displaystyle\frac{\partialG(S_{3})}{\partialS_{3}}~,~~~A_{2}=-A_{3}=2i\displaystyle\frac{\partial^{2}G(S_{3})}{\partial^{2}S_{3}}~,~~~A_{4}=i~,~~~A_{5}=-4\displaystyle\frac{\partial^{3}G(S_{3})}{\partial^{3}S_{3}}
A_{hh}^{0}+C_{t}^{0}+C_{s}^{0}={\frac{-i\kappa^{2}cT_{p}}{2}}\bigg\{Tr(\epsilon_{1}.D)Tr(\epsilon_{2}.D)+Tr(\epsilon_{1}.\epsilon_{2})-Tr(\epsilon_{1}.D.\epsilon_{2}.D)\bigg\},
S_{BH}=2\pi\left(\sqrt{N_{5}^{(1)}}+\sqrt{\bar{N}_{5}^{(1)}}\right)\left(\sqrt{N_{5}^{(2)}}+\sqrt{\bar{N}_{5}^{(2)}}\right)\left(\sqrt{N_{5}^{(3)}}+\sqrt{\bar{N}_{5}^{(3)}}\right)\left(\sqrt{n}+\sqrt{\bar{n}}\right)
{\calK}={\frac{1}{2{\mathrm{dim}}({\widetildeG})}}\sum_{a=1}^{{\mathrm{\small{dim}}}({\widetildeG})}{\calK}_{a}={\frac{1}{2{\mathrm{dim}}({\widetildeG})}}\sum_{a=1}^{{\mathrm{\small{dim}}}({\widetildeG})}{\mathrm{Tr}}\left(\Omega{\widetildeJ}_{a}q^{L_{0}}{\overline{q}}^{{\overline{L}}_{0}}\right)~.
\begin{array}{rl}{g_{\mu\nu}}&{\rightarrow(3,3)+58(1,1)\nonumber}\\{\phi}&{\rightarrow(1,1)\nonumber}\\{A_{\mu}}&{\rightarrow(2,2)\nonumber}\\{B_{\mu\nu}}&{\rightarrow(3,1)+(1,3)+22(1,1)\nonumber}\\{C_{\mu\nu\rho}}&{\rightarrow23(2,2).}\\\end{array}
\begin{array}{rclrcl}{\chi_{0}(\tau)}&{=}&{\displaystyle\frac12\left(\frac{\Theta_{1,2}(\tau)}{\eta(\tau)}+\eta(\tau)^{2}\right),}&{\chi_{1}(\tau)}&{=}&{\displaystyle\eta(\tau)^{-1}\Theta_{0,2}(\tau),}\\{\chi_{2}(\tau)}&{=}&{\displaystyle\frac12\left(\frac{\Theta_{1,2}(\tau)}{\eta(\tau)}-\eta(\tau)^{2}\right),}&{\chi_{3}(\tau)}&{=}&{\displaystyle\eta(\tau)^{-1}\Theta_{2,2}(\tau).}\\\end{array}
{\calH}_{\mathrm{fast}}^{n=2}\simeq2g\phi\;(a_{j}^{\dagger}a_{j}+\theta_{a}^{\dagger}\theta_{a}+b_{j}^{\dagger}b_{j})+(\phi-m_{I})\chi_{I}^{\dagger}\chi_{I}+g\phi\;(8-N)\.
\chi_{i}^{\dagger}\gamma_{0}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\ldots\gamma_{\mu_{n-1}}\gamma_{\mu_{n}}\psi_{j}=-\epsilon_{il}\epsilon_{jk}\psi_{k}^{\dagger}\gamma_{0}\gamma_{\mu_{n}}\gamma_{\mu_{n-1}}\ldots\gamma_{\mu_{2}}\gamma_{\mu_{1}}\chi_{l}.
C=\left(\begin{array}{ccccc}{0}&{0}&{\cdots}&{0}&{C^{1N_{1}}}\\{0}&{0}&{\cdots}&{C^{2(N_{1}-1)}}&{C^{2N_{1}}}\\{\vdots}&{\vdots}&{{}}&{\vdots}&{\vdots}\\{0}&{C^{(N_{1}-1)2}}&{\cdots}&{C^{(N_{1}-1)(N_{1}-1)}}&{C^{(N_{1}-1)N_{1}}}\\{C^{N_{1}1}}&{C^{N_{1}2}}&{\cdots}&{C^{N_{1}(N_{1}-1)}}&{P^{N_{1}N_{1}}}\\\end{array}\right)
h_{~\nu}^{\mu}=\left[\begin{array}{ccccc}{h_{1}}&{h_{3}}&{0}&{0}&{0}\\{-d^{2}h_{3}/f}&{h_{2}}&{0}&{0}&{0}\\{0}&{0}&{h_{~\chi}^{\chi}}&{h_{~\theta}^{\chi}}&{h_{~\phi}^{\chi}}\\{0}&{0}&{h_{~\chi}^{\theta}}&{h_{~\theta}^{\theta}}&{h_{~\phi}^{\theta}}\\{0}&{0}&{h_{~\chi}^{\phi}}&{h_{~\theta}^{\phi}}&{h_{~\phi}^{\phi}}\\\end{array}\right]
dl^{2}=\frac{1}{2f^{2}}\left\{df^{2}+(d\chi+vdu-udv)^{2}\right\}-\frac{1}{f}\left\{e^{-2\phi}dv^{2}+e^{2\phi}(du-\kappadv)^{2}\right\}+2d\phi^{2}+\frac{1}{2}e^{4\phi}d\kappa^{2}.
