harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\tilde{\Psi}}_{\alpha,\beta}^{\gamma}={\Psi}_{\alpha,\beta}^{\gamma}q^{(-\alpha^{r}-\beta^{l}+\gamma^{r})}e^{-i\pi(\alpha^{r}+\beta^{l}-\gamma^{r})}\frac{v_{\beta^{r}}^{1/4}v_{\gamma^{l}}^{1/4}}{v_{\beta^{l}}^{1/4}v_{\gamma^{r}}^{1/4}}\frac{\nu_{1}(d_{\alpha^{r}})\nu_{1}(d_{\beta^{l}})}{\nu_{1}(d_{\gamma^{r}})}
\psi_{(1)}^{c}\,{\gamma^{\mu}}_{cb}\,\psi_{(2)}^{b}=\phi_{(1)}^{M}\,{\sigma^{\mu{\dot{L}}}}_{M}\,\varphi_{(2){\dot{L}}}+\varphi_{(1){\dot{M}}}\,{\sigma^{\mu{\dot{M}}}}_{L}\,\phi_{(2)}^{L}\,.
I[A_{+},\tilde{J}_{\pm},\tilde{X},C_{-}]=i\intd^{2}\xid^{2}\theta[A_{+}^{a}(\alpha^{-1})_{a}^{~b}\tilde{J}_{-b}+(A_{+}^{a}(\alpha^{-1})_{a}^{~c}E_{cb}^{0}-A_{+}^{a}\beta_{ab}^{t}-\tilde{J}_{+b})C_{-}^{b}],
S_{gh}=\mathrm{tr}\intd\zeta^{(-4)}du{\bfb}(\nabla^{++})^{2}{\bfc}-\mathrm{i}\,g\,\mathrm{tr}\,\intdu\,d\zeta^{(-4)}\,\nabla^{++}{\bfb}\;[v^{++},{\bfc}]\;.
\lambda=\frac{4\piMus_{t}}{e^{2}}\\sqrt{s_{t}^{2}-s_{y}^{2}}=\frac{4\piMus_{t}}{e^{2}}\\frac{e^{2}}{4\piMu^{2}}=\frac{s_{t}}{u}=\frac{R(\tau^{*})}{u}
\hat{H}_{1}=\phi_{(\tilde{u}+2,\bar{\tilde{u}})}^{7_{3}7_{3}}\phi_{(\tilde{u},\overline{{\tilde{u}+1}})}^{7_{3}7_{3}}\phi_{(1,\bar{\tilde{u}})}^{37_{3}}\phi_{(\tilde{u},\bar{2})}^{7_{3}3}M_{s}^{-3}
\rho(r)+p(r)=\frac{b+b^{\prime}r}{2r^{3}}+\left(1-\frac{b(r)}{r}\right)\left[\psi^{\prime\prime}+{\psi^{\prime}}^{2}+\frac{\psi^{\prime}}{r}+\frac{b-b^{\prime}r}{2r(r-b)}\psi^{\prime}\right]\ge0
\hat{H}_{\mathrm{IIE}}=\frac{1}{2m}[\hat{F}^{k}\!_{i}(\hat{f}=0),\hat{\Pi}_{k}-\hat{A}_{k}]_{+}\hat{g}^{ij}[\hat{F}^{l}\!_{j}(\hat{f}=0),\hat{\Pi}_{l}-\hat{A}_{l}]_{+}+V(\hat{q})+{\hbar}^{2}\hat{Q}\;,
\left\{{\frac{d^{2}}{dz^{2}}}+(1+k^{2})\left({\frac{\omega^{2}}{2a^{2}\mu^{2}r}}+{\frac{1}{r}}\right)-2(2+r)k^{2}\mathrm{sn}^{2}(z)\right\}\,\eta(z)=0~.
j_{3}(\alpha,m,L)=\frac{2\mu^{2-2\omega}}{L}\frac{\Gamma\left(\frac{1}{2}-\omega\right)}{\sqrt{\pi}\left(4\pi\right)^{\omega}}\sum_{k_{3}}k_{3}\left(m^{2}+k_{3}^{2}\right)^{\omega-\frac{1}{2}}.
{\calL}=\frac{2f^{2}}{(1+\theta\bar{\theta})^{2}}\left[(\dot{\theta}+i\dot{\alpha}\theta)(\dot{\bar{\theta}}-i\dot{\alpha}\bar{\theta})+g^{0k}\partial_{k}\bar{\theta}(\dot{\theta}+i\dot{\alpha}\theta)+g^{0k}\partial_{k}\theta(\dot{\bar{\theta}}-i\dot{\alpha}\bar{\theta})\right].
\mathcal{H}=\left(a^{\dagger},\,b\right)\left(\begin{matrix}{\alpha^{\dagger}}&{\beta^{\dagger}}\\{-\beta^{T}}&{\alpha^{T}}\\\end{matrix}\right)\left(\begin{matrix}{\omega}&{0}\\{0}&{-\omega}\\\end{matrix}\right)\left(\begin{matrix}{\alpha}&{-\beta^{*}}\\{\beta}&{\alpha^{*}}\\\end{matrix}\right)\left(\begin{matrix}{a}\\{b^{\dagger}}\\\end{matrix}\right).
\left[-\left({\frac{\partial^{2}}{\partial\rho_{1}^{2}}}+{\frac{\partial^{2}}{\partial\rho_{2}^{2}}}+{\frac{\partial^{2}}{\partial\rho_{3}^{2}}}\right)+{\alpha^{2}}(\rho_{1}^{2}\rho_{2}^{2}+\rho_{2}^{2}\rho_{3}^{2}+\rho_{1}^{2}\rho_{3}^{2})+\frac{{\hat{l}_{1}}^{2}}{\rho_{1}^{2}}+\frac{{\hat{l}_{2}}^{2}}{\rho_{2}^{2}}+\frac{{\hat{l}_{3}}^{2}}{\rho_{3}^{2}}\right]\Psi=2\,E\Psi
f_{3}(r)=2\sqrt{\frac{\bar{s}^{2}-Qc}{Qc}}e^{-\tilde{c}\sqrt{\bar{s}^{2}-Qc}}\frac{\left(1-\frac{r_{0}^{2}}{r^{2}}\right)^{(1+\sqrt{(\bar{s}^{2}-Qc)/D_{2}}\)/2}}{1-e^{-2\tilde{c}\sqrt{\bar{s}^{2}-Qc}}\left(1-\frac{r_{0}^{2}}{r^{2}}\right)^{\sqrt{(\bar{s}^{2}-Qc)/D_{2}}}}.
