harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int[d\phi^{A}]\Big(\partial_{\mu}J_{a}^{\mu}(x)-\hbar(\,J_{a}\,)_{Reg.}\vert_{_{\chi\,=\,0}}\,(x)\,-\,J^{r}{\overline{R}}_{a}^{r}\,(x)\,\,\Big)exp{\frac{i}{\hbar}}\Big(S_{BV}+\,\,J^{A}\,\phi_{A}\Big)\,\,\,=\,\,0
L_{CS}=\frac{1}{2e}(\dot{x}_{\mu}-\frac{i}{2}v\epsilon_{\mu\nu\lambda}\xi_{a}^{\nu}\xi_{a}^{\lambda})^{2}-\frac{e}{2}m^{2}+2i\numv\xi_{1}^{\mu}\xi_{2\mu}-\frac{i}{2}\xi_{a\mu}\dot{\xi}_{a}{}^{\mu}.
e^{-2\phi(x^{+},x^{-})}=\frac{N}{48}\frac{\displaystyle\left(C_{3}^{+}+\frac{\Lambda}{8}C_{4}^{-}\right)x^{+}+\left(C_{3}^{-}+\frac{\Lambda}{8}C_{4}^{+}\right)x^{-}}{\displaystyle1+\frac{\Lambda}{8}x^{+}x^{-}}+a.
\Psi({\bfr})\stackrel{\scriptscriptstyle(r\rightarrow\infty)}{\sim}\frac{\lambda\,\mu^{\epsilon}\,\Psi({\bf0})}{4\pi}\left(\frac{\kappa}{2\pi}\right)^{({D}-3)/2}\,\frac{e^{-\kappar}}{r^{({D}-1)/2}}\left[1+O(1/r)\right]\;,
2\kappa^{2}S_{\Phi}={\frac{\kappa^{2}N}{2\pig_{s}}}\,{\frac{1}{\sqrt{G_{\Omega\Omega}}}}\left({\frac{3}{4}}{e^{3\Phi(r)/4}}\mu_{6}+{\frac{1}{4}}{\frac{e^{-\Phi(r)/4}}{V_{\mathrm{E}}(r)}}\mu_{2}\right)\,.
\Pi_{\mu\nu}^{ij}(p)=-\int\!\frac{d^{2}k}{i(2\pi)^{2}}\frac{(2k+p)_{\mu}(2k+p)_{\nu}}{(k^{2}-m_{i}^{2})[(k+p)^{2}-m_{j}^{2}]}+\int\!\frac{d^{2}k}{i(2\pi)^{2}}\frac{2g_{\mu\nu}}{k^{2}-m_{i}^{2}}.
S\,=\,{\frac{1}{4\pi\alpha^{\prime}}}\int_{\Sigma}d^{2}\sigma\Big(\eta_{ij}\partial_{a}X^{i}\partial_{b}X^{j}g^{ab}\,+\,\epsilon_{ab}{\calB}_{ij}\partial_{a}X^{i}\partial_{b}X^{j}\Big)
C_{\Omega}=2^{5/3}3^{-1}\pi^{10/3}\kappa^{-4/3}N^{1/2}m^{1/2}r_{+}\frac{6r_{+}^{4}+l^{2}r_{+}^{2}-2l^{4}}{3r_{+}^{4}+5l^{2}r_{+}^{2}+2l^{4}}\frac{3r_{+}^{2}+2l^{2}}{3r_{+}^{2}-2l^{2}}.
\begin{array}{cl}{}&{H^{(nl)}(z_{2},\cdots,z_{n+1}|z_{n+2},\cdots,z_{N},z_{1}\tau^{4})}\\{=}&{\displaystyle\prod_{j=2}^{n+1}(z_{1}\tau^{4}-z_{j}\tau^{2})^{-1}\displaystyle\prod_{j=2}^{n+1}\displaystyle\prod_{i=n+2}^{N}(z_{i}-z_{j}\tau^{2})^{-1}I^{(nl)}(z_{2},\cdots,z_{n+1}|z_{n+2},\cdots,z_{N},z_{1}\tau^{4}),}\\\end{array}
{y}=y_{(3)}={\frac{1}{\pi_{C^{\prime}}{\bar{\pi}}_{C}t^{C{C^{\prime}}}}}[{\frac{1}{\sqrt{2}}}({\bar{\alpha}}_{A}{\bar{\alpha}}^{B}+{\bar{\beta}}_{A}{\bar{\beta}}^{B}){\bar{\pi}}_{B}\omega^{A}+{\frac{1}{\sqrt{2}}}(\alpha_{A^{\prime}}\alpha^{B^{\prime}}+\beta_{A^{\prime}}\beta^{B^{\prime}})\pi_{B^{\prime}}{\bar{\omega}}^{A^{\prime}}].
x_{A\alpha\beta}=x_{\alpha\beta}+i\theta_{(\alpha}^{++}\theta_{\beta)}^{--}+i\theta_{(\alpha}^{(++)}\theta_{\beta)}^{(--)}+i\theta_{(\alpha}^{[+]\{+\}}\theta_{\beta)}^{[-]\{-\}}+i\theta_{(\alpha}^{[+]\{-\}}\theta_{\beta)}^{[-]\{+\}}\;.
\begin{array}{ccccccc}{}&{\circ_{0}}&{\left(\frac{1}{g}\right)}&{}&{}&{}&{}\\{}&{\mid}&{}&{}&{}&{}&{}\\{\left(\frac{R_{1}}{g}\right)\;\;\circ_{1}-}&{\circ_{2}}&{-}&{\circ_{3}}&{-\dots-}&{\circ_{d-1}\;\;\left(\frac{1}{R_{d}}\right)}\\\end{array}
\chi^{(b)}\;:=\;\frac{\varepsilon_{\mu\nu}\partial_{\mu}}{\triangle}\left(\frac{e\pi}{\pi+gN}A_{\nu}\;+\;g^{1/2}h_{\nu}^{\prime}\right)\;+\;e\frac{\varepsilon_{\mu\nu}\partial_{\mu}}{\triangle}a_{\nu}^{(b)}\;.