T^{L}=\left(\begin{matrix}{1}&{0}\\{0}&{0}\\\end{matrix}\right),~~~T^{R}=\left(\begin{matrix}{0}&{0}\\{0}&{1}\\\end{matrix}\right),~~~Q=\left(\begin{matrix}{0}&{1}\\{0}&{0}\\\end{matrix}\right),~~~Q^{\dagger}=\left(\begin{matrix}{0}&{0}\\{1}&{0}\\\end{matrix}\right).
A_{\mu}^{(cl)}=\left(\begin{array}{ccc}{X_{\mu}^{(1)}}&{}&{}\\{}&{X_{\mu}^{(2)}}&{}\\{}&{}&{X_{\mu}^{(3)}}\\\end{array}\right)=\sum_{a=0}^{2}X_{\mu}^{a}\otimesU^{a}.
\frac{(2\pi)^{D/2}}{GN\Lambda^{D-2}}=\frac{1-(eH/\Lambda^{2})^{D/2-1}}{D/2-1}+(eH/\Lambda^{2})^{D/2-1}\int_{\rho^{2}/eH}^{\infty}\frac{ds}{s}e^{-s}.
\begin{array}{rcrl}{\left[L_{0}^{(i)}\,,\,L_{\pm}^{(i)}\right]}&{=}&{\pm\,L_{\pm}^{(i)}}&{i=1,2,3}\\{\left[L_{+}^{(i)}\,,\,L_{-}^{(i)}\right]}&{=}&{2\,L_{0}^{(i)}}&{i=1,2,3\nonumber}\\{\left[L_{A}^{(i)}\,,\,L_{B}^{(j)}\right]}&{=}&{0}&{i\neqj}\\\end{array}
\Sigma_{i}=\frac{i}{4}f_{\hat{\alpha}\hat{\beta}}^{i}\gamma^{\hat{\alpha}}\gamma^{\hat{\beta}}=S_{i}-\frac{1}{2}\gamma^{5}\gamma^{i}=\left(\begin{array}{cc}{\sigma_{i}}&{0}\\{0}&{0}\\\end{array}\right)\,,
\tilde{I}_{\mathrm{D}}=\frac{1}{16\pi}\intd^{D}x\sqrt{-g}\left(R-\frac{1}{2}(\nabla\phi)^{2}-\frac{e^{\alpha\phi}}{2(\tilde{d}+1)!}\tilde{F}_{\tilde{d}+1}^{2}\right)-\frac{1}{8\pi}\intd^{D-1}x\sqrt{-h}K.
\Phi=(m+\frac{1}{2})\pi,\,\,\,\,\mathrm{and}\,\,\,\,\,\,\,\,\gamma=m^{\prime}+\frac{1}{2},\,\,\,\,\mathrm{with}\,\,\,\,\,\,m,m^{\prime}\in\,\,\,\,\mathrm{{\bf~Z}},
I_{N}={\frac{1}{N}}\,\int_{0}^{N}\!dy\,\Bigl({\frac{1}{\sqrt{1-e^{-y}}}}-1\Bigr)-{\frac{1}{4N^{2}}}\,\int_{0}^{N}\!dy\,{\frac{y^{2}e^{-y}}{(1-e^{-y})^{3/2}}}+{\calO}\bigl(1/N^{3}\bigr).
