harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T_{D9}^{ferm}=T_{D25~wr}^{bosonic}={\frac{\sqrt\pi}{2^{4}\kappa_{26}}}(2\pi\alpha^{\prime}{}^{1/2})^{2}={\frac{\sqrt\pi}{\kappa_{10}}}(2\pi\alpha^{\prime}{}^{1/2})^{-6}{\calN}^{-1/4}\.
({}^{(3)}\bigtriangleupA)_{k}={\frac{\partial^{2}A_{k}}{\partialr^{2}}}+{\frac{1}{r^{2}}}({}^{(2)}\bigtriangleupA)_{k}-{\frac{1}{r^{2}}}A_{k}+{\frac{2}{r}}\partial_{k}A_{r}.
I_{2}^{(d)}=-\frac{(d-4)}{(d-3)}I_{1}^{(d)}+\frac{m^{2}}{(d-3)}\left(1-\frac{q^{2}}{4m^{2}}\right)\left(I_{1}^{(d)}\right)^{\prime}+\frac{m^{2d-10}}{(d-3)(d-4)}H_{1}^{(d)},
M_{CM}=\left(\begin{array}{lll}{\mathcal{A}_{CM}}&{\mathcal{B}_{CM}}&{\mathcal{E}_{CM}}\\{\mathcal{B}_{CM}}&{\mathcal{A}_{CM}}&{\mathcal{E}_{CM}}\\{(\mathcal{E}_{CM})^{t}}&{(\mathcal{E}_{CM})^{t}}&{\mathcal{G}_{CM}}\\\end{array}\right),
\begin{aligned}{{}}&{{}R_{i,i+1}(1)R_{1\dotsN,0}(u)=R_{1\dotsN,0}^{i,i+1}(u)R_{i,i+1}(1),}\\{{}}&{{}R_{i,i+1}(1)L_{1\dotsN}(u)=L_{1\dotsN}^{i,i+1}(u)R_{i,i+1}(1),}\\\end{aligned}
\dot{\epsilon}\approx\left[{\frac{2\left({\frac{\dot{\varepsilon}}{\varepsilon}}\right)^{2}}{\left({\frac{\dot{\varepsilon}}{\varepsilon}}\right)^{2}-{\frac{\hat{\calC}c_{0}^{2}\varepsilon^{4}}{\hat{a}_{0}^{4}}}}}-(1+\epsilon)(1+3w)\right]\epsilon{\frac{\dot{\varepsilon}}{\varepsilon}}.
\begin{array}{ccc}{\mathrm{Diagram}\;G}&{g=k+2}&{j\in\mathrm{Exp}(G)}\\{A_{n}}&{n+1}&{1,2,3,\ldots,n}\\{D_{2n+1}}&{4n}&{1,3,5,\ldots,4n-1,2n}\\{D_{2n}}&{4n-2}&{1,3,5,\ldots,4n-3,2n-1}\\{E_{6}}&{12}&{1,4,5,7,8,11}\\{E_{7}}&{18}&{1,5,7,9,11,13,17}\\{E_{8}}&{30}&{1,7,11,13,17,19,23,29}\\\end{array}
S=-\frac{1}{2\pi}\intd^{2}\sigma\left[\partial_{\alpha}X^{\mu}\partial^{\alpha}X_{\mu}-i\bar{\psi}^{\mu,j}\rho^{\alpha}\partial_{\alpha}\psi_{\mu,j}+i\bar{\phi}^{\mu,k}\rho^{\alpha}\partial_{\alpha}\phi_{\mu,k}\right].\,
\left(D_{i}\right)_{\alpha\beta}v^{\beta}(x)\rightarrow\frac1a\left({\calD}_{i}\right)_{\alpha\beta}v^{\beta}(x)=\frac1a\left\{\frac12\left(v^{\alpha}(x+\widehat{i})-v^{\alpha}(x-\widehat{i})\right)-f_{\alpha\gamma\beta}\theta^{\gamma}(x,x+\widehat{i})v^{\beta}(x)\right\}
g_{\mathrm{I^{\prime}}}=\frac{l_{\mathrm{H}}}{g_{\mathrm{H}}^{1/2}R_{\mathrm{H}}}\,\quadl_{\mathrm{I^{\prime}}}=g_{\mathrm{H}}^{1/2}l_{\mathrm{H}}\,\quad\frac{R_{\mathrm{I^{\prime}}}}{l_{\mathrm{I^{\prime}}}}=g_{\mathrm{H}}^{1/2}\frac{l_{\mathrm{H}}}{R_{\mathrm{H}}}\,
\frac{d^{2}h_{k}^{(i)}}{dr^{2}}+(\frac{1}{r}-\frac{4(p-1)ag^{2}}{r(1-g^{2})})\frac{dh_{k}^{(i)}}{dr}-(\frac{k^{2}}{r^{2}}+2(2p-1)\eta^{2}g^{2}+\frac{4(p-1)a^{2}g^{2}}{r^{2}(1-g^{2})^{2}})h_{k}^{(i)}=0.