{\calM}_{T}\equiv{\frac{1}{T_{2}}}\left(\begin{matrix}{1}&{T_{1}}\\{T_{1}}&{|T|^{2}}\\\end{matrix}\right),\\\\\{\calM}_{U}\equiv{\frac{1}{U_{2}}}\left(\begin{matrix}{1}&{U_{1}}\\{U_{1}}&{|U|^{2}}\\\end{matrix}\right),
\frac{\delta^{2}\Gamma}{\delta\phi^{2}\delta\phi^{2}}=-\frac{g^{4}}{4\,(2\pi)^{3}}\intd^{4}p\,\theta(p_{0})\delta(p^{2}+m^{2})N(p_{0})\,\frac{k^{2}}{2(p\cdotk)^{2}}\,,
\tilde{C}_{\mu}^{\alpha^{\prime}}M_{\alpha^{\prime}\beta^{\prime}}\tilde{C}^{\alpha^{\prime}\mu}=\sum_{(a)}\bar{\tilde{\phi}}_{V}^{(a)}\left[-\tilde{g}_{3}\tilde{C}_{\mu}^{\alpha}\frac{\lambda_{\alpha}}{2}-\tilde{g}_{1}(\tilde{n}+1/3)\tilde{B}_{\mu}\right]^{2}\tilde{\phi}_{V}^{(a)},
\partial_{\bar{z}}{\tt\frac{1}{z}}=\frac{1}{2\theta}\left[z,{\tt\frac{1}{z}}\right]=\frac{1}{2\theta}|0\rangle\langle0|=\frac{1}{\theta}e^{-r^{2}/\theta}\,,\quad\partial_{z}{\tt\frac{1}{\bar{z}}}=\frac{1}{2\theta}\left[{\tt\frac{1}{\bar{z}}},\bar{z}\right]=\frac{1}{2\theta}|0\rangle\langle0|=\frac{1}{\theta}e^{-r^{2}/\theta}\,.
\int\frac{d^{4}q}{(2\pi)^{4}}\,\Bigg(\frac{(k+q)^{2}+q^{2}-k^{2}}{\Big((k+q)^{2}+M^{2}\Big)^{2}\Big(q^{2}+M^{2}\Big)^{2}}-\frac{4M^{2}}{\Big((k+q)^{2}+M^{2}\Big)\Big(q^{2}+M^{2}\Big)^{3}}\Bigg)=0.
\begin{array}{rcl}{\hat{f}_{0}}&{=}&{-4c{\hat{g}^{2}}(1+xp_{x}+x^{2}p_{xx})+2c\hat{g}x(3-c)(1+p_{x}x)-2c{x^{2}}(1-c^{2})-{\hat{g}^{2}}{x^{2}}}\\{\hat{f}_{1}}&{=}&{4c{\hat{g}^{2}}-2c\hat{g}x(3-c)+2c{x^{2}}(1-c^{2})+\hat{g}{x^{3}}(1-3c)-2{\hat{g}^{2}}p_{x}{x^{3}}}\\{\hat{f}_{2}}&{=}&{x^{2}\left({\hat{g}^{2}}-\hat{g}{x}(1-5c)-2c{x^{2}}(1+c)\right)}\\{\hat{f}_{3}}&{=}&{\hat{g}x^{5}}\\\end{array}
\widetilde{\psi}_{-}(x^{+},x^{-},k_{\bot})=\Biggl({\frac{m-k_{\bot}\cdot\gamma_{\bot}}{\omega_{\bot}^{2}}}\Biggr)\gamma^{+}i\partial_{+}\widetilde{\psi}_{+}(x^{+},x^{-},k_{\bot})\;.
\sum_{n=0}^{\infty}\epsilon(n)=\int_{0}^{\infty}{dn}\,\epsilon(n)+{\frac{1}{2}}\,\epsilon(0)-{\frac{1}{12}}\,\epsilon^{\prime}(0)+{\frac{1}{720}}\,\epsilon^{\prime\prime\prime}(0)-{\frac{1}{30240}}\,\epsilon^{(v)}(0)+\cdots,
\begin{array}{l}{e^{+2}\nabla\psi_{+2q}^{2+}+e^{-2}\nabla\psi_{-2q}^{2+}+e_{p}^{+}\nabla\chi_{pq}^{2}+e_{\dot{p}}^{-}\nablah_{-\dot{p}q}^{2+}+T^{+2}\psi_{+2q}^{2+}+T^{-2}\psi_{-2q}^{2+}+T_{p}^{+}\chi_{pq}^{2}+T_{\dot{p}}^{-}h_{-\dot{p}q}^{2+}}\\{=-\frac12\gamma_{q\dot{q}}^{i}\Omega^{+2i}e_{\dot{q}}^{-}.}\\\end{array}
\hat{\cal{L}}_{DJ}=-\frac{1}{2}(\partial_{\mu}\hat{A}_{\nu}-\partial_{\nu}\hat{A}_{\mu})\partial^{\mu}\hat{A}^{\nu}-(\partial_{\mu}\hat{A}_{\nu}-\partial_{\nu}\hat{A}_{\mu})\partial^{\mu}\hat{a}^{\nu}+\frac{m}{2}\epsilon_{\mu\alpha\beta}(\hat{A}^{\mu}\partial^{\alpha}\hat{A}^{\beta}+2\hat{A}^{\mu}\partial^{\alpha}\hat{a}^{\beta}).
\partial_{+}g_{L}g_{L}^{-1}=\left(\begin{array}{cc}{v_{L}\partial_{+}u_{L}+b_{L}\partial_{+}a_{L}}&{u_{L}\partial_{+}a_{L}-a_{L}\partial_{+}u_{L}}\\{b_{L}\partial_{+}v_{L}-v_{L}\partial_{+}b_{L}}&{u_{L}\partial_{+}v_{L}+a_{L}\partial_{+}b_{L}}\\\end{array}\right),
M=\left(\begin{array}{ccccc}{0}&{P_{3}}&{-P_{2}}&{E_{1}}&{B_{1}}\\{-P_{3}}&{0}&{P_{1}}&{E_{2}}&{B_{2}}\\{P_{2}}&{-P_{1}}&{0}&{E_{3}}&{B_{3}}\\{-E_{1}}&{-E_{2}}&{-E_{3}}&{0}&{N}\\{-B_{1}}&{-B_{2}}&{-B_{3}}&{-N}&{0}\\\end{array}\right).