S_{0}[B_{\mu\nu}(x),A_{\mu}(x)]=\intd^{4}x\;\Bigg\{\frac{1}{4}\epsilon^{\mu\nu\rho\sigma}B_{\mu\nu}(x)F_{\rho\sigma}(x)+\frac{1}{2}A_{\mu}(x)A^{\mu}(x)\Bigg\},
\begin{array}{rclcrclcrcl}{\tilde{\gamma}\,^{m}}&{\equiv}&{\gamma_{ijl}\,\epsilon^{ijlm}}&{;}&{\Pi_{1}^{m}}&{\equiv}&{e^{\phi/2}\mathcal{F}^{lm}\gamma_{l}}&{;}&{\Pi_{2}^{m}}&{\equiv}&{e^{-\phi/2}\tilde{F}^{lm}\gamma_{l}}\\\end{array}
\gamma_{ij}=\Pi_{i}^{a}\,\Pi_{j}^{b}\,\eta_{\underline{{ab}}}\quad;\quadG_{ij}=V_{i}^{\underline{{a}}}\,V_{j}^{\underline{{b}}}\,\eta_{\underline{{ab}}}\quad;\quad\widehat{G}=\eta\,G\,\eta
\intd^{4}x\int_{0}^{L}dyM^{3}\partial_{M}\left(\sqrt{-G}W^{M}\right)=\intd^{4}x\sqrt{-G}M^{3}W^{5}\Big|_{y=L}-\intd^{4}x\sqrt{-G}M^{3}W^{5}\Big|_{y=0}\,
R_{i_{1}i_{2}i_{3}}^{j_{1}j_{2}j_{3}}=\omega^{j_{1}(i_{3}-j_{3})}{\frac{\displaystylew(p_{2},p_{12},p_{1}|i_{3}-j_{3})w(p_{4},p_{34},p_{3}|i_{1}-j_{1})}{\displaystylew(p_{6},p_{56},p_{5}|j_{2}-i_{2})}}
\Psi^{(+)}({\bfr})=e^{i{\bfk}\cdot{\bfr}}+\intd^{{D}}r^{\prime}G_{{D}}^{(+)}({\bfr},{\bfr}^{\prime};k)V({\bfr}^{\prime})\Psi^{(+)}({\bfr}^{\prime})\;,
{\calL}=\bar{\psi}i\gamma^{\mu}D_{\mu}\psi+{\frac{G}{2N}}\left[\left(\bar{\psi}\psi\right)^{2}+\left(\bar{\psi}i\gamma_{5}\psi\right)^{2}+\left(\bar{\psi}\gamma_{3}\psi\right)^{2}\right]
\varrho_{*}\Big(\sum_{k=1}^{n}\,f_{k}\,d^{\prime}f_{k}^{\prime}\Big)\,=\,\sum_{k=1}^{n}\,\big(r_{U_{\iota}\capU}^{U}(f_{k})\,\#\,1\big)\,d\big(r_{U_{\iota}\capU}^{U}(f_{k}^{\prime})\,\#\,1\big),
\{G_{Q}(\eta),G_{Q}(\rho)\}^{*}=\int_{\Sigma}[\eta,\rho]^{a}G_{a}+\int_{\partial\Sigma}\omega\mathrm{\tiny~\wedge\,~}[\eta,\rho]_{a}A^{a}+\int_{\partial\Sigma}\omega\mathrm{\tiny~\wedge\,~}\eta_{a}d\rho^{a}
<A_{2}|\sum_{k_{1},k_{2}=-\infty}^{+\infty}\hat{U}(k_{1},k_{2})|A_{1}>=\frac{2\pi}{k}\,\frac{1}{\sqrt{k}}\,\sum_{n_{1},n_{2}=-\infty}^{+\infty}\,e^{-\frac{2i\pi}{k}n_{1}n_{2}}\,\delta\left(A_{2}-\frac{2\pin_{1}}{k}\right)\,\delta\left(A_{1}-\frac{2\pin_{2}}{k}\right)\\\.
\Psi\left(x\rightarrow+\infty\right)=e^{ipx}\left[C\left(\begin{array}{c}{0}\\{1}\\\end{array}\right)-\frac{im}p\left(\begin{array}{c}{1}\\{0}\\\end{array}\right)\intV\left(x^{\prime}\right)\psi^{+}\left(x^{\prime}\right)\psi^{-}\left(x^{\prime}\right)dx^{\prime}\right],
\sum_{n=1}^{\infty}\frac{1}{n^{2}z^{2}}\,\frac{1}{(t^{2}+n^{2}z^{2}/4)}=\frac{\pi^{2}}{6z^{2}t^{2}}+\frac{1}{8t^{4}}-\frac{\pi}{4zt^{3}}\,\frac{1}{\mathrm{th}(2\pit/z)}.