\tilde{S}=-\frac{1}{4\pi}\int\!{d^{2}}\xi[{\frac{\alpha^{\prime}}{R^{2}}}{\partial_{\alpha}}{\chi}{\partial^{\alpha}}{\chi}+\frac{1}{\alpha^{\prime}}g_{\mu\nu}{\partial_{\alpha}}{x^{\mu}}{\partial^{\alpha}}{x^{\nu}}+2\epsilon^{\alpha\beta}\frac{k}{R}A_{\mu}{\partial_{\alpha}}{\chi}{\partial_{\beta}}{x^{\mu}}].
dQ_{n}(\zeta)\left[1-\sqrt{1-{\frac{c_{1}^{(n)}}{d\zeta}}-{\frac{c_{2}^{(n)}}{d\zeta^{2}}}}\right]\;\equiv\;\sum_{i=1}^{n}p_{i}\zeta^{i}+p_{0}+\sum_{j=1}^{\infty}p_{-j}\zeta^{-j}\;,
\begin{array}{rcl}{\mathrm{Tr}[T_{a},\cdots]=0,}&{{\qquad}}&{\mathrm{Tr}[\Sigma_{\alpha},\cdots]=0,}\\{\mathrm{Str}[T_{a},\cdots]=0,}&{{\qquad}}&{\mathrm{Str}\{\Sigma_{\alpha},\cdots\}=0,}\\\end{array}
V(y)={\frac{2(D-2p-1)K^{2}}{(D-2)^{4}a^{4}}}{\frac{2(D-2p-1)-(D-2)^{2}a^{2}}{(1-K|y|)^{2}}}-{\frac{4(D-2p-1)K}{(D-2)^{2}a^{2}}}\delta(y).
C_{2}=i\,e\left(\begin{array}{cccc}{0}&{0}&{\phi_{1}}&{\phi_{2}}\\{0}&{0}&{0}&{0}\\{-\,\phi_{1}}&{0}&{0}&{A_{2}}\\{-\,\phi_{2}}&{0}&{-\,A_{2}}&{0}\\\end{array}\right)
Z_{S}\left(i\frac{H}{T_{K}},\frac{H}{T}\right)Z_{S}\left(-i\frac{H}{T_{K}},\frac{H}{T}\right)=1+Z_{S+1/2}\left(\frac{H}{T_{K}},\frac{H}{T}\right)Z_{S-1/2}\left(\frac{H}{T_{K}},\frac{H}{T}\right)
\langle\overline{{\phi}}_{1},\overline{{\phi}}_{2}\rangle=\overline{{\phi}}_{2}(\phi_{1}^{*})=\int\dv\\overline{{\phi}}_{2}(x)\phi_{1}^{*}(x)=\int\dv\\phi_{2}(x)\overline{{\phi_{1}^{*}}}(x).
\tilde{H}^{ab}\longrightarrow\left(\begin{array}{cc|cc}{0}&{2\,sh\,ch}&{sh^{2}+ch^{2}}&{0}\\{-2\,sh\,ch}&{0}&{0}&{-(sh^{2}+ch^{2})}\\\hline{-(sh^{2}+ch^{2})}&{0}&{0}&{-2\,sh\,ch}\\{0}&{sh^{2}+ch^{2}}&{2\,sh\,ch}&{0}\\\end{array}\right),
{\calA}^{\mu}\Sigma_{\mu}=\left(\begin{matrix}{-\Sigma_{y}}&{\Sigma_{0}}\\{-(1+B^{2})\Sigma_{0}+E_{x}B\Sigma_{y}}&{E_{x}B\Sigma_{0}+(1+E_{x}^{2})\Sigma_{y}}\\\end{matrix}\right)\,.
\frac{\partial_{1}\beta}{\beta}+\frac{\gamma}{8\Omega}+\frac{\beta^{2}}{2}\OmegaV(\Omega)-\frac{\Omega^{\delta}}{4}\{\frac{\beta^{2}}{\alpha^{2}}(\partial_{0}f)^{2}+(\partial_{1}f)^{2}\}=0,
(z_{1}-z_{2})^{2\alpha^{\prime}p_{1\cdot}p_{2}}(z_{1}-z_{3})^{2\alpha^{\prime}p_{1\cdot}p_{3}}(z_{2}-z_{3})^{2\alpha^{\prime}p_{2\cdot}p_{3}}\prod_{i=1}^{3}\left[V_{i}^{\prime}(0)\right]^{\alpha^{\prime}p_{i}^{2}}=const.
\left({\frac{\dot{a}}{a}}\right)^{2}={\frac{(p-3)^{2}}{16L_{p}^{2}}}\left[\tilde{E}^{2}\tilde{P}_{\phi}^{-2{\frac{p-1}{p-3}}}\left({\frac{a}{L_{p}}}\right)^{2{\frac{p+1}{p-3}}}-\left({\frac{L_{p}}{a}}\right)^{2}\right],
S=\frac{1}{2\kappa^{2}}\int{d\eta\left[ne^{-\phi}\left(-6\frac{s^{2}}{n^{2}}-\frac{\phi^{2}}{n^{2}}s^{2}+6\frac{\phi^{\prime}s^{\prime}s}{n^{2}}+6ks^{2}-\Lambdas^{4}-\frac{q^{2}}{2s^{2}}\right)\right]}.