X_{L}(a,b)=\sum_{j\geq0}\Bigl\{G_{L}(\underbrace{\downarrow\uparrow\cdots\downarrow\uparrow}_{2j})+G_{L}(\underbrace{\uparrow\downarrow\cdots\uparrow\downarrow}_{2j+2})-G_{L}(\underbrace{\downarrow\uparrow\cdots\uparrow\downarrow}_{2j+1})-G_{L}(\underbrace{\uparrow\downarrow\cdots\downarrow\uparrow}_{2j+1})\Bigr\},
\frac{1}{2\lambdaf^{2}}\int\;d^{4}X\,\frac{d^{4}q}{\left(2\pi\right)^{4}}\left(\varphi(X)\right)^{2}\tilde{\pi}_{0}(q)\left[\partial_{q}^{2}+\frac{4i\lambda}{q^{2}-\Sigma^{2}(q)}\right]\tilde{\pi}_{0}(q),
\begin{array}{lcr}{(|X_{1}|^{2}-|X_{2}|^{2})-(|Y_{1}|^{2}-|Y_{2}|^{2})=R_{2}}\\{X_{1}\overline{{Y}}_{1}-X_{2}\overline{{Y}}_{2}=\overline{{X}}_{1}Y_{1}-\overline{{X}}_{2}Y_{2}=0.}\\\end{array}
\begin{array}{rl}{}&{<0|\cdotsT(\pi_{1}\pi_{2}\pi_{5}\stare^{\pmip\cdotx_{5}}\star\pi_{5})\cdots|0>}\\{=}&{\displaystyle\sum_{\lambda_{1}\lambda_{2}}\tau_{15}^{\lambda_{1}}\tau_{25}^{\lambda_{2}}(D_{15}^{\lambda_{1}}\stare^{\pmip_{+}\cdotx_{5}}\starD_{25}^{\lambda_{2}}+(1\leftrightarrow2))<0|\cdots|0>.}\\\end{array}
\frac{4\lambda^{2}}{tanh^{2}(\lambday)}\frac{\partial^{2}\Psi}{\partial\theta^{2}}+\frac{1}{tanh(\lambday)}\frac{\partial}{\partialy}(tanh(\lambday)\frac{\partial\Psi}{\partialy})+\tilde{\kappa}^{2}\Psi=0
L=\bar{\psi}i\gamma_{\mu}\partial_{\mu}\psi-j_{\mu}(A_{\mu}+B_{\mu}+\epsilon_{\mu\nu}\partial_{\nu}\theta)+\frac{1}{\pi}A_{\mu}\epsilon_{\mu\nu}\partial_{\nu}\theta+\frac{1}{2\pi}\partial_{\nu}\theta\partial_{\nu}\theta\;,
\mathrm{ch}(E)=n+\frac{1}{2}\mathrm{tr}(\thetam)+q\cdot\mathrm{Pfaff}(\theta)+\frac{1}{2}(m+q\cdot\tilde{\theta})_{jk}\alpha^{j}\alpha^{k}+q\alpha^{1}\alpha^{2}\alpha^{3}\alpha^{4}
l_{3}=\left(\frac{N_{2}^{(1)}\bar{N}_{2}^{(1)}N_{5}^{(2)}\bar{N}_{5}^{(2)}}{N_{2}^{(2)}\bar{N}_{2}^{(2)}N_{5}^{(1)}\bar{N}_{5}^{(1)}}\right)^{\frac14}l_{2}
\frac{\partial}{\partialq_{tA}}=\frac{\partialr}{\partialq_{tA}}\frac{\partial}{\partialr}+\frac{\partiale_{B}}{\partialq_{tA}}\frac{\partial}{\partiale_{B}}+\frac{\partialE_{s}}{\partialq_{tA}}\frac{\partial}{\partialE_{s}}+\frac{\partialy_{sB}}{\partialq_{tA}}\frac{\partial}{\partialy_{sB}}\;.
{\calA}_{V_{\propto}+\mathrm{NO}}=-{\frac{ig^{2}}{(2\pi)^{26}}}2^{12}\pi\int_{0}^{1}{\frac{dq^{\prime}}{q^{3}}}\int_{0}^{1}d\nu_{12}^{\prime}\,\left[\,f(-q^{2})\,\right]^{-24}\left[\,F\left(\nu_{12}^{\prime},-q^{2}\right)\,\right]^{-2},
D^{\sigma\lambda,\alpha\mu}\equiv\frac{1}{2}(g^{\sigma\lambda}R^{\alpha\mu}-g^{\alpha\mu}R^{\sigma\lambda}+g^{\alpha\sigma}R^{\lambda\mu}-g^{\lambda\mu}R^{\alpha\sigma}+g^{\alpha\lambda}R^{\sigma\mu}-g^{\sigma\mu}R^{\alpha\lambda}.