\left[e^{2k|z|}\Box^{(4)}+\partial_{z}^{2}-4k^{2}+4k\delta(z)-4k\delta(z-z_{c})\right]G_{R}(x,z;x^{\prime},z^{\prime})=\delta^{(4)}(x-x^{\prime})\delta(z-z^{\prime}).
g^{\mu\nu}=\left(\begin{array}{lc}{e^{U}\omega_{i}\omega_{j}\gamma^{ij}+e^{-U}}&{-e^{U}\omega_{i}\gamma^{ij}}\\{-e^{U}\omega_{j}\gamma^{ij}}&{e^{U}\gamma^{ij}}\\\end{array}\right)\,
F[x,y]=\int{\frac{d^{2}\vec{\alpha}}{\pi^{N}}}\int{\frac{d^{2}\vec{\beta}}{\pi^{N}}}\int{\frac{d^{2}\vec{\beta}^{\prime}}{\pi^{N}}}\;\rho_{R}(\vec{\beta}^{*},\vec{\beta}^{\prime})\;e^{-\mid\vec{\alpha}\mid^{2}-\frac{\mid\vec{\beta}\mid^{2}}{2}-\frac{\mid\vec{\beta}^{\prime}\mid^{2}}{2}}
ds^{2}=\left(1+\frac{D}{r_{1}}\right)^{1/2}\left(1-\frac{D}{r_{2}}\right)^{1/2}dt^{2}-\left(1+\frac{D}{r_{1}}\right)^{-1/2}\left(1-\frac{D}{r_{2}}\right)^{-1/2}d\vec{x}^{2}\,.
\begin{array}{rclcc}{\ell(r)}&{=}&{r^{3}\,\frac{2\,\sqrt{a^{2}+1}}{a^{2}-1}\,\frac{d}{dr}\,\left[H(r)\right]^{\frac{1-a^{2}}{1+a^{2}}}}&{;}&{\mathrm{electric~0~-brane}}\\{p}&{=}&{-\frac{4}{\sqrt{a^{2}+1}}\,k}&{;}&{\mathrm{magnetic~solitonic~0~-brane}}\\\end{array}
(X,Y)=\intd^{n}x\left(\frac{\delta^{R}X}{\delta\Phi^{M}(x)}\frac{\delta^{L}Y}{\delta\Phi_{M}^{*}(x)}-\frac{\delta^{R}X}{\delta\Phi_{M}^{*}(x)}\frac{\delta^{L}Y}{\delta\Phi^{M}(x)}\right)
M_{BPS}^{2}={\textstyle{\frac{1}{2}}}e^{-2\phi_{\infty}}{\vec{\beta}}^{T}\mu_{R}{\vec{\beta}}+{\textstyle{\frac{1}{2}}}e^{2\phi_{\infty}}{\vec{\tilde{\alpha}}^{T}}\mu_{R}{\vec{\tilde{\alpha}}}+\left[({\vec{\beta}}^{T}\mu_{R}{\vec{\beta}})({\vec{\alpha}}^{T}\mu_{R}{\vec{\alpha}})-({\vec{\beta}}^{T}\mu_{R}{\vec{\alpha}})^{2}\right]^{\frac{1}{2}},
1-\frac{e^{\widehat{u}}}{S_{1}}-\frac{e^{\widehat{v}}}{S_{2}}-S_{1}e^{-t_{1}-\widehat{u}}-\frac{S_{2}}{S_{1}^{2}}e^{-t_{2}+2\widehat{u}-\widehat{v}}+\frac{S_{1}}{S_{2}}e^{-t_{3}+\widehat{v}-\widehat{u}}+\frac{S_{2}}{S_{1}}e^{-t_{4}+\widehat{u}-\widehat{v}}=0\,.
b^{\pm}(\sigma)=\sum_{n=-\infty}^{\infty}\hat{b}_{n}e^{\pmin\sigma}=\pi_{c}(\sigma)\mpib(\sigma)\,,\qquadc^{\pm}(\sigma)=\sum_{n=-\infty}^{\infty}\hat{c}_{n}e^{\pmin\sigma}=c(\sigma)\pmi\pi_{b}(\sigma)\,.
\left(\begin{matrix}{\Gamma_{S}^{11}(p)}&{\Gamma_{S}^{12}(p)}\\{\Gamma_{S}^{21}(p)}&{\Gamma_{S}^{22}(p)}\\\end{matrix}\right)={\bf\sfM}^{-1}\left(\begin{matrix}{\Gamma_{RF}^{\phi_{S}}(p)}&{0}\\{0}&{{\Gamma_{RF}^{\phi_{S}}}^{*}(p)}\\\end{matrix}\right){\bf\sfM}^{-1},
0=\intd^{N^{2}}Xd^{N^{2}}Y\sum_{\alpha}\frac{\partial}{\partialX_{\alpha}}\left[\mathrm{tr}\frac{1}{\xi^{\prime}-X}\mathrm{tr}\left(\frac{1}{\xi-X}Yt^{\alpha}Y\right)\mathrm{e}^{-S_{D}}\right],
Z^{(1)}(k;l_{1},l_{2})=(-1)^{2l_{1s}+2l_{2s}-1}16^{2l_{1s}}\prod_{p=1}^{\infty}\frac{[1+(-1)^{2l_{2}}k^{p}k^{(2l_{1}-1)/2}]^{8}}{[1-k^{p}]^{8}}\,.