\begin{array}{rcl}{}&{}&{\pi_{l}^{\sigma}(e)v_{k}^{l}=\sigma[l-k+1]v_{k-1}^{l}~,}\\{}&{}&{\pi_{l}^{\sigma}(f)v_{k}^{l}=[k+1]v_{k+1}^{l}~,}\\{}&{}&{\pi_{l}^{\sigma}(t)v_{k}^{l}=\sigmaq^{l-2k}v_{k}^{l}~,}\\\end{array}
(\deltag,\deltag)=2\int\sqrt{g}\,d^{D}x\:\deltag_{\mu\nu}^{T}g^{\mu\mu^{\prime}}g^{\nu\nu^{\prime}}\deltag_{\mu^{\prime}\nu^{\prime}}^{T}+C\int\sqrt{g}\,d^{D}x\:\deltag_{\lambda}^{\lambda}\deltag_{\rho}^{\rho}
\intd^{3}x\intd^{3}x^{\prime}\phi_{-}(x)D_{+}(x,x^{\prime})\phi_{-}(x^{\prime})+\intd^{3}x\intd^{3}x^{\prime}\phi_{+}(x)D_{-}(x,x^{\prime})\phi_{+}(x^{\prime})
j_{3}(\alpha,m,L)=\langle\Omega|\bar{\chi}_{\alpha}\left(x\right)\gamma_{3}\chi_{\alpha}\left(x\right)|\Omega\rangle=\frac{1}{L}\sum_{k_{3}}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{2k_{3}}{E_{k}}
\left(C_{(5,1)(5,1)}^{(5,1)}\right)^{2}=c_{p,1}(-1)^{p-1}\frac{(2p-1)!^{3}(p-1)!^{3}(7p-2)!^{2}(6p-2)!(6p-3)!}{6(4p-1)!^{3}(3p-1)!^{3}(5p-2)!(4p-2)!(3p-2)!(2p-2)!}\,,
\langleT_{\mu}^{\mu}\rangle_{ren}=\frac{1}{(4\pi)^{4}}\left[\frac{896}{243}\Lambda^{4}-\frac{4352}{81}\left(\frac{\Lambda^{3}}{r^{2}}\right)+\frac{12304}{45}\left(\frac{\Lambda^{2}}{r^{4}}\right)-\frac{15700}{27}\left(\frac{\Lambda}{r^{6}}\right)+\frac{41657}{90}\left(\frac{1}{r^{8}}\right)\right],
\bar{g}^{vu}(\deltaR_{uu}+{\frac{1}{2}}e^{4\bar{\Phi}}\bar{H}_{u\rho\sigma}\bar{H}_{u\alpha}^{~~~\sigma}h^{\rho\alpha})-h(\bar{R}_{vu}-{\frac{1}{4}}e^{4\bar{\Phi}}\bar{H}_{v\rho\sigma}\bar{H}_{u}^{~~\rho\sigma})=0.
\timese^{-{i}[\tilde{\bfr}^{\prime\prime}\times\tilde{\bfr}+\tilde{\bfr}^{\prime\prime\prime}\times\tilde{\bfr}^{\prime\prime\prime\prime}]}f(\tilde{\bfr}^{\prime\prime\prime})g(\tilde{\bfr}^{\prime\prime\prime\prime})h(\tilde{\bfr}^{\prime\prime}).
L=E^{-1}\left(\begin{array}{cc}{\sqrt{3}\frac{\sqrt{1-U_{oo}}}{\sqrt{1+U_{oo}}}}&{0}\\{0}&{\frac{1}{\sqrt{3}}\frac{\sqrt{1-U_{ee}}}{\sqrt{1+U_{ee}}}}\\\end{array}\right)E^{-1}\,.
\begin{array}{rcl}{\delta_{\omega}V^{a}}&{=}&{\omega^{b}\partial_{b}V^{a}-V^{b}\partial_{b}\omega^{a}+{\textstyle{\frac{1}{2}}}\partial_{b}\omega^{b}V^{a},}\\{\delta_{\omega}X}&{=}&{\omega^{a}\partial_{a}X.}\\\end{array}
G_{{\calQ}^{4}}(x_{p})=\frac{3^{3}\,\Gamma(11)}{2^{7}\left(\pi^{3}\Gamma(4)\right)^{4}}\:g^{8}e^{-\frac{8\pi^{2}}{g^{2}}+i\theta}\,x_{12}^{4}x_{34}^{4}\prod_{p<q}\frac{\partial}{\partialx_{pq}^{2}}B(x_{pq})\,,
P^{-1}{}_{\!\!\mu_{1}\cdots\mu_{k},\nu_{1}\cdots\nu_{k}}(p)\,P_{\nu_{1}\cdots\nu_{k},\lambda_{1}\cdots\lambda_{k}}(p)={\calE}^{A}{}_{\!\!\mu_{1}\cdots\mu_{k},\lambda_{1}\cdots\lambda_{k}}\,
a_{0}=-{\frac{6}{g}}{\textstyle\sum_{3}}+{\frac{6}{g^{2}}}\sum_{n\ne0}\bigl({\textstyle\sum_{3}}a_{n}a_{-n}+a_{n}a_{-n}{\textstyle\sum_{3}}+a_{n}{\textstyle\sum_{3}}a_{-n}-{\frac{3}{2}}{\textstyle\sum_{3}}\bigr)+\cdots.
ds^{2}\begin{array}{c}{\approx}\\{{{r\rightarrow\infty}}}\\\end{array}\ell^{2}{\frac{dr^{2}}{r^{2}}}\,+\,\frac{r^{2}}{\ell^{2}}\stackrel{(0)}{\bfg}(x^{i})\,+\,\stackrel{(2)}{\bfg}(x^{i})\,+\,\cdots\qquad.