I_{CWZW}[\alpha]=\intdx^{0}\left(-\sum_{n\neq0}\frac{\dot{\alpha}_{0}^{a}\alpha_{a\:n}}{ikn}+\sum_{n\ge1}\frac{2\dot{\alpha}_{-n}^{a}\alpha_{a\:n}}{ikn}+\tauL_{0}-2\pi\alpha_{0}^{3}\right),
\begin{array}{lcl}{C(\Lambda)E_{-\alpha}(\mu)\omega}&{=}&{\frac{2}{\Lambda-\mu}\left(E_{-\alpha}(\Lambda)h_{\alpha}(\mu)\right.}\\{}&{}&{\left.-E_{-\alpha}(\mu)h_{\alpha}(\Lambda)\right)\omega+E_{-\alpha}(\Lambda)C(\Lambda)\omega}\\\end{array}
{\calL}_{\mathrm{spin}}^{(2)}[B]={(4\pi)}^{-D}e^{2}\int_{0}^{\infty}{\frac{dT}{T}}e^{-m^{2}T}T^{2-D}\int_{0}^{\infty}d\hat{T}\int_{0}^{1}du_{a}\,J(z,u_{a},\hat{T},D)\,\nonumber\,
S_{E}={\frac{1}{2\kappa_{10}^{2}}}\intd^{10}x\sqrt{-G^{E}}\left[{\calR}_{G^{E}}-{\frac{1}{2}}(\partial\Phi)^{2}-{\frac{1}{2\cdot(p+2)!}}e^{2b\Phi}F_{p+2}^{2}\right],
X(1-X)e^{i(p_{1}+p_{2}+p_{3})2\pi\delta}\Bigl(\bigl(1-e^{i(p_{1}-p_{2})2\pi\delta}\bigr)\bigl(1-e^{i(p_{2}-p_{3})2\pi\delta}\bigr)\bigl(1-e^{i(p_{3}-p_{1})2\pi\delta}\bigr)\Bigr)=0
f^{(4)}=(N_{0},N_{1}^{\mathrm{SL}},N_{1}^{\mathrm{TL}},N_{2}^{\mathrm{SL}},N_{2}^{\mathrm{TL}},N_{3}^{\mathrm{SL}},N_{3}^{(3,1)},N_{3}^{(2,2)},N_{4}^{(4,1)},N_{4}^{(3,2)}).
\begin{array}{ll}{({\calF}^{k})^{ij}={\calF}^{il_{1}}{\calF}_{l_{1}l_{2}}\cdots{\calF}^{l_{k-1}j}}&{k>0\,,}\\{({\calF}^{k})^{ij}=g^{ij}}&{k=0\,.}\\\end{array}
\frac{\partial\Gamma}{\partial\theta}=\frac{\delta\Gamma}{\delta\Phi_{i}^{\prime}}\Phi_{i,\theta}^{\prime}+\frac{\delta\Gamma}{\delta<\zeta>}<\zeta_{,\theta}>+\frac{\delta\Gamma}{\deltaA_{i,\lambda}^{\prime}}\frac{\delta\Gamma}{\deltaA_{i}}\eta_{i}
\int_{-1/K_{p}}^{1/K_{p}}dy\,u_{0}^{2}(y)w(y)\sim\left.(1-K_{p}|y|)^{\frac{8(q+1)(n-1)+4(p+1)(n+1)+(p+n)^{2}a_{p}^{2}}{(p+n)^{2}a_{p}^{2}+4(p+1)n}}\right|_{-1/K_{p}}^{1/K_{p}}
{\calL}_{\mathrm{CS}}=-\frac{g_{5}^{2}}{128\pi^{2}}\,\frac{1}{2}\,\mathrm{Sgn}_{\pi}(\pi/2-y/R)\,A_{5}(x,y)\,\,F_{\mu\nu}(x,y)\widetilde{F}^{\mu\nu}(x,y)\,
S_{2}(\phi,\phi)=-\intd^{D}X\sqrt{-G}\left\{(D_{M}\phi)^{\dagger}D^{M}\phi+\frac{1}{2}\phi\frac{\partial^{2}U}{\partial\Phi^{2}}\phi+\frac{1}{2}\phi^{*}\frac{\partial^{2}U}{\partial{\Phi^{*}}^{2}}\phi^{*}+\phi^{*}\frac{\partial^{2}U}{\partial\Phi\partial\Phi^{*}}\phi\right\}~,
\omega_{i}^{2}\partial_{z}(\omega_{j}/\omega_{i})=\sum_{p=1}^{5h-5}a_{ij}^{p}\psi_{p+s(3)-1}^{(3)},\qquada_{ij}^{p}={\frac{1}{2\pii}}\oint_{{\calC}}\psi_{-p-s(3)+1}^{(-2)}
b=\frac{2\mu}{3}\frac{i}{2}\left(\begin{array}{ccc}{-2}&{}&{}\\{}&{1}&{}\\{}&{}&{1}\\\end{array}\right)-\frac{g}{12\pir}\frac{i}{2}\left(\begin{array}{ccc}{-2}&{}&{}\\{}&{1}&{}\\{}&{}&{1}\\\end{array}\right).
ds_{\mathrm{inv}}^{2}=\sum_{i,j}g_{ij}g_{ij}dg^{ii}dg^{jj}+\frac{1}{2}\sum_{i<j;k<l}\left(g_{ik}g_{jl}+g_{il}g_{jk}\right)dg^{ij}dg^{kl}+2\sum_{i;k<l}g_{ik}g_{il}dg^{ii}dg^{kl}
\phi^{\prime\prime}+(D-1)\frac{a^{\prime}}{a}\phi^{\prime}-\frac{b^{\prime}}{b}\phi^{\prime}-\frac{\lambda\Lambda}{8}b^{2}e^{\lambda\phi/(D-2)}\,\nonumber-\frac{\lambdaV}{2b}e^{\lambda\phi/2(D-2)}\delta(y)=0.