\left(\begin{array}{c}{x^{\prime}}\\{t^{\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{\frac{1}{\sqrt{1-\beta^{2}}}}&{\frac{\beta}{\sqrt{1-\beta^{2}}}}\\{\frac{\beta}{\sqrt{1-\beta^{2}}}}&{\frac{1}{\sqrt{1-\beta^{2}}}}\\\end{array}\right)\left(\begin{array}{c}{x}\\{t}\\\end{array}\right)
V_{restored}^{ring}(H_{Y},T)=-\frac{T}{12\pi}[\frac{2}{3}(g^{\prime}/2)^{2}T^{2}+m_{D_{f}}^{2}-\frac{((g^{\prime}/2)H_{Y})^{1/2}T}{2\pi}-\frac{1}{8\pi^{2}}(g^{\prime}/2)H_{Y}]^{3/2},
S_{,ij}^{G}=-{\calG}_{ij}^{-1}+\left\{\begin{array}{c}{k}\\{i\j}\\\end{array}\right\}{\calE}_{k}+\frac{\delta\chi^{\alpha}}{\delta\Phi^{i}}c_{\alpha\beta}\frac{\delta\chi^{\beta}}{\delta\Phi^{j}}.
\begin{array}{ccc}{\varepsilon^{i}=\left(\begin{array}{cc}{\bar{\varepsilon}_{a}}&{0}\\{0}&{\varepsilon^{a}{}^{t}}\\\end{array}\right)~~~~}&{\mathrm{or~}}&{~~~\varepsilon^{i}=\left(\begin{array}{cc}{0}&{-\bar{\varepsilon}_{a}}\\{\varepsilon^{a}{}^{t}}&{0}\\\end{array}\right)}\\\end{array}
V=\sum_{n,m,m^{\prime}}\sum_{k_{1},\ldotsk_{n}}^{\prime}\sum_{j_{1},\ldotsj_{n},}V_{\left\{m\right\},\left\{m^{\prime}\right\}}^{n}\left(k_{1},\ldotsk_{n}\right)\prod_{i=1}^{n}\left(a_{k_{i}}^{+}\left(j_{i}\right)\right)^{m_{i}}\left(a_{k_{i}}\left(j_{i}\right)\right)^{m_{i}^{\prime}},
\begin{array}{rcl}{G_{\sigma}^{(n)}(\zeta^{\prime\prime},\zeta_{2n-1},\zeta_{2n})|_{\zeta_{2n}=\epsilonx^{-1}\zeta_{2n-1}}}&{=}&{G_{\epsilon\sigma}^{(n-1)}(\zeta^{\prime\prime})\otimesu_{\epsilon}~~~~(\epsilon=\pm),}\\\end{array}
{\frac{(a_{1}+\alpha)D(1-i\xi_{1}-i\xi_{2})^{2}}{i\xi_{2}(1-\lambda)P(\xi_{1})}}ln\biggl[{\frac{-i(\xi_{1}+\xi_{2})}{{\frac{-i\xi_{2}\lambda}{1-i\xi_{2}(1-\lambda)}}-\sqrt{[{\frac{1-i\xi_{2}}{1-i\xi_{2}(1-\lambda)}}]^{2}-{\frac{D}{1-i\xi_{2}(1-\lambda)}}}}}\biggr]
\chi_{T}=2^{4/3}\pi^{2/3}\kappa^{-2/3}Nmr_{+}^{-2}\frac{r_{+}^{2}+2l^{2}}{3r_{+}^{4}+5l^{2}r_{+}^{2}+2l^{4}}\frac{3r_{+}^{2}+2l^{2}}{3r_{+}^{2}-2l^{2}}.
\ddot{{\calA}}+{\calA}_{\alpha}^{\prime\prime}+2{\calH}{\calA}_{\alpha}^{\prime}-\varepsilon^{2}\Box{\calA}_{\alpha}-\biggl[\frac{P^{\prime\prime}}{P}+2{\calH}\frac{P^{\prime}}{P}\biggr]{\calA}_{\alpha}=0,
\begin{array}{rclrcl}{\left[L_{1},L_{-2}\right]}&{=}&{L_{-1},}&{\quad\left[L_{1},L_{2}\right]}&{=}&{0,}\\{\left[L_{2},L_{-2}\right]}&{=}&{N+\frac{D}{2},}&{\quad\left[L_{0},L_{n}\right]}&{=}&{0,}\\{\left[L_{1},L_{-1}\right]}&{=}&{L_{0},}&{\quad\left[N,L_{n}\right]}&{=}&{{}-nL_{n},\quadn=0,\pm1,\pm2.}\\\end{array}
\hat{S}=\frac{(4\pi)^{(n-2)/2}\beta}{16\pi^{3/2}\ell^{2}}\Gamma\left(\frac{3-n}{2}\right)\Gamma\left(\frac{n}{2}\right)N^{n-3}\left[(n-1)(n-2)N^{2}-(n-3)\ell^{2}\right]=S_{NUT}
1\,\mathrm{eV}\sim{\frac{10^{-3}\mathrm{cm}\,2\times10^{-5}\mathrm{eV-cm}}{\tau_{f}^{2}(3\times10^{10}\mathrm{cm\,s}^{-1})^{2}}}\sim{\frac{10^{-29}\mathrm{eV}}{\tau_{f}^{2}(\mathrm{s}^{2})}}.