L^{(+)}=\left(\begin{array}{cc}{q^{-\left(\begin{array}{cc}{1}&{0}\\{0}&{0}\\\end{array}\right)}}&{(q-q^{-1})(-1)^{F}Z^{+}}\\{0}&{(-1)^{F}q^{\left(\begin{array}{cc}{0}&{0}\\{0}&{1}\\\end{array}\right)}}\\\end{array}\right)
X=\left[\begin{array}{cccccc}{0}&{}&{}&{}&{}&{1}\\{1}&{0}&{}&{}&{}&{}\\{}&{1}&{\cdots}&{}&{}&{}\\{}&{}&{}&{\cdots}&{0}&{}\\{}&{}&{}&{}&{1}&{0}\\\end{array}\right],\quadY=\mathrm{diag}(q,\cdots,q^{2Q})
\sqrt{-G}G^{55}\partial_{5}\Phi\Big|_{y=0}=-\sqrt{-g^{0}}\frac{\partialV_{0}}{\partial\Phi}\Big|_{y=0}\,\qquad\sqrt{-G}G^{55}\partial_{5}\Phi\Big|_{y=L}=\sqrt{-g^{L}}\frac{\partialV_{L}}{\partial\Phi}\Big|_{y=L}\.
P=(\partial^{2}+2u)^{3}+3\{w_{1},\partial^{3}\}-3\{w_{2},\partial^{2}\}+\{-\mathrm{~\frac{1}{2}~}w_{1}^{\prime\prime}+6uw_{1},\partial\}-2w_{2}^{\prime\prime}+6w_{1}^{2}-12uw_{2}
\xi\frac{\partialV^{(1)}}{\partial\xi}=\frac12(m_{2}^{2}\varphi)\xie^{2}\int\frac{d^{d}k}{(2\pi)^{d}}\frac1{D_{N}\,(k^{2}+m_{gh}^{2})}\left\{(\varphi-2u)k^{2}-\xiue^{2}\varphi^{2})\right\}.
\begin{array}{l}{\deltaC=\frac{i}{\xi}g^{\dagger}\partial_{\mu}A^{\mu}-\frac{i}{\sqrt{\xi}}g^{\dagger}E,}\\{\delta\bar{C}=-\frac{i}{\xi}f^{\dagger}\partial_{\mu}A^{\mu}+\frac{i}{\sqrt{\xi}}f^{\dagger}E,}\\{\deltaE=\frac{1}{\sqrt{\xi}}\partial_{\mu}\partial^{\mu}(fC+g\bar{C}),}\\\end{array}
\partial_{\mu}<{\bar{\psi}}\gamma^{\mu}i\gamma_{5}\psi>=\left[\partial_{-}i{\frac{\delta}{\deltaJ_{-}}}-\partial_{+}i{\frac{\delta}{\deltaJ_{+}}}\right]e^{i\Gamma_{f}[A+J]}\vert_{J=0}\,\,=\,\,{\frac{e}{\pi}}(\partial_{+}A_{-}-\partial_{-}A_{+})\,,
\alpha^{\prime}\left(\frac{c}{\hbar}\right)m^{2}=24\sum_{n>0}\frac{2n^{2}-H^{2}m^{2}\frac{\alpha^{\prime2}}{c^{2}}}{\sqrt{n^{2}-H^{2}m^{2}\frac{\alpha^{\prime2}}{c^{2}}}}+2N
\delta_{i_{1}}^{n_{1}}P_{i_{2}}^{n_{2}}(\alpha_{3})P_{i_{3}}^{n_{3}}(\alpha_{3}+\beta_{3})=\delta_{i_{1}}^{n_{1}}P_{i_{2}}^{n_{2}}(\alpha_{3})P_{i_{3}}^{n_{3}}(\alpha_{3}+\beta_{3})\.
\frac{\left(-k\right)^{\nu-\frac{1}{2}}}{F_{\ell}(k)}W\left[\dot{f}_{\ell}(k,R),\phi_{\ell}(k,R)\right]=\frac{\nu-\frac{1}{2}}{k}+\frac{\dot{F}_{\ell}(k)}{F_{\ell}(k)}+2k\int_{0}^{R}dr\,G_{\ell}(r,r,k)\,,
{\calB}^{r}(t)\sim\frac{1}{(\Omega!)^{2}}\left[\Phi^{*}(t)\right]^{r}\sum_{s=0}^{r}{\binom{r}{s}}(\Omega-s)!\left[\Omega-(r-s)\right]!x^{2(r-s)}\Phi^{s}(t)\Phi^{r-s}(t-1)\;.
\eta_{4}\left(\chi,\sigma,\tau,t\right)=\omega^{-2}\left(\chi,\sigma,\tau,t\right),\eta_{5}\left(\chi,\sigma,\tau,t\right)=h_{(0)}^{-2}\frac{b(\sigma,\tau)}{a(\tau)}\left[\intdt\omega^{-1}\left(\chi,\sigma,\tau,t\right)\right]^{2},
e^{{\frac{1}{2}}\epsilon\varphi_{\alpha}-(p+1)A}=H_{\alpha},\\ds^{2}=\prod_{\alpha=1}^{N}H_{\alpha}^{-{\frac{\tilde{p}+1}{D-2}}}dx^{\mu}dx^{\nu}\eta_{\mu\nu}+\prod_{\alpha=1}^{N}H_{\alpha}^{\frac{p+1}{D-2}}dy^{m}dy^{m},
\begin{array}{l}{\displaystyle{\frac{A_{n}}{B_{n}}}\rightarrow_{r_{0}\rightarrow0}0\qquad\mathrm{for}\quadn\leqk}\\{\displaystyle{\frac{B_{n}}{A_{n}}}\rightarrow_{r_{0}\rightarrow0}0\qquad\mathrm{for}\quadn\geqk+1}\\\end{array}
H=\frac{1}{2}\mathrm{tr}M_{1}^{2}+\frac{1}{2}\mathrm{tr}M_{2}^{2}+a\mathrm{tr}M_{1}M_{2}+\frac{g}{N}(\mathrm{tr}M_{1}^{4}+\mathrm{tr}M_{2}^{4})
\frac12\epsilon^{\mu\nu\lambda\sigma}\,e_{\lambda}^{a}\,e_{\sigma}^{b}\,\hat{\xi}_{ab}^{+}=\frac12\epsilon^{\mu\nu\lambda\sigma}\,\frac{\partialq^{\gamma}}{\partialx^{\lambda}}e_{\gamma}^{c}\,\frac{\partialq^{\delta}}{\partialx^{\sigma}}e_{\delta}^{d}\,\hat{\xi}_{cd}^{+}
U_{4}\equiv{\frac{I+i{\not\!\!E}}{\sqrt{1+E^{2}}}}=U_{3}^{-1}U_{2}^{-1}U_{1}^{-1}={\frac{(I-i{\not\!\!D})(I-i{\not\!\!B})(I-i{\not\!\!A})}{\sqrt{1+D^{2}}\sqrt{1+B^{2}}\sqrt{1+A^{2}}}},
L_{0}=\frac{1}{4}\left(\begin{array}{cc}{\sigma_{3}+1}&{0}\\{0}&{-\sigma_{3}-1}\\\end{array}\right)\;,\;\;\;L_{+}=\frac{1}{2}\left(\begin{array}{cc}{0}&{\sigma_{3}+1}\\{0}&{0}\\\end{array}\right)\;,\;\;\;L_{-}=\frac{1}{2}\left(\begin{array}{cc}{0}&{0}\\{\sigma_{3}+1}&{0}\\\end{array}\right),
\varepsilon_{0}=\frac{E_{0}}{AL_{z}}=\frac{eB}{(2\pi)^{2}}\int_{-\infty}^{\infty}dp_{z}\left(\sqrt{m^{2}+p_{z}^{2}}-2\sum_{n=0}^{\infty}\sqrt{m^{2}+p_{z}^{2}+2eBn}\right).