(gt_{0})^{A}{}_{B}\equiv(d^{-1})^{AC}\eta_{CB}=\left(\begin{matrix}{0}&{0}&{0}\\{0}&{0}&{(d^{-1})^{\bar{I}K}\eta_{K\beta}}\\{(d^{-1})^{\alpha\gamma}\eta_{\gammaJ}}&{0}&{(d^{-1})^{\alpha\gamma}\eta_{\gamma\beta}}\\\end{matrix}\right),\quadT^{A}=\left(\begin{matrix}{V^{I}}\\{T^{\bar{I}}}\\{T^{\alpha}}\\\end{matrix}\right),
\begin{aligned}{[m]H_{3}}&{{}=e^{2\Phi}\star_{10}\left(H_{7}-A\wedgeH_{6}\right)}\\{H_{4}}&{{}=\star_{10}H_{6}-e^{2\Phi}A\wedge\star_{10}H_{7}+e^{2\Phi}\star_{10}\imath_{A^{\sharp}}(A\wedgeH_{6})~,}\\\end{aligned}
\begin{array}{rcl}{{\calH}^{(2)}}&{=}&{2\partiald^{(1)}+\frac{1}{2\pi\alpha^{\prime}}(i_{k}C^{(3)})-2(i_{k}B^{(2)})\partialc^{(0)}}\\{}&{}&{}\\{{\calG}^{(1)}}&{=}&{\partialc^{(0)}+\frac{1}{2\pi\alpha^{\prime}}(DXC^{(1)})\,,}\\\end{array}
Z(t)=\sum_{j\geq0}(j+1)^{2}e^{-t(j+1)^{2}+t}=\frac{e^{t}}{2}\sum_{j\inZ}(j+1)^{2}e^{-t(j+1)^{2}}=-\frac{e^{t}}{2}\frac{d}{dt}\sum_{j\inZ}e^{-tj^{2}}
d\tilde{s}_{\mathrm{10D}}^{2}={\frac{r^{2}}{R^{2}}}\left[-dudv+{\frac{p(u)}{r^{2}}}du^{2}+{\frac{R^{4}}{r^{4}}}dr^{2}\right]+R^{2}d\Omega_{3}^{2}+dy_{i}^{2}
U(f,h,u)=\frac{(u^{2}-1)^{2}}{2x^{4}}+\frac{u^{2}f^{2}}{x^{2}}+\frac{f^{2}}{x^{2}}+\frac\beta4(h^{2}-1)^{2}+\frac\beta4(f^{2}-1)^{2}\.
\partial_{\bar{a}}C_{i_{1}...i_{s}}=fact({\bar{C}}_{\bar{a}})-\sum_{i=1}^{s}{\bar{C}}_{{\bar{a}}i}^{l}C_{\sigma_{1}(l)i_{1}..{\hat{i}}..i_{s}}+{\bar{C}}_{{\bar{a}}0}^{l}C_{\sigma_{2}(l)i_{1}...i_{s}}
\begin{array}{rcl}{\displaystyle\left[a_{q},a_{q}^{\dagger}\right]}&{=}&{\left[N+1+b\gamma\right]_{q}-\left[N+b\gamma\right]_{q}~,}\\{\displaystyle\left[N,a_{q}\right]}&{=}&{-a_{q}~,~~~\displaystyle\left[N,a_{q}^{\dagger}\right]=a_{q}^{\dagger}~.}\\\end{array}
{\bfH_{1}}=-{\bf1}\frac{d^{2}}{d\rho^{2}}+\left(\begin{array}{llll}{\frac{m^{2}-\frac{1}{4}}{\rho^{2}}}&{\;\;\;\frac{-2F}{\rho}}\\{\frac{-2F}{\rho}}&{\frac{(m+1)^{2}-\frac{1}{4}}{\rho^{2}}}\\\end{array}\right).
\sum_{\ell=1}^{5}x_{\ell}^{\frac{D}{w_{\ell}}}+a_{0}\prod_{\ell=1}^{5}\left(x_{\ell}\right)+\sum_{i=7}^{r+3}a_{i}\prod_{\ell=1}^{5}x_{\ell}^{n_{i}^{\ell}}=0,
|\intd^{3}x^{\prime}\int_{o}^{2\pi}d\phi^{\prime}G_{o}(K)V(x^{\prime},\phi^{\prime})f(\vec{x}^{\prime},\phi)|\leqC|K|^{2}\intd^{3}x^{\prime}\frac{\tilde{V}(x^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}e^{-ImK|\vec{x}-\vec{x^{\prime}}|}\cdot\|f\|.
\displaystyle{\mathcal{G}_{K,L}\left(A,B;\omega^{m}\right)\equiv\frac{1}{\sqrt{(K+L)!K!L!}}\partial_{x}^{K}\partial_{y}^{L}\prod_{s=0}^{K+L-1}\left(\omega^{-s/2}x\mathbf{A}+\omega^{s/2}y\mathbf{B}\right)\vert_{x=y=0}.}
\frac{\partial_{\hat{\mu}}\hat{\Phi}\partial^{\hat{\mu}}\hat{\Phi}^{*}}{(1-\hat{\Phi}\hat{\Phi}^{*})^{2}}=\frac{1}{4}\frac{\partial_{\hat{\mu}}\hat{\lambda}\partial^{\hat{\mu}}\hat{\lambda}^{*}}{(\Im\mathrm{m}\hat{\lambda})^{2}}\,,
\int_{\calM}dV(g_{\tau\tau})^{-D/2}e^{-m_{i}^{2}g_{\tau\tau}t}={\frac{\beta\beta_{H}}{4\pi}}(m_{i}^{2}t)^{\fracD2-1}\Gamma\left(1-\fracD2,~\epsilon^{2}m_{i}^{2}\left({\frac{2\pi}{\beta_{H}}}\right)^{2}t\right){\calA}^{H}~~~,
L_{f}(x_{f})=J_{\psi}L(x)=-\frac{1}{4}J_{\psi}F_{\mu\nu}^{+}F^{\mu\nu}=-\frac{1}{4}J_{\psi}S(B_{f})^{2}(F_{\mu\nu}^{(f)})^{+}F_{(f)}^{\mu\nu},
\partial_{+}\partial_{+}S\psi_{1}\psi_{2}{\widetilde\psi}_{1}{\,\bar{\!\widetilde\psi}}_{2}+\partial_{-}\partial_{-}S{\bar{\psi}}_{1}{\bar{\psi}}_{2}{\,\bar{\!\widetilde\psi}}_{1}{\widetilde\psi}_{2}+\partial_{+}\partial_{l}S\psi_{1}\psi_{2}({\widetilde\psi}_{1}{\,\bar{\!\widetilde\psi}}_{1}-{\widetilde\psi}_{2}{\,\bar{\!\widetilde\psi}}_{2})+\cdots