W=\frac{\Lambda_{1d}^{5}\Omega_{(2,\,r-1)}}{\Omega_{(1,\,r-1)}}+\frac{\Lambda_{rd}^{5}\Omega_{(1,\,r-2)}}{\Omega_{(1,\,r-1)}}\pm2\frac{(\Lambda_{1d}^{5}\Lambda_{rd}^{5}\prod_{i=2}^{r-1}\Lambda_{i}^{4})^{1/2}}{\Omega_{(1,\,r-1)}}.
\lambda_{1}^{2}=(1-x)^{\frac{2}{1+\beta^{2}}}\left[{\frac{2\beta^{2}}{(1+\beta^{2})(1-\beta^{2})}}x-{\frac{2(1+\beta^{2})}{(1+3\beta^{2})(1-\beta^{2})}}\left(1-(1-x)^{\frac{2\beta^{2}}{1+\beta^{2}}}\right)\right].
\left(\begin{array}{c}{{\hat{\phi}}_{1n}}\\{{\hat{\phi}}_{2n}}\\\end{array}\right)=U\left(\begin{array}{c}{C_{1n}}\\{C_{2n}}\\\end{array}\right);\;\;\;\;\;\;\;\;U=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{i}&{-i}\\{1}&{1}\\\end{array}\right).
\begin{array}{lcl}{K_{m+\alpha}((m+\alpha)z^{\prime})\}&{\sim\}&{\sqrt{\frac{\pi}{2(m+\alpha)}}\frac{e^{-(m+\alpha)\eta}}{(1+\left(z\frac{m}{m+\alpha}\right)^{2})^{1/4}}}\\{}&{}&{\left\{1+\sum_{j=1}^{\infty}(-1)^{j}\frac{u_{j}([1+\left(z\frac{m}{m+\alpha}\right)^{2}]^{-1/2})}{m^{j}\left(1+\frac{\alpha}{m}\right)^{j}}\right\}}\\\end{array}
F[f](x)-c_{1}^{\prime}\epsilon(x^{0})\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).
\triangle_{P+1,P}=\sum_{g_{P}=1}^{[N/P]}\frac{1}{g_{P}!}\sum_{\begin{array}{l}{{g_{1},g_{2},....,g_{P-1}\geq0}}\\{\sum_{j=1}^{P-1}j.g_{j}=N-Pg_{P}}\\\end{array}}\frac{G!}{g_{1}!g_{2}!....,g_{P-1}!(G-\sum_{j=1}^{P}g_{j})!}
S_{\mathrm{gauge~field}}=\intdx^{0}\left\{-{\frac{in}{2r}}\overline{{\pi}}_{a}\pi_{a}\sum_{i}(\phi_{i}\widetilde{D}_{0}\overline{{\phi}}_{i}-\overline{{\phi}}_{i}\widetilde{D}_{0}\phi_{i})\right\}
a_{4,4}={\frac{1}{4!}}\int_{\Sigma_{r}}{\bar{h}}^{1/2}d^{3}x\left[{\frac{1}{12}}(\bar{F}^{ij}\bar{F}_{ij})^{2}+{\frac{4}{15}}\bar{F}_{ij}\bar{F}_{pk}\bar{F}^{ik}\bar{F}^{pj}\right]~~~,
\begin{array}{l}{\hat{\calR}=q{\calP}_{S}-q^{-1}{\calP}_{A}+q^{-4}{\calP}_{1}~,}\\{\hat{\calR}^{-1}=q^{-1}{\calP}_{S}-q{\calP}_{A}+q^{4}{\calP}_{1}~,}\\\end{array}
V=\frac14\mathrm{e}^{\mathcalK}\left[\left(4h_{b_{1}b_{1}}-2\omega_{b_{1}}^{i}\omega_{b_{1}}^{i}\right)g_{1}^{2}+\left(4h_{b_{2}b_{2}}-2\omega_{b_{2}}^{i}\omega_{b_{2}}^{i}\right)g_{2}^{2}\right],