[a^{\lambda\lambda^{\prime}}(\xi),a^{\dagger\lambda^{\prime\prime}\lambda^{\prime\prime\prime}}(\xi^{\prime})]=\left[\eta^{\lambda\lambda^{\prime\prime}}\eta^{\lambda^{\prime}\lambda^{\prime\prime\prime}}+\eta^{\lambda\lambda^{\prime}}\eta^{\lambda^{\prime\prime}\lambda^{\prime\prime\prime}}\right]\xi^{0}\delta^{3}(\vec{\xi}-\vec{\xi}^{\prime}).
\left(\begin{array}{ccc}{-1}&{0}&{0}\\{0}&{0}&{-1}\\{0}&{-1}&{0}\\\end{array}\right),\quad\left(\begin{array}{ccc}{0}&{-1}&{0}\\{-1}&{0}&{0}\\{0}&{0}&{-1}\\\end{array}\right),\quad\left(\begin{array}{ccc}{0}&{0}&{-1}\\{0}&{-1}&{0}\\{-1}&{0}&{0}\\\end{array}\right)
\phi_{\psi}^{+}(0)=-e^{\frac{k\pir_{c}}{2}}\phi_{\psi}^{-}(0)\,\qquad\phi_{\psi}^{-}(\pir_{c})=e^{\frac{k\pir_{c}}{2}}\phi_{\psi}^{+}(\pir_{c})\.
\frac{1}{\tilde{g}_{4}(k)}=\frac{\left(\frac{k}{\Lambda}\right)^{4-d}}{\tilde{g}_{4}(\Lambda)}+\frac{3\Omega_{d}\left(1-\left(\frac{k}{\Lambda}\right)^{4-d}\right)}{(4-d)2(2\pi)^{d}}.
\begin{aligned}{Z_{1}=F(r_{x},r_{y},}&{{}r_{u},r_{v})r_{x}e^{i\theta_{x}},~~Z_{2}=F(r_{x},r_{y},r_{u},r_{v})r_{y}e^{i\theta_{y}},~~Z_{3}=F(r_{x},r_{y},r_{u},r_{v})r_{x},}\\{}&{{}U=F(r_{x},r_{y},r_{u},r_{v})r_{u}e^{i\theta_{u}},~~V=F(r_{x},r_{y},r_{u},r_{v})r_{v}e^{i\theta_{v}}.}\\\end{aligned}
\Psi^{c}(x)\,=\,\sum_{\sigma=0,\pm1}\int{\frac{d^{3}p}{(2\pi)^{3}}}{\frac{1}{2\,\omega_{\vec{p}}}}\Big[S(C)\,u_{\sigma}^{\ast}(\vec{p}\,)\,a_{\sigma}^{\dagger}(\vec{p}\,)\,e^{ip\cdotx}+S(C)\,v_{\sigma}^{\ast}(\vec{p}\,)\,b_{\sigma}(\vec{p}\,)\,e^{-ip\cdotx}\Bigr]\quad.