\varphi^{\prime\prime}+{\frac{2}{r}}\varphi^{\prime}+{\frac{(r+1)^{4}\omega^{2}}{r^{4}}}\varphi-{\frac{l(l+1)}{r^{2}}}\varphi-{\frac{2}{r^{2}(r+1)^{2}}}\varphi=0,
ds^{2}=-\Big(1+{\frac{(q_{0}q_{1}q_{2})^{\frac{1}{3}}}{r^{2}}}\Big)^{-2}dt^{2}+\Big(1+{\frac{(q_{0}q_{1}q_{2})^{\frac{1}{3}}}{r^{2}}}\Big)(dr^{2}+r^{2}d\Omega_{3}^{2}).
\bar{M}_{i}=\left(\begin{matrix}{\Omega_{1}^{(i)}}&{}&{}&{}\\{}&{.}&{}&{}\\{}&{}&{.}&{}\\{}&{}&{}&{\Omega_{K}^{(i)}}\\\end{matrix}\right)\left(\begin{matrix}{m^{(i)}}&{}&{}&{}\\{}&{.}&{}&{}\\{}&{}&{.}&{}\\{}&{}&{}&{m^{(i)}}\\\end{matrix}\right)\left(\begin{matrix}{\Omega_{1}^{\dagger(i)}}&{}&{}&{}\\{}&{.}&{}&{}\\{}&{}&{.}&{}\\{}&{}&{}&{\Omega_{K}^{\dagger(i)}}\\\end{matrix}\right).
B_{2}^{f}=-\frac{V}{16\pi^{2}}\frac{2^{1-2\nu}}{\nu\Gamma(\nu)^{2}}\int_{0}^{\infty}dt\,t^{2\nu+3}\,\frac{K_{\nu}^{\prime}(t)}{I_{\nu}^{\prime}(t)}>0,
\begin{array}{rcl}{[\hat{S}_{+}(\vec{x}),\hat{S}_{-}(\vec{y})]}&{=}&{2\hat{S}_{3}(\vec{x})\delta(\vec{x}-\vec{y})\,,}\\{[\hat{S}_{3}(\vec{x}),\hat{S}_{\pm}(\vec{y})]}&{=}&{\pm\hat{S}_{\pm}(\vec{x})\delta(\vec{x}-\vec{y})\,,}\\\end{array}
\sum_{n\in{\bfZ}}nN_{n}=0,\quad\sum_{n\in{\bfZ}}n(q)\left(N_{n(q)}-\overline{{N}}_{-n(q)}\right)=\sum_{n\in{\bfZ}}\left(n(q)N_{n(q)}+n(-q)\overline{{N}}_{n(-q)}\right)=0,
X_{(3t_{1})}={\frac{e^{{\frac{5}{2}}i\lambda_{1}}}{(2\mathrm{sin}{\frac{g_{s}}{2}})^{2}}}(e^{i\lambda_{1}}-1)(2e^{i\lambda_{1}}-1)+{\frac{1}{3(2\mathrm{sin}{\frac{3g_{s}}{2}})^{2}}}(-e^{{\frac{3}{2}}i\lambda_{1}}+e^{{\frac{9}{2}}i\lambda_{1}})+e^{{\frac{7}{2}}i\lambda_{1}}-e^{{\frac{9}{2}}i\lambda_{1}}.
Q_{b}^{(n)}(w|\zeta)^{\varepsilon}=Q^{(n)}(w|\zeta)^{\varepsilon}q_{b}^{(n)}(w|\zeta),~~~~q_{b}^{(n)}(w|\zeta)=\prod_{a\inA}\prod_{j=1}^{2n}(xz_{j}-w_{a}^{-1}).
\langleC^{a}(x)\bar{C}^{b}(y)\rangle=i\int{\frac{d^{4}k}{i(2\pi)^{4}}}{\frac{-k^{2}\delta^{ab}-v\epsilon^{ab}}{(-k^{2})^{2}+v^{2}}}e^{ik(x-y)}.