{g^{\prime}}_{\mu\nu}(x,l)=[f(l)^{\star}g_{\mu\nu}(l)](x)=\frac{\partial{x^{\prime}}^{\alpha}(x,l)}{\partialx^{\mu}}g_{\alpha\beta}(x^{\prime}(x,l),l)\frac{{x^{\prime}}^{\beta}(x,l)}{\partialx^{\nu}}
ds^{2}~=~-f^{-2/3}\left(1-\frac{a^{2}}{\bar{r}^{2}}\right)+f^{1/3}\left[\left(1-\frac{a^{2}}{\bar{r}^{2}}\right)^{-1}d\bar{r}^{2}+r^{2}d\Omega_{3}^{2}\right]~,
\langle\beta\rangle={\frac{\intd^{3}{\bfk}\,\beta({\bfk})e^{-{\omega_{k}/\sigma}}}{\intd^{3}{\bfk}\,e^{-{\omega_{k}/\sigma}}}}={\frac{1}{\sigma}}+{\frac{{\sum}_{n=1}^{\infty}{\frac{(-1)^{n+1}}{n(n+1)}}K_{1}({(n+1){\cal{M}}/\sigma})}{{\cal{M}}K_{2}({{\cal{M}}/\sigma})}}\,,
Z_{N}^{inf}(x_{1},\cdots,x_{M})=(x_{1}+\cdots+x_{M})^{N}=\sum_{{n_{i}}\atop{\Sigman_{i}=N}}~{\frac{N!}{n_{1}!n_{2}!\cdotsn_{M}!}}~x_{1}^{n_{1}}x_{2}^{n_{2}}\cdotsx_{M}^{n_{M}}\,\,\,\,.
S=-\frac{1}{4\pi\alpha^{\prime}}\intd^{2}\xi\sqrt{-g\:}g^{\alpha\beta}\:\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\:G_{\mu\nu}+e_{0}\:\inta_{\mu\:}dX^{\mu}+e_{\pi}\:\inta_{\mu\:}dX^{\mu}
T_{(\mu\nu)\beta}=-\frac{1}{2e}\epsilon_{\,\,\,\,\,\,\,\mu\nu}^{\alpha\sigma}\partial_{\alpha}h_{\sigma\beta}-\frac{1}{2eM^{2}}\epsilon_{\,\,\,\,\mu\nu\beta}^{\sigma}\partial_{\eta}\Theta_{\sigma}^{\eta}\,,
d\mu=\mathrm{const}\intd^{4}a\frac{d\rho}{\rho^{5}}\frac{d^{3}\omega}{2\pi^{2}}M^{8}\rho^{8}\times\frac{1}{M^{4}\rho^{4}}\intd^{2}\epsilon_{1}d^{2}\epsilon_{2}d^{2}\bar{\beta}_{1}d^{2}\bar{\beta}_{2}
\begin{array}{l}{{\calD}_{-k}{\calX}^{-k}-{\calX}^{-k}{\calD}_{-k}=\mu_{-k}^{-1}~,~~~~~~\mu_{-k}{\calX}^{-k}=q^{2}{\calX}^{-k}\mu_{-k}~,}\\{\mu_{-}{\calD}_{-k}=q^{-2}{\calD}_{-k}\mu_{-k}~,~~~~~~\mu_{-}^{-1}\equiv1+(q^{-2}-1){\calX}^{-k}{\calD}_{-k}~,}\\\end{array}
\left\{i\left[\tilde{e}_{a}^{\mu}\gamma^{a}\left(\partial_{\mu}-ieA_{\mu}\right)+\gamma^{a}\tilde{C}_{ab}^{b}+\frac{i}{4}\tilde{\omega}_{\mu}^{cd}\varepsilon_{abcd}\gamma^{5}\gamma^{b}\tilde{e}^{a\mu}\right]-\frac{m}{\sqrt{\zeta+b_{g}}}\left(1+\frac{h}{\zeta}\right)\right\}\Psi^{\prime}=0
\int_{-1}^{1}{d\nu}Z_{\mathrm{str}}(\beta,L,\nu)\propto\frac{\beta^{2}\left(\beta^{2}-8\pi^{2}\alpha^{\prime}\right)^{3}}{L^{8}}e^{-L\sqrt{\beta^{2}-8\pi^{2}\alpha^{\prime}}/\pi\alpha^{\prime}}.