(f_{\lambda}^{\rho})=\left[\begin{array}{cc}{f_{1}^{1}}&{f_{2}^{1}}\\{f_{1}^{2}}&{f_{2}^{2}}\\\end{array}\right]=\left[\begin{array}{cc}{F_{3}}&{F_{1}+iF_{2}}\\{F_{1}-iF_{2}}&{-F_{3}}\\\end{array}\right]-
(\Theta\delta^{\prime})(t)\rightarrow\frac{1}{2\pi}(\hat{\Theta}\ast\delta^{\prime})(\omega)=\frac{1}{(2\pi)^{2}}\int_{-\infty}^{+\infty}d\omega^{\prime}\,\frac{\omega-\omega^{\prime}}{\omega^{\prime}-i0}
F[\Phi,X,\varphi,\chi]=\int\!dx\;\left(\Phi^{j}J^{1~j}[\varphi]-\fraci2~\overline{X}^{j}\gamma^{1}\chi^{j}\right)=\intdx\Bigl(\Phi^{i}(\sqrt{1-{\varphi}^{2}}\frac{\stackrel{\leftrightarrow}{\partial}}{\partialx}\varphi^{i}+\varepsilon^{ijk}\varphi^{j}\frac\partial{\partialx}\varphi^{k})-\fraci2\overline{X}^{j}\gamma^{1}\chi^{j}\Bigr)~.
{\cal{S}}^{[p_{1},...,p_{N}]}\;\;=\;\;-\left(\prod_{i=1}^{N}\frac{1}{p_{i}!}\right)\intd^{D}x\;A^{{\mu}_{1}^{1}...{\mu}_{p_{1}}^{1}...{\mu}_{1}^{i}...{\mu}_{p_{i}}^{i}...{\mu}_{p_{N}}^{N}}E_{{\mu}_{1}^{1}...{\mu}_{p_{1}}^{1}...{\mu}_{1}^{i}...{\mu}_{p_{i}}^{i}...{\mu}_{p_{N}}^{N}}
\frac{M^{2}(K)}{K}=\frac{\kappa^{2}+k_{1,\perp}^{2}}{n_{1}}+\frac{\kappa^{2}+k_{2,\perp}^{2}}{n_{2}}+\frac{\kappa^{2}+k_{3,\perp}^{2}}{n_{3}}\,,
{\bfD}_{\mu}(\sigma)F={\frac{\deltaF}{\deltaz^{\mu}(\sigma)}}=D_{\mu}\,f_{\rho_{1}\rho_{2}}{\calD}z^{\prime\rho_{1}}{\calD}z^{\prime\rho_{2}}+2{\calD}^{2}(f_{\mu\rho_{2}}{\calD}z^{\prime\rho_{2}})-2f_{\rho_{1}\rho_{2}}z^{\prime\kappa}z^{\prime\lambda}R_{\mu\kappa\lambda}^{{\,\,\,\,\,\,\,\,}\rho_{1}}{\calD}z^{\prime\rho_{2}}
{\calL}_{2}=\alpha\\\{((b_{-a})^{2}+(b_{+a})^{2}+{\frac{1}{2}}(b_{Ir})^{2})+\frac{2}{\alpha}(b_{-a}({\partial}_{\mu}A_{-a}^{\mu}-2k{\varphi}_{a})+b_{+a}({\partial}_{\mu}A_{+a}^{\mu})+{\frac{1}{2}}b_{Ir}({\partial}_{\mu}Y^{\mu}))\},
S[A,\overline{{\psi}},\psi]=\int\mathrm{~\,~d^3x~\,~}[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\overline{{\psi}}\gamma_{\mu}(i\partial^{\mu}+eA^{\mu})\psi+m\overline{{\psi}}\psi],
\bar{n}_{\mathrm{eq}}=2\tilde{g}\int_{0}^{\infty}dx\frac{x(x^{2}+1)}{\left[4(x^{2}-1)^{2}+\pi^{2}\tilde{g}^{2}x^{2}\right](\mathrm{e}^{\tilde{\beta}x}-1)}\;.