<BR>\{\Pi_{i}(x),\theta(y)\}^{*}=3D{\frac{\partial=_{i}}{=20{\nabla^{2}}}}\delta(x-y);~~\{\Pi_{i}(x),\Pi_{0}(y)\}^{*}=3D-m^{2}{\frac{\partial=_{i}}{=20{\nabla^{2}}}}\delta(x-y).<BR><BR>
{\calR}(-i\infty)\sim\left[\begin{array}{cccc}{1}&{}&{}&{}\\{}&{q^{-1}}&{0}&{}\\{}&{1-q^{-2}}&{q^{-1}}&{}\\{}&{}&{}&{1}\\\end{array}\right]~,~~~~{\calR}(+i\infty)\sim\left[\begin{array}{cccc}{1}&{}&{}&{}\\{}&{q}&{1-q^{2}}&{}\\{}&{0}&{q}&{}\\{}&{}&{}&{1}\\\end{array}\right]~,
2h_{\calO}\langle\!\langle{\calO}|{\calO}\rangle=2\langle\!\langle{\calO}|L_{0}|{\calO}\rangle=\langle\!\langle{\calO}|[L_{1},L_{-1}]|{\calO}\rangle=\|L_{-1}|{\calO}\rangle\|^{2}\geq0,
\Omega=-{\calH}\,\mathrm{d}z\wedge\mathrm{d}\bar{z}+\bar{p}\,\mathrm{d}z\wedge\mathrm{d}\varphi+p\,\mathrm{d}\varphi\wedge\mathrm{d}\bar{z}\stackrel{!}{=}\mathrm{d}S^{z}\wedge\,\mathrm{d}S^{\bar{z}}\,,
{}^{\sigma}\!C_{\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}}^{N_{1}N_{2}N_{3}N_{4}\,\Lambda}\=\C_{\gamma_{\sigma(1)}\gamma_{\sigma(2)}\gamma_{\sigma(3)}\gamma_{\sigma(4)}}^{N_{\sigma(1)}N_{\sigma(2)}N_{\sigma(3)}N_{\sigma(4)}\,\Lambda}
P_{L}^{-}=\sum_{N>0}\left({\frac{N\pi}{L}}\right)\alpha{_N^{\dagger}}\alpha_{N}+\sum_{n>0}\left({\frac{n\pi}{L}}\right)\left[\beta{_n^{\dagger}}\beta_{n}+\delta{_n^{\dagger}}\delta_{n}\right]-{\frac{\pi}{2L}}z_{L}^{2}\;.
S=\frac{1}{4\pi\alpha^{\prime}}\intd^{2}\sigma\left(\sqrt{g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\mu}-2\pi\alpha^{\prime}iB_{\mu\nu}^{E}\epsilon^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\right)
\sigma^{\mathrm{reg}}(\xi)=-\,{\frac{1}{12}}+\int_{0}^{|\xi|}x\,dx\int_{0}^{\infty}{\frac{1}{\sqrt{\pi}}}\,\,{\frac{dt}{\sqrt{t}}}\,e^{-\,{tx^{2}}}\Big({\frac{\textstyle1}{2}}\sqrt{{\frac{\pi}{t}}}\,-\,{\frac{1}{2}}+\sqrt{{\frac{\pi}{t}}}\sum_{n=1}^{\infty}e^{-\,{\frac{n^{2}\pi^{2}}{t}}}\Big)\.
B(p^{2})\,=\,m+\frac{e^{2}}{\pi}\int_{0}^{\infty}dk\frac{kB(k^{2})}{k^{2}+{B(k^{2})}^{2}}\frac{1}{\sqrt{(p^{2}+k^{2}+\mu^{2})^{2}-4p^{2}k^{2}}}\;,
L^{2}g_{ind}^{-1}=\left(\begin{matrix}{\frac{1-x^{2}+A^{2}x^{2}}{A^{2}b^{2}(1-x^{2})}}&{-\frac{x}{b^{2}(1-x^{2})}}&{0}&{0}\\{-\frac{x}{b^{2}(1-x^{2})}}&{\frac{1}{b^{2}(1-x^{2})}}&{0}&{0}\\{0}&{0}&{\frac{1-x^{2}}{b^{2}}}&{0}\\{0}&{0}&{0}&{-1}\\\end{matrix}\right)
\frac{d}{d\tilde{y}}\left(\begin{matrix}{f}\\{g}\\\end{matrix}\right)=\left(\begin{matrix}{2\tilde{u}^{\prime}}&{\left(4\tilde{y}^{2}+2\tilde{u}^{2}+\tilde{t}\right)+4\tilde{y}\tilde{u}}\\{-\left(4\tilde{y}^{2}+2\tilde{u}^{2}+\tilde{t}\right)+4\tilde{y}\tilde{u}}&{-2\tilde{u}^{\prime}}\\\end{matrix}\right)\left(\begin{matrix}{f}\\{g}\\\end{matrix}\right).
{\frac{d}{dt}}\left({\frac{(M-g_{s}X_{\mu}H^{\mu}){\bfV}}{\sqrt{1-{\bfV}^{2}}}}\right)=g({\bfB}_{0}-{\bfV}\times{\bfE}_{0})+q({\bfE}_{0}+{\bfV}\times{\bfB}_{0})+g_{s}{\bfH}_{0}\sqrt{1-{\bfV}^{2}}.
\alpha_{\phi}^{ij}(x;y)=\alpha^{ij}(x)+\nabla_{r}\alpha^{ij}(x)\,y^{r}+\left(\frac12\,\nabla_{r}\nabla_{s}\alpha^{ij}(x)-\frac16\,R_{rks}^{[i}(x)\alpha^{j]k}(x)\right)\,y^{r}y^{s}+\cdots,
\Sigma_{\mu_{1}\mu_{2}...\mu_{2p}}=\Sigma_{[\mu_{1}}\tilde{\Sigma}_{\mu_{2}}...\Sigma_{\mu_{2p-1}}\tilde{\Sigma}_{\mu_{2p}]},\quad\tilde{\Sigma}_{\mu_{1}\mu_{2}...\mu_{2p}}=\tilde{\Sigma}_{[\mu_{1}}\Sigma_{\mu_{2}}...\tilde{\Sigma}_{\mu_{2p-1}}\Sigma_{\mu_{2p}]}
{\begin{array}{l}{Q_{1}\R1}\\{Q_{5}\R5}\\{Q_{p}\W}\\\end{array}}\left[{\begin{array}{llllllllll}{0}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{5}&{\cdot}&{\cdot}&{\cdot}&{\cdot}\\{0}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{5}&{6}&{7}&{8}&{9}\\{0}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{5}&{\cdot}&{\cdot}&{\cdot}&{\cdot}\\\end{array}}\right]\quad.