\sum_{a=1}^{3}\sum_{j=1}^{3}Z_{j}^{a}K_{j}^{a}=\sum_{a=1}^{3}\sum_{j=1}^{3}\left\{Z_{j}^{a}\tilde{K}_{j}^{a}(1)+\cdots+Z_{j}^{a}\tilde{K}_{j}^{a}(N/4)\right\},
L_{\\nu}^{\mu}(\mathbf{p})=L_{\\nu}^{\mu}(\hat{\mathbf{p}})=\left(\begin{array}{cc}{\frac{p^{0}}{m}}&{-\frac{p_{j}}{m}}\\{\frac{p^{i}}{m}}&{\delta_{\j}^{i}-\frac{\frac{p^{m}}{m}\frac{p_{n}}{m}}{1+\frac{p^{0}}{m}}}\\\end{array}\right).
a^{-1}(sin{\theta_{n}^{0}})^{m_{1}+l_{1}}C_{N-m_{1}}^{m_{1}+\frac{n-1}{2}}(cos{\theta_{n}^{0}})C_{N^{\prime}-l_{1}}^{l_{1}+\frac{n-1}{2}}(cos{\theta_{n}^{0}})\oint_{\partialC}d^{n-1}{\eta}Y_{a}^{m_{1}}(\eta^{\theta_{n}^{0}})Y_{a^{\prime}}^{l_{1}}(\eta^{\theta_{n}^{0}})
X_{(0)}^{\mu}(x,y)=x^{\mu}+\frac{1}{2}\alpha_{0}^{\mu}\left((x+y)-(x-y)\right)+i\sum\frac{1}{m}\alpha_{m}^{\mu}e^{im(x-y)}+i\sum\frac{1}{m}\alpha_{m}^{\mu}e^{-im(x+y)}
{\calH}=i\intd\sigma\left(\bar{\calZ}_{L}{\calZ}_{L}^{\prime}-\bar{\calZ}_{R}{\calZ}_{R}^{\prime}\right)=\sum_{n=-\infty}^{\infty}n(\bar{Z}_{Ln}Z_{Ln}+\bar{Z}_{Rn}Z_{Rn})\,.
\sum_{\langle\Gamma_{q_{0}}\rangle\in\pi_{1}(Q)}\psi_{m^{\prime}}^{S}(\langle\Gamma_{q}\rangle)u_{m^{\prime}m}^{*}(\langle\Gamma_{q_{0}}\rangle)\chi_{F}^{m^{\prime\prime}}(\langle\Gamma_{q}\rangle)u_{m^{\prime\prime}m}(\langle\Gamma_{q_{0}}\rangle).
{\calL}_{gf}=-\sqrt{-{\bar{g}}}{\bar{g}}^{\mu\nu}{\bar{g}}^{\rho\sigma}{\bar{g}}^{\alpha\beta}\left({\bar{D}}_{\rho}g_{\mu\sigma}-{\frac{1}{2}}{\bar{D}}_{\mu}g_{\rho\sigma}\right)\left({\bar{D}}_{\alpha}g_{\nu\beta}-{\frac{1}{2}}{\bar{D}}_{\nu}g_{\alpha\beta}\right).
{\calD}_{1}\left(x^{2}\right)=\frac{m}{2\pi^{2}x^{2}}\Biggl[\frac{K_{1}(m|x|)}{\left|x\right|}+\frac{m}{2}\Biggl(K_{0}(m|x|)+K_{2}(m|x|)\Biggr)\Biggr].
3\frac{\dot{a}}{a}\frac{\partial\Omega}{\partialr}+\frac{\partial^{2}\Omega}{\partialt\partialr}=0,\qquad3\frac{\dot{a}}{a}\frac{\partial\Omega}{\partial\theta}+\frac{\partial^{2}\Omega}{\partialt\partial\theta}=0,
2\pix\sum_{m=1}^{\infty}\int_{0}^{1}{\frac{\pi^{2}B_{2}(t)}{(x^{2}+(2\pi(t+m))^{2})^{{\frac{3}{2}}}}}dt=2\pi^{3}x\sum_{k=0}^{\infty}{\binom{-{\frac{3}{2}}}{k}}{\frac{x^{2k}}{(2\pi)^{3+2k}}}\sum_{m=1}^{\infty}\int_{0}^{1}{\frac{B_{2}(t)}{(t+m)^{3+2k}}}dt\.
S=S_{0}^{L}+\intd^{4}x\left(A^{*\alpha(\lambda)}\partial_{\alpha}C_{(\lambda)}+\varepsilon^{\alpha\beta\gamma\delta}B_{\alpha\beta(\lambda)}^{*}\partial_{\gamma}\eta_{\delta}^{\;\;(\lambda)}+\eta_{\;\;\;(\lambda)}^{*\alpha}\partial_{\alpha}\eta^{(\lambda)}\right).
{\calE}_{1}\;=\;{\frac{1}{2}}\int{\frac{d^{3}k}{(2\pi)^{3}}}\left[\epsilon(k)\;-\;{\frac{k^{2}}{2m}}\left(1+{\frac{gm\rho}{k^{2}}}-{\frac{g^{2}m^{2}\rho^{2}}{2k^{4}}}\right)\right]\;,
{\calS}=\intd^{4}x\sqrt{-g}\left[{\frac{{\calR}-2\Lambda}{16\piG}}-{\frac{1}{2}}\partial_{\mu}\Phi^{a}\partial^{\mu}\Phi^{a}-{\frac{\lambda}{4}}(\Phi^{a}\Phi^{a}-\eta^{2})^{2}\right]\,,
\nabla_{\mu}\nabla^{\mu}T-\frac{\nabla_{\mu}\nabla_{\nu}T\nabla^{\mu}T\nabla^{\nu}T}{1+\nabla_{\alpha}T\nabla^{\alpha}T}-\nabla_{\mu}T\nabla^{\mu}\Phi-\frac{1}{V}\frac{dV}{dT}=0,
V(\rho)\approx\frac{3E\rho_{*}^{4}}{2(\rho^{2}-l^{2})^{2}}+\frac{L^{2}}{2m_{p}\rho^{2}}\left(1-\frac{2\rho_{*}^{4}}{(\rho^{2}-l^{2})^{2}}\right).
\Biggl[-\frac{1}{2}\frac{d^{2}}{dx_{\parallel}^{2}}-\frac{1}{2}\Biggl(\omega+qE_{0}x_{\parallel}\Biggr)^{2}\Biggr]\phi_{\omega,{\bfk}_{\perp}}(x_{\parallel})=-\frac{1}{2}(m^{2}+{\bfk}_{\perp}^{2})\phi_{\omega,{\bfk}_{\perp}}(x_{\parallel}).