\begin{array}{l}{e_{i}=q_{i}^{H_{i}}X_{i}^{+}~,~~~~t_{i}=q_{i}^{2H_{i}}~,}\\{f_{i}=X_{i}^{-}q_{i}^{-H_{i}}~,~~~~\bar{t}_{i}=q_{i}^{2H_{i}}~.}\\\end{array}
\nu_{+}-\nu_{-}=\frac{N}{24\cdot8\pi^{2}}\int_{\calM}\mathrm{tr}(R\wedgeR)-\frac{g^{2}N_{f}}{32\pi^{2}}\int_{\calM}F_{\mu\nu}^{a}\tilde{F}^{\mu\nua}d^{4}x,
{\pmb{\mathbbT}}(t)={\mathbbT}^{(vac)}(t)\prod_{k=1}^{L}{\frac{(1+2n_{k}-2ate^{i\pi/6})(1+2n_{k}-2ate^{-i\pi/6})}{(1+2n_{k}+2ate^{i\pi/6})(1+2n_{k}+2ate^{-i\pi/6})}},
\begin{array}{l}{e\vertjm\rangle=[j+m+1]\vertj,m+1\rangle~,}\\{f\vertjm\rangle=[j-m+1]\vertj,m-1\rangle~,}\\{\displaystyle\frac{e^{a}}{[a]!}\vertjm\rangle=\displaystyle\frac{[j+m+a]!}{[a]![j+m]!}\vertj,m+a\rangle~,}\\{\displaystyle\frac{f^{a}}{[a]!}\vertjm\rangle=\displaystyle\frac{[j-m+a]!}{[a]![j-m]!}\vertj,m-a\rangle~.}\\\end{array}
\sigma_{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\sigma_{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\,,\sigma_{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)\,.
G(x,y;A,h^{\prime})\;=\;\frac{1}{2\pi}\frac{1}{(x\!-\!y)^{2}}\!\left(\begin{array}{cc}{0}&{e^{-[\chi(x)-\chi(y)]}\;\overline{{(\tilde{x}\!-\!\tilde{y})}}}\\{e^{+[\chi(x)-\chi(y)]}\;(\tilde{x}\!-\!\tilde{y})}&{0}\\\end{array}\right)\;,
W_{\infty}^{3,C}=-{\frac{1}{16}}(1-\gamma)^{-1}\sum_{k=0}^{\infty}(-1)^{k}\beta^{k}(k+3)(k+5)(k+7)\sum_{m=0}^{\infty}(m^{4}-m^{3})m^{k}\alpha_{m}^{-k-9}.
\tilde{\Gamma}_{t,s}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})=\int_{0}^{t}d\taug_{\tau}^{2}e^{-(D-4)(t-\tau)}I_{s+t-\tau}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})\;.
N^{\calP}~=~{\frac{2}{k+2}}~\sum_{r=0}^{k/2}~{\frac{\prod_{l=1}^{p}sin\left({\frac{(2j_{l}+1)(2r+1)\pi}{k+2}}\right)}{\left[sin\left({\frac{(2r+1)\pi}{k+2}}\right)\right]^{p-2}}}~.
i\alpha\mu^{-1}\pi^{2}\delta({\bfx},{\bfx^{\prime}})=\intd^{2}{\bfx^{\prime\prime}}d^{2}{\bfx^{\prime\prime\prime}}[\rho({\bfx^{\prime\prime}}),\rho({\bfx^{\prime\prime\prime}})]\epsilon_{lj}\frac{({\bfx}-{\bfx^{\prime\prime}})^{l}}{|{\bfx}-{\bfx^{\prime\prime}}|^{2}}\frac{({\bfx^{\prime}}-{\bfx^{\prime\prime\prime}})^{j}}{|{\bfx^{\prime}}-{\bfx^{\prime\prime\prime}}|^{2}}.
R_{A}=\left[\frac{\sqrt{|M|^{2}+({\tilde{M}}_{i}+M_{W})^{2}}+{\tilde{M}}_{i}+M_{W}}{\sqrt{|M|^{2}+({\tilde{M}}_{i}-M_{W})^{2}}+{\tilde{M}}_{i}-M_{W}}\right]^{-k}\.