\left(4\frac{d}{dx}+\frac{5}{x}\right)\,F_{1}^{[4]}(x)+x\frac{d}{dx}\,F_{2}^{[4]}(x)=\sum_{k=1}^{\infty}\frac{1}{8\pi^{3/2}}(2k+1)\frac{\Gamma(2k+1/2)}{\Gamma(2k)}\,{\calB}_{2k+4}\;(2x)^{2k-3}
\hat{Z}_{f}[j]=\int{\calD}A{\calD}\psi{\calD}\bar{\psi}{\frac{\delta(F(A))\delta(G(A))}{Vol({\calG}_{A})}}e^{\int_{\Sigma}d^{2}x\sqrt{g}\bar{\psi}(-i{\partial\!\!/}+{j\!\!/}+{A\!\!/})\psi},
V(\phi,\beta)=\mu^{D}\sum_{s=1}^{\infty}a^{\frac{D}{2}-s}f(D,s)(\gamma+g\phi)^{2s}\sum_{n=-\infty}^{\infty}\frac{1}{((n+\frac{1}{2})^{2})^{s-\frac{d}{2}}}.
_{[a_{1},\ldots,a_{i}]}F(x)_{[b_{1},\ldots,b_{j}]}=\sum_{k=0}^{\infty}\frac{\left(\prod_{m=1}^{i}\frac{\Gamma(a_{m}+k)}{\Gamma(a_{m})}\right)}{\left(\prod_{n=1}^{j}\frac{\Gamma(b_{n}+k)}{\Gamma(b_{n})}\right)}x^{k}
\begin{array}{rcl}{{\dot{x}_{1}}=\{x_{1},x_{5}\}=x_{2}x_{3}x_{4}x_{6}}\\{{\dot{x}_{2}}=\{x_{2},x_{5}\}=x_{1}x_{3}x_{4}x_{6}}\\{{\dot{x}_{3}}=\{x_{3},x_{5}\}=x_{1}x_{2}x_{4}x_{6}}\\{{\dot{x}_{4}}=\{x_{4},x_{5}\}=g^{2}x_{1}x_{2}x_{3}x_{6}}\\{{\dot{x}_{5}}=\{x_{5},x_{5}\}\equiv0}\\{{\dot{x}_{6}}=\{x_{6},x_{5}\}=0.}\\\end{array}
T_{F}={\frac{1}{2\sqrt{7}}}\left(e^{i\rho_{1}}\sum_{\alpha=1}^{7}e^{iQ_{\alpha}^{1}\cdot\phi}+e^{i\rho_{2}}\sum_{\alpha=1}^{7}e^{iQ_{\alpha}^{2}\cdot\phi}+e^{i\rho_{3}}\sum_{\alpha=1}^{7}e^{-iQ_{\alpha}^{3}\cdot\phi}\right)+{\mathrm{H.c.}}~,
\mathrm{Det}\left[{\calD}\right]=\left(p_{0}^{2}-p_{1}^{2}-p_{2}^{2}-p_{3}^{2}-m^{2}\right)^{2}\frac{(p_{0}^{2}-p_{1}^{2}-p_{2}^{2}-p_{3}^{2}+3m^{2}+4mp_{0})^{2}}{16m^{4}(p_{0}+m)^{4}}\quad.
|\{c_{qi}^{a}\},p_{q},\cdots,p_{1},I,k\rangle=\prod_{a=1}^{4}\prod_{j=1}^{N_{q}}(P_{q}^{a\daggerj})^{c_{qj}^{a}}|\{0\},p_{q},\cdots,p_{1},I,k\rangle,
\widetilde{K}_{l}^{(\alpha)}(0)=\frac{1}{(8\pi)^{l}}\;\frac{(-1)^{\alpha}}{\alpha!\Gamma(\alpha+3/2)}\;\int_{0}^{\infty}\mathrm{d}x\;x^{\alpha+1/2}\;e^{-x}\left\{\frac{\mathrm{erf}(\sqrt{x})}{\sqrt{x}}\right\}^{l}.
\tilde{J}_{0}=\left(\begin{array}{ccc}{-aD-Da}&{\frac{1}{2}D(a-b)-bD}&{Db-cD}\\{\frac{1}{2}(a-b)D-Db}&{\frac{1}{2}(b-c)D+\frac{1}{2}D(b-c)}&{\frac{1}{2}(c-e)D+Dc}\\{bD-Dc}&{\frac{1}{2}D(c-e)+cD}&{\frac{3}{2}D+De+eD}\\\end{array}\right),
\tau_{+}=\left(\begin{array}{c}{0\;\;1}\\{0\;\;0}\\\end{array}\right),\;\;\;\;\;\;\tau_{-}=\left(\begin{array}{c}{0\;\;0}\\{1\;\;0}\\\end{array}\right),\;\;\;\;\;\;\tau_{3}=\left(\begin{array}{c}{1\;\;\;\;\;0}\\{0\;-1}\\\end{array}\right)
{\calL}=e\left({\frac{1}{2}}R+g^{2}V-{\frac{1}{4}}G_{IJ}F_{\mu\nu}{}^{I}F^{\mu\nuJ}-{\frac{1}{2}}{\calG}_{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}\,\right),
\alpha_{00}=\sum_{\rho=0}^{N}\frac{\left(\eta_{0\rho}\right)^{2}}{4}\left[\left(2+\frac{\tilde{\omega}}{\Omega_{\rho}}+\frac{\Omega_{\rho}}{\omega}\right)\mathrm{e}^{i\Omega_{\rho}t}+\left(2-\frac{\tilde{\omega}}{\Omega_{\rho}}-\frac{\Omega_{\rho}}{\tilde{\omega}}\right)\mathrm{e}^{-i\Omega_{\rho}t}\right]\;,
S_{A_{1}A_{2}}^{\,k\,\,p}(\theta_{12})S_{kA_{3}}^{B_{1}r}(\theta_{13})S_{\,p\,\,r}^{B_{2}B_{3}}(\theta_{23})=S_{A_{1}k}^{rB_{3}}(\theta_{13})S_{A_{2}A_{3}}^{\,p\,\,k}(\theta_{23})S_{\,r\,\,p}^{B_{1}B_{2}}(\theta_{12}).