\left[\begin{array}{ccc}{f_{L}^{1}}&{f_{L}^{2}}&{f_{L}^{3}}\\{f_{R}^{1}}&{f_{R}^{2}}&{f_{R}^{3}}\\\end{array}\right]\longrightarrow\left[\begin{array}{ccc}{(J_{1},K_{1}^{1})}&{(J_{2},K_{2}^{1})}&{(J_{3},K_{3}^{1})}\\{(P_{1},K_{1}^{2})}&{(P_{2},K_{2}^{2})}&{(P_{3},K_{3}^{2})}\\\end{array}\right]
\left.\frac{dC_{V}}{dT}\right|_{T_{c}^{+}}-\left.\frac{dC_{V}}{dT}\right|_{T_{c}^{-}}=\left.\OmegaT_{c}^{+}\mu^{\prime\prime}(T_{c}^{)}\frac{\partial\varrho(T,\mu)}{\partialT}\right|_{T=T_{c}^{+}}
\Omega_{\pm}^{A}\vertn\colonp\rangle=exp\biggl[-e\int\sum_{\lambda=1,2}{\frac{dk^{+}}{\sqrt{2k^{+}}}}\int{\frac{d^{2}k_{\perp}}{(2\pi)^{\frac{3}{2}}}}[f(k,\lambda\colonp)a^{\dagger}(k,\lambda)-f^{*}(k,\lambda\colonp)a(k,\lambda)]\biggr]\vertn\colonp\rangle\;.
\left(\!\begin{array}{c}{z_{1}^{\prime}}\\{z_{2}^{\prime}}\\{z_{3}^{\prime}}\\{z_{4}^{\prime}}\\\end{array}\!\right)={\frac{1}{2}}\left(\!\begin{array}{rrrr}{1}&{1}&{1}&{1}\\{1}&{1}&{-1}&{-1}\\{1}&{-1}&{1}&{-1}\\{1}&{-1}&{-1}&{1}\\\end{array}\!\right)\!\!\left(\!\begin{array}{c}{z_{1}}\\{z_{2}}\\{z_{3}}\\{z_{4}}\\\end{array}\!\right).
\gamma_{r}=e^{-i\gamma_{5}\theta}\gamma_{0}=\left(\begin{array}{cc}{0}&{e^{-i\theta}}\\{e^{i\theta}}&{0}\\\end{array}\right),\qquad\gamma_{\theta}=e^{-i\gamma_{5}\theta}\gamma_{1}=\left(\begin{array}{cc}{0}&{-ie^{-i\theta}}\\{ie^{i\theta}}&{0}\\\end{array}\right).
\nu~=~N\int_{0}^{\infty}dt\,e^{-Nt}\,\left.\left[1-\frac{16t}{3\pi^{2}}+\frac{32(27\pi^{2}+608)t^{2}}{81\pi^{4}}\right]^{-1}\right|_{N\,=\,8}~=~0.98
\xi_{FI,2}^{5_{3}}\sim(i\sqrt{3})\mathrm{Tr}\tilde{B}_{1}^{5_{3}}(2\phi_{1,1,1}-\phi_{1,1,2}-\phi_{1,1,3})+(-3i)\mathrm{Tr}\tilde{B}_{1}^{5_{3}}(\rho_{1,1,2}-\rho_{1,1,3})\;,
A_{z}=\frac{\langlea~\partial_{z}g~g^{-1}\rangle}{1+\langlea~g~a~g^{-1}\rangle},~~~~~~A_{\bar{z}}=\frac{\langlea~g^{-1}~\partial_{\bar{z}}g\rangle}{1+\langlea~g~a~g^{-1}\rangle}.
2i\|z_{13}|^{-3/2}|z_{14}|^{-3/2}|z_{15}|^{-1}|z_{23}|^{-3/2}|z_{24}|^{-3/2}|z_{25}|^{-1}|z_{35}|^{-1}|z_{45}|^{-1}\cdot\biggl(\frac{z_{14}z_{23}-z_{13}z_{24}}{z_{12}z_{34}}\biggr)
\partial_{z}\partial_{\bar{z}}\psi-(\partial_{z}\psi)(\partial_{\bar{z}}\psi)-\frac{1}{2}\partial_{\varphi}^{2}V(\varphi_{0})=0\,,\quad\partial_{z}\partial_{\bar{z}}\psi+(\partial_{z}\psi)(\partial_{\bar{z}}\psi)-m^{2}=0\,.
{\calL}_{\mathrm{YM}}={\calL}_{\mathrm{YM}}^{(i)}+{\calL}_{\mathrm{YM}}^{(a)},\quad{\calL}_{\mathrm{YM}}^{(i)}=-\frac14\left({\calF}_{\mu\nu}^{i}\right)^{2},\quad{\calL}_{\mathrm{YM}}^{(a)}=-\frac14\left({\calF}_{\mu\nu}^{a}\right)^{2}.
\hat{a}_{a}^{\dagger}A_{\;\;\;cb}^{ab}\hat{a}^{c}+\hat{a}_{a}^{\dagger}A_{\;\;\;bd}^{ab}\hat{a}^{d}+\hat{a}_{b}^{\dagger}A_{\;\;\;ca}^{ab}\hat{a}^{c}+\hat{a}_{a}^{\dagger}A_{\;\;\;ad}^{ab}\hat{a}^{d}
0=\mathbf{\tilde{Q}}_{m}^{a}\frac{\delta_{L}}{\deltaB^{\alpha}}\Gamma_{m}\equiv\mathbf{\tilde{Q}}_{m}^{a}\Big(\frac{\delta_{L}}{\deltaB^{\alpha}}\Gamma_{m}+\epsilon^{cd}E_{\alphadc}^{*}\Big)-\epsilon^{ab}\Big(\frac{\delta_{L}}{\deltaC^{\alphab}}\Gamma_{m}-H_{\alphab}\Big);
\frac{g_{*}^{2}}{C_{\tilde{O}}}{\cal{F}}_{V,1}^{(\tilde{\eta}_{o})}(Y,W)=H^{-1}(\eta,x)\left[{\cal{G}}_{1}^{(\tilde{\eta}_{o})}(x_{1},x_{3},x_{2},x_{4})-{\cal{G}}_{1}^{(\tilde{\eta}_{o})}(x_{1},x_{4},x_{3},x_{2})\right].