\begin{array}{lcll}{\mathrm{floor}}&{\mathrm{SU(3/1)}}&{\mathrm{field}}&{\mathrm{helicity}}\\{\mid\mathrm{gnd}>}&{(0~0~+\frac{3}{2})}&{\lambda}&{-\frac{1}{2}}\\{\mid\mathrm{1st}>}&{(0~1~+\frac{3}{2})}&{\overline{4}A_{\mu}}&{-1}\\{\mid\mathrm{2nd}>}&{(1~0~+\frac{5}{2})}&{4\Psi_{\mu}}&{-\frac{3}{2}}\\{\mid\mathrm{3rd}>}&{(0~0~+\frac{7}{2})}&{e_{\mu}^{a}}&{-2.}\\\end{array}
ds_{B}^{2}=a\prod_{i=1}^{3}(z-z_{i})^{-\frac{2}{3}}\prod_{i=1}^{3}(\bar{z}-\bar{z_{i}})^{-\frac{2}{3}}dzd\bar{z}+a^{\prime}\prod_{i=1}^{3}(z^{\prime}-z_{i}^{\prime})^{-\frac{2}{3}}\prod_{i=1}^{3}(\bar{z^{\prime}}-\bar{z_{i}^{\prime}})^{-\frac{2}{3}}dz^{\prime}d\bar{z^{\prime}}
\overline{{{\cal{A}}_{i}^{\gamma}}}({\bf{r}})={\delta}_{{i}\,{\gamma}}\,\left({\varphi}_{A}+\Phi\right)+\frac{r_{i}\,r_{\gamma}}{r^{2}}\,\left({\varphi}_{B}+r\Phi^{\prime}\right)+{\epsilon}_{i{\gamma}n}\frac{r_{n}}{r}\,{\varphi}_{C}\,,
\int\frac{d^{2}\rho}{\pi}\e^{-|\rho|^{2}}e^{\alpha^{\prime}k_{\mu}^{2}/4n}(\rho|e^{\sqrt{\alpha^{\prime}/2}\tilde{k}_{\mu}\alpha_{-n}^{\mu}z^{n}/n}e^{-\sqrt{\alpha^{\prime}/2}\tilde{k}_{\mu}\alpha_{n}^{\mu}z^{-n}/n}|-q^{n}\rho)
\delta({\mathrm{~\sum~}}\vec{p}_{i})\delta(p_{1}^{-}+p_{4}^{-})\delta(p_{2}^{+}+p_{3}^{+})\,(\vec{p}_{1}+\vec{p}_{4})^{-2}(p_{1}^{-}p_{2}^{+})^{2-(\vec{p}_{1}+\vec{p}_{4})^{2}\alpha^{\prime}/4}.
{\calA}^{\pm}=\pm\mathrm{{\bf~I}}\frac{d}{dz}+\mathrm{{\bf~W}}(z),\quad\Psi_{\mathrm{SUSY}}^{(n)}(z)=\left(\begin{array}{cc}{\psi_{-}^{(n)}(z)}\\{\psi_{+}^{(n)}(z)}\\\end{array}\right),
\lbraceA,B\rbrace{}^{*}=\lbraceA,B\rbrace-i[\lbraceA,\pi_{\theta\,i}+{\frac{i}{2}}\theta_{i}^{*}\rbrace\lbrace\pi_{\theta^{*}\,i}+{\frac{i}{2}}\theta_{i},B\rbrace+\lbraceA,\pi_{\theta^{*}\,i}+{\frac{i}{2}}\theta_{i}\rbrace\lbrace\pi_{\theta\,i}+{\frac{i}{2}}\theta_{i}^{*},B\rbrace]
=\int_{W\left(t\right)}d^{2q}\sigma\,\frac{\epsilon^{0\mu_{1}\ldots\mu_{2q-2k}\nu_{1}\ldots\nu_{2k}}}{\left(2q-2k\right)!\,2^{k}\cdotk!}\cdot\partial_{\mu_{1}}X^{m_{1}}\cdots\partial_{\mu_{2q-2k}}X^{m_{2q-2k}}\cdotF_{\nu_{1}\nu_{2}}\cdotsF_{\nu_{2k-1}\nu_{2k}}\quad.
p_{a\gamma}(f\starg)\leq\sum_{b\leqa}\sum_{\epsilon\leq\gamma}\left(\begin{array}{c}{a}\\{b}\\\end{array}\right)\left(\begin{array}{c}{\gamma}\\{\epsilon}\\\end{array}\right)q_{a-b,\gamma-\epsilon}(f)q_{0,\eta-\epsilon}(g)
D_{\mu}^{ij}[A]\frac{\deltaW}{\deltaA_{\mu}^{j}}+gf^{ijk}(J_{\mu}^{j}\frac{\deltaW}{\deltaJ_{\mu}^{k}}+\xi^{j}\frac{\deltaW}{\delta\xi^{k}}+\overline{{\xi}}^{j}\frac{\deltaW}{\delta\overline{{\xi}}^{k}})=0.