\mu_{n}\rightarrow_{\eta\rightarrow+\infty}\frac{\alpha_{n}}{\sqrt{2\Omega_{n}^{out}}}e^{-i\Omega_{n}^{out}\eta}+\frac{\beta_{n}}{\sqrt{2\Omega_{n}^{out}}}e^{+i\Omega_{n}^{out}\eta}\,,
e^{\sqrt{2}t_{1}a^{\dagger}}\vert0\rangle*_{\mathrm{point}}e^{\sqrt{2}t_{2}a^{\dagger}}\vert0\rangle=\pi^{-1/4}\sqrt{\frac{2}{3}}e^{{\frac{1}{3}}(t_{1}^{2}+t_{2}^{2})-{\frac{4}{3}}t_{1}t_{2}}e^{-{\frac{2\sqrt{2}}{3}}(t_{1}+t_{2})a^{\dagger}+{\frac{1}{6}}(a^{\dagger})^{2}}\vert0\rangle.
{}\bar{g}^{\mu\nu}(Y)={\left[{\bfg}_{Y}\left(\frac{\partial}{\partialX^{\mu}},\frac{\partial}{\partialX^{\nu}}\right)\right]}^{-1}=X_{\rho}^{\mu}(Y)X_{\sigma}^{\nu}(Y)g^{\rho\sigma}(Y).
\Psi^{0}=R^{2}(\sum_{s=1}^{N}|\Pi^{s}|^{2}+R^{2})^{-1/2},...,\quad\Psi^{i}=\Pi^{i}R(\sum_{s=1}^{N}|\Pi^{s}|^{2}+R^{2})^{-1/2}.
\phi(x^{0}\,,\,x^{1})=\frac{1}{2R}\sum_{n\geq0}\left\{\left[a_{n}(x^{0})+ib_{n}(x^{0})\right]e^{i\omega_{n+\frac{1}{2}}x^{1}}+\left[a_{n}(x^{0})-ib_{n}(x^{0})\right]e^{-i\omega_{n+\frac{1}{2}}x^{1}}\right\},
\langle{\cal{O}}_{n}^{(\mathrm{o)}}(x)\bar{\cal{O}}_{n^{\prime}}^{(\mathrm{o)}}(0)\rangle_{(\mathrm{free)}}=\delta_{n,n^{\prime}}\left(\frac{g_{\mathrm{QGT}}^{2}}{8\pi^{2}}\frac{1}{|x^{2}|}\right)^{2(J_{R}+1)}~.
\widetilde\Omega_{(p)}^{\prime}=\mathrm{sTr}\{F^{p-1}\widetilde{{d\omega}^{m-p}{\omega}}\}=(-1)^{m-p}\mathrm{sTr}\{F^{p-1}(\widetilde\omega^{2})^{m-p}\widetilde{\omega}\}\quad.
D_{\mu\nu}(x,y)=\intd^{3}k\int_{C}dk^{0}e^{-ik\cdot(x-y)}D_{\mu\nu}^{0}(k)+\intd^{3}k\int_{-\infty}^{\infty}e^{-ik\cdot(x-y)}D_{\mu\nu}^{\mathrm{extra}}(k),
\biggl(f_{1}*_{1}(f_{2}f_{3})\biggr)(\xi)+f_{1}(\xi)\biggl(f_{2}*_{1}f_{3}\biggr)(\xi)-\biggl(f_{1}*_{1}f_{2}\biggr)(\xi)f_{3}(\xi)-\biggl((f_{1}f_{2})*_{1}f_{3}\biggr)(\xi)=0.
c(p)a(-p)\left\{\Gamma_{S}^{11}(p)+\left[\frac{b(-p)}{a(-p)}+\frac{d(p)}{c(p)}\right]\Gamma_{S}^{12}(p)+\frac{d(p)}{c(p)}\frac{b(-p)}{a(-p)}\Gamma_{S}^{22}(p)\right\}=0.
\left.\hat{F}_{\alpha\beta}^{I}\right|_{\mathrm{reg}}=\frac{(\partial\Pi)^{2}}{2\Pi^{2}}e_{\alpha}^{a\,}\,\hat{\xi}_{ab}^{+}\,e_{\beta}^{b\,}-\frac{D_{\alpha}\,\partial_{\gamma}\Pi}{2\Pi}e^{a\,\gamma}\,\hat{\xi}_{ab}^{+}\,e_{\beta}^{b}-e_{\alpha}^{a\,}\,\hat{\xi}_{ab}^{+}\,e^{b\,\delta}\frac{D_{\beta}\,\partial_{\delta}\Pi}{2\Pi}
\psi_{k}(z,\bar{z})={\frac{1}{\sqrt2}}\left[\left({\frac{f_{n,m}(z)}{\overline{f}_{n,m}(z)}}\right)^{k}+\left({\frac{\overline{{f}}_{n,m}(z)}{\f_{n,m}(z)}}\right)^{k}\right],\qquadk=0,1,2,\ldots,
D_{i}E_{i}^{A}(x)\,\Psi_{ab}=ig\Bigl(\delta(x-x_{2})T_{b^{\prime}b}^{A}\delta_{aa^{\prime}}-\delta(x-x_{1})T_{aa^{\prime}}^{A}\delta_{b^{\prime}b}\Bigr)\Psi_{a^{\prime}b^{\prime}},