{\frac{\langle:p_{i}p^{i}:\rangle}{pr_{1}^{2}}}\sim{\frac{Q_{1}}{(Q_{1}Q_{5})^{2}}};\\{\frac{\langle:\gamma^{2}:\rangle}{pr_{1}^{2}r_{5}^{2}}}\sim{\frac{1}{Q_{1}Q_{5}}}.
\begin{array}{ccc}{\Gamma^{\mu}}&{=}&{\left(\begin{array}{cc}{\gamma^{a}e_{1a}^{\mu}(x)}&{0}\\{0}&{\gamma^{a}e_{2a}^{\mu}(x)}\\\end{array}\right)}\\{\Gamma^{5}}&{=}&{\left(\begin{array}{cc}{-\gamma^{a}e_{1a}^{\mu}a_{1\mu}}&{\gamma^{5}\phi_{2}^{-1}(x)}\\{\gamma^{5}\phi_{1}^{-1}(x)}&{-\gamma^{a}e_{2a}^{\mu}a_{2\mu}}\\\end{array}\right).}\\\end{array}
f_{-}^{n}=\frac{D_{0}^{n-1}}{D_{0}^{n}},\;f_{0}^{n}=\frac{\dot{D}_{0}^{n}}{D_{0}^{n}}+\tau,\;f_{+}^{0}=\frac{D_{0}^{n+1}}{D_{0}^{n}}.
{\calK}^{\pm}\,(\psi^{0})=({\calK}_{ab}^{\pm}\,(\psi^{0}))=\left(\begin{array}{ccc}{\left(\frac{1}{2}\,\delta_{ij}\pm\frac{i}{2}\,\epsilon_{ij3}-2\,{\calW}_{i}^{\pm}(\psi^{0})\,({\calW}_{j}^{\pm}\,(\psi^{0}))\right)_{3\times3}}&{}&{}\\{}&{\left(\Sigma_{\alpha\beta}^{\pm}\right)_{4\times4}}&{}\\{}&{}&{0}\\\end{array}\right)_{8\times8}
\Sigma_{a_{1},\ldots,a_{p};\;b_{1},\ldots,b_{q};c}^{\;i_{1},\ldots,i_{p};\;j_{1},\ldots,j_{q}}(k)\equiv\sum_{n=1}^{\infty}\frac{(n!)^{2}}{(2n)!}\frac{k^{n}}{n^{c}}[S_{a_{1}}(n\!-\!1)]^{i_{1}}\ldots[S_{a_{p}}(n\!-\!1)]^{i_{p}}\;[S_{b_{1}}(2n\!-\!1)]^{j_{1}}\ldots[S_{b_{q}}(2n\!-\!1)]^{j_{q}},
\phi_{n_{1}}^{n_{3}}(\tau)=\int_{0}^{1}\int_{0}^{1}\phi\left(\left[\begin{array}{ccc}{1}&{0}&{\xi_{3}}\\{0}&{1}&{\xi_{1}}\\{0}&{0}&{1}\\\end{array}\right)\tau\right]e^{-2\pii(n_{1}\xi_{1}+n_{3}\xi_{3})}d\xi_{1}d\xi_{3}.
T=-\frac{1}{2}(\partialX)^{2}-\frac{1}{2}(\partial\phi)^{2}-\frac{1}{\sqrt{2k}}\partial^{2}\phi-\frac{1}{2}(\partialH^{\prime})^{2}+\left\{Q_{U(1)},~\frac{1}{\sqrt{2(k+2)}}\eta(-\partialu+i\partialv^{\prime})\right\}.
O_{F_{22}}\xi_{n}\equiv-\frac{d^{2}}{dz^{2}}\xi_{n}-\frac{2m^{2}\alpha}{\lambda^{2}}(2\alpha+\lambda)sech^{2}(mz)\xi_{n}+m^{2}(\frac{2\alpha}{\lambda^{2}}(2\alpha+\lambda)-\frac{2\alpha}{\lambda})\xi_{n}=\tilde{\omega}_{n}^{2}\xi_{n}.
ds^{2}=X(\frac{d\eta^{2}}{1-\eta^{2}}-\frac{d\mu^{2}}{1-\mu^{2}})-(1-\mu^{2})^{1/2}(1-\eta^{2})^{1/2}[\chidy^{2}+\frac{1}{\chi}(dx-\lambdady)^{2}],
D_{\mu\nu}^{prop}(B)=-(g_{\mu\rho}^{\perp}-n_{\mu}(nKn)^{-1}(nK)_{\rho}^{\perp})\tilde{K}_{\rho\sigma}^{-1}(g_{\sigma\nu}^{\perp}-(Kn)_{\sigma}^{\perp}(nKn)^{-1}n_{\nu})
S=\intd^{4}x\sqrt{|\hat{g}_{E}|}\left[\hat{R}(\hat{g}_{E})-{\textstyle\frac{1}{2}}(\partial\hat{\phi})^{2}-{\textstyle\frac{1}{2}}e^{2\hat{\phi}}(\partial\hat{a})^{2}-{\textstyle\frac{1}{4}}e^{-\hat{\phi}}\hat{F}^{I}\hat{F}^{I}+{\textstyle\frac{1}{4}}\hat{a}\hat{F}^{I}{}^{\star}\hat{F}^{I}\right]\,.