\hat{J}_{0}=z\frac{\partial}{\partialz}+\alpha,\quad\hat{J}_{1}=-\frac{1+z^{2}}{2}\frac{\partial}{\partialz}-\alphaz,\quad\hat{J}_{2}=-i\frac{1-z^{2}}{2}\frac{\partial}{\partialz}+i\alphaz.
{\calQ}_{\mu\nu}=\frac{1}{2}g_{\mu\nu}R_{\mathrm{EGB}}^{2}-2R~R_{\mu\nu}+4R_{\mu\alpha}~R_{~~\nu}^{\alpha}+4R_{\alpha\beta}~R_{\,\,\,\mu\,\,\,\nu}^{\alpha\,\,\,\beta}-2R_{\mu\alpha\beta\gamma}~R_{\nu}^{~~\alpha\beta\gamma}.
J_{\perp}={\frac{m_{f}H_{p}z^{2}\dot{\theta}_{\perp}}{\sqrt{1-H_{p}H_{F}v_{\perp}^{2}}}},\\\\\E={\frac{m_{f}}{H_{F}}}\left[{\frac{1}{\sqrt{1-H_{p}H_{F}v_{\perp}^{2}}}}-1\right].
=-\int_{\Sigma_{t}}\nabla_{\nu}\left[t^{\nu}\theta_{(g)}^{\mu}(\deltag)-t^{\mu}\theta_{(g)}^{\nu}(\deltag)\right]d\Sigma_{\mu}=\int_{C_{\infty}}\sqrt{\gamma}d^{2}yNu_{\nu}\theta_{(g)}^{\nu}(\deltag)~~~.
\Sigma(z)=\frac{\int_{-\infty}^{z}\sqrt{-g^{(4)}}dz^{\prime}\left(T_{\;\;t}^{t}-T_{\;\;z}^{z}-T_{\;\;x}^{x}-T_{\;\;y}^{y}\right)\intdxdy}{\sqrt{g^{(2)}}\intdxdy}.
G_{D}({\bfr}^{\prime\prime},{\bfr}^{\prime};E)=\sum_{n}\frac{\psi_{n}({\bfr}^{\prime\prime})\,\psi_{n}^{*}({\bfr}^{\prime})}{E-E_{n}+i\epsilon}+\intd\alpha\,\frac{\psi_{\alpha}({\bfr}^{\prime\prime})\,\psi_{\alpha}^{*}({\bfr}^{\prime})}{E-E_{\alpha}+i\epsilon}\;,
\Biggl[\,\mu\frac{\partial}{\partial\mu}+\beta_{m^{2}}\frac{\partial}{\partialm^{2}}+\beta_{\lambda}\frac{\partial}{\partial\lambda}+\beta_{\nu}\frac{\partial}{\partial\nu}-2\gamma_{\Phi}\,\Biggr]\,{\Sigma}(p)+\beta_{m^{2}}+2\gamma_{\Phi}(p^{2}-m^{2})=0.
\Psi(z^{i},dz^{j})=F_{0}(z^{i})+F_{k}(z^{i})dz^{k}+F_{kl}(z^{i})dz^{k}dz^{l}+...\quad\in\Lambda{\bfC}_{\mathrm{h}}^{D}=\bigoplus_{p=0}^{D}\Lambda^{p}{\bfC}_{\mathrm{h}}^{D},
O=\left(\begin{array}{cccc}{x_{3}}&{0}&{\sqrt{1-x_{3}^{2}}}&{0}\\{0}&{1}&{0}&{0}\\{-x_{1}}&{0}&{\frac{x_{1}x_{3}}{\sqrt{1-x_{3}^{2}}}}&{\frac{x_{2}}{\sqrt{1-x_{3}^{2}}}}\\{x_{2}}&{0}&{-\frac{x_{2}x_{3}}{\sqrt{1-x_{3}^{2}}}}&{\frac{x_{1}}{\sqrt{1-x_{3}^{2}}}}\\\end{array}\right).