\left(\begin{array}{c}{{\mathbf{E}^{\prime}}}\\{\mathbf{B}^{\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{c}\\{b}&{d}\\\end{array}\right)\left(\begin{array}{c}{\mathbf{E}}\\{\mathbf{B}}\\\end{array}\right)
S_{0}=\intd^{3}\sigma\,\left(\frac{1}{g^{2}}-\frac{1}{4}F_{ab}F^{ab}-\frac{1}{2}D_{a}Y^{A}D^{a}Y^{A}-\frac{g^{2}}{4}\left(\left\{Y^{A},Y^{B}\right\}\right)^{2}\right)\,,
I_{H}^{BH}={\frac{\beta}{r_{+}}}\left[-r_{+}^{4}+r_{+}^{2}(1+\sum_{i<j}^{4}q_{i}q_{j})+r_{+}(\sum_{i=1}^{4}q_{i}+2\sum_{i<j<k}^{4}q_{i}q_{j}q_{k})+3\prod_{i=1}^{4}q_{i}\right]\,
\begin{array}{c}{\displaystyleZ_{N}(\vartheta)=N\left[\phi(\vartheta+\Theta,\displaystyle\frac{1}{2})+\phi(\vartheta-\Theta,\displaystyle\frac{1}{2})\right]+\displaystyle\sum_{k=1}^{N_{H}}\phi(\vartheta-h_{k},1)+}\\{-\displaystyle\sum_{k=1}^{M_{C}+M_{W}}\phi(\vartheta-\xi_{k},1)-\displaystyle\sum_{k=1}^{M_{R}+N_{H}}\phi(\vartheta-x_{k},1)}\\\end{array}
I=\intdx^{+}dx^{-}(4\Omega\partial_{+}\partial_{-}\rho+\frac{\mu}{2}e^{2\rho}\Omega^{1-\lambda-\gamma/4}+\Omega\partial_{+}f\partial_{-}f-e^{\chif-2\rho}\Omega^{1+\gamma/4-\epsilon/2}F_{-+}^{2}),
{\calL}=\frac{1}{4\pi^{2}g}\bigg\{-\sqrt{R^{2}(1-{\dot{R}}^{2}+{R^{\prime}}^{2}-E_{z}^{2})+B^{2}(1-{\dot{R}}^{2})}+\frac{h}{2}R^{2}\bigg\}.
\phi^{\mathrm{f}\to\mathrm{inter}}={\frac{1}{u}}\left[\alpha\left({\frac{u}{2}}\right)^{\nu}{\frac{1}{\Gamma(\nu+1)}}+\beta\left({\frac{u}{2}}\right)^{-\nu}{\frac{1}{\Gamma(-\nu+1)}}\right].
e^{-1}\mathcal{L}=\frac{1}{2}R-{\frac{1}{4}}G_{IJ}F_{\mu\nu}{}^{I}F^{\mu\nuJ}-\frac{1}{2}\mathcal{G}_{ij}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{-1}}{48}}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^{I}F_{\rho\sigma}^{J}A_{\lambda}^{K}.
\sum_{c_{3}^{\prime},c_{2}^{\prime},c_{1}^{\prime}}\phi_{c_{3},c_{2},c_{1}}^{C_{3},C_{2},C_{1};c_{3}^{\prime},c_{2}^{\prime},c_{1}^{\prime}}e_{c_{3}^{\prime}}^{C_{3}}\otimes(e_{c_{2}^{\prime}}^{C_{2}}\otimese_{c_{1}^{\prime}}^{C_{1}})=\phi((e_{c_{3}}^{C_{3}}\otimese_{c_{2}}^{C_{2}})\otimese_{c_{1}}^{C_{1}}
Z=\sum_{h_{ij}}\prod_{ni}\int_{-\infty}^{\infty}\!dA_{i}(n)\;exp\:(-\frac{1}{4\kappa^{2}}{\sum_{nij}\:[\bigtriangleup_{i}A_{j}(n)-\bigtriangleup_{j}A_{i}(n)+h_{ij}(n)]^{2}}).
H=Tr(a_{0}^{2})+2\sum_{k=1}^{\infty}Tr(a_{k}^{+}a_{k}^{-})+{\frac{\pi}{L}}\sum_{k=1}^{\infty}kTr(a_{k}^{+}b_{k}^{-}+b_{k}^{+}a_{k}^{-}).
A^{i_{1}\ldotsi_{n}}A^{j_{1}\ldotsj_{n}}\bar{B}_{k_{1}\ldotsl_{m}}\bar{B}_{l_{1}\ldotsl_{m}}D_{i_{1}j_{1},\ldots,i_{n}j_{n}}^{k_{1}l_{1},\ldots,k_{m}l_{m}}
{V(\phi,\psi)=-\frac{1}{2\kappa^{2}}e^{-\frac{3}{7}\sqrt{14}\kappa\psi}\left(\frac{3}{2}e^{-\frac{4}{21}\sqrt{21}\kappa\phi}+12e^{\frac{1}{7}\sqrt{21}\kappa\phi}-3e^{\frac{10}{21}\sqrt{21}\kappa\phi}\right)+\frac{1}{2}F^{2}e^{-\sqrt{14}\kappa\psi}.}
b_{m}(t)=b_{m}(0)+i\sum_{n\neqm}\int_{0}^{t}dt^{\prime}e^{-i\lambda\int_{0}^{t^{\prime}}dt^{\prime\prime}[E_{n}(t^{\prime\prime})-E_{m}(t^{\prime\prime})]}g_{mn}(t^{\prime})b_{n}(t^{\prime}),
\Pi_{\mu\nu}=\left(\begin{array}{ccc}{\mathit{\Pi}_{\mathit{0}}+\mathit{\Pi}_{\mathit{0}}\,^{\prime}\,k^{2}}&{0}&{\mathit{\Pi}_{\mathit{1}}k}\\{0}&{0}&{0}\\{-\mathit{\Pi}_{\mathit{1}}k}&{0}&{\mathit{\Pi}_{\,\mathit{2}}k^{2}}\\\end{array}\right)