H=\frac{1}{2}\sum_{n=0}^{\infty}\frac{1}{n+1}H_{n}^{\dagger}H_{n}=\frac{1}{2}\sum_{n=0}^{\infty}\frac{1}{n+1}\sum_{i=0}^{n}a_{i}^{\dagger}a_{n-i}^{\dagger}\sum_{j=0}^{n}a_{j}a_{n-j}
{\calL}_{\Lambda}={\calL}_{\Lambda}^{S}+{\calL}_{\Lambda}^{F}=\sum_{\kappa\in{(I,~II,~T_{I},~T_{II})}}\left\{N_{\kappa}^{F}{\calL}_{\Lambda}^{F}(\kappa)+N_{\kappa}^{S}{\calL}_{\Lambda}^{S}(\kappa)\right\}~.
\sum_{\sigma_{i},\tau_{i}}S(\{\sigma_{i}\}_{i=1}^{k},\{\tau_{i}\}_{i=1}^{k})F_{\sigma_{2}\tau_{2}}^{\sigma_{1}\tau_{1}}F_{\sigma_{3}\tau_{3}}^{\sigma_{2}\tau_{2}}\dotsF_{\sigma_{1}\tau_{1}}^{\sigma_{k}\tau_{k}}
A_{{\sfGI}\,i}({\bf{r}})=[\,A_{{\sfGI}\,i}^{b}({\bf{r}})\,{\textstyle\frac{\lambda^{b}}{2}}\,]=V_{\cal{C}}({\bf{r}})\,[\,A_{i}^{b}({\bf{r}})\,{\textstyle\frac{\lambda^{b}}{2}}\,]\,V_{\cal{C}}^{-1}({\bf{r}})+{\textstyle\frac{i}{g}}\,V_{\cal{C}}({\bf{r}})\,\partial_{i}V_{\cal{C}}^{-1}({\bf{r}})\;,
e^{-\phi}k^{{\frac{1}{2}}}\left[\mathrm{det}\left(\begin{matrix}{{\dot{X}}^{\mu}{\dot{X}}^{\nu}(g_{\mu\nu}-k^{-2}B_{\mu}B_{\nu})}&{}&{-k^{-2}{\dot{X}}^{\mu}B_{\mu}}\\{-k^{-2}B_{\mu}}&{}&{-k^{-2}}\\\end{matrix}\right)+{\hat{\calF}}_{01}^{2}\right]^{{\frac{1}{2}}}\,.
[\mathsf{L}_{1},\mathsf{u}_{2}]=-i\hbar\mathsf{\mathcal{C}}_{12}\mathsf{u}_{2},\qquad[\mathsf{R}_{1},\mathsf{v}_{2}]=i\hbar\mathsf{v}_{2}\mathsf{\mathcal{C}}_{12}
\omega_{0}(x)=2\int_{0}^{\infty}{\frac{x\rho(\lambda)d\lambda}{x^{2}-\lambda^{2}}}=\int_{0}^{\infty}\rho(\lambda)d\lambda\left({\frac{1}{x-\lambda}}+{\frac{1}{x+\lambda}}\right)=\int_{-\infty}^{\infty}{\frac{\rho(\lambda)d\lambda}{x-\lambda}},
\left[-\omega^{2}\gamma^{tt}(r)f(r)+{\frac{1}{\sqrt{\frac{-\gamma(r)}{f(r)}}[r^{2}h(r)]^{\frac{n+1}{2}}}}{\frac{d}{dr}}\sqrt{\frac{-\gamma(r)}{f(r)}}[r^{2}h(r)]^{\frac{n+1}{2}}{\frac{d}{dr}}\right]\phi_{\omega}(r)=0.
A_{L,N^{\prime},N^{\prime\prime}}^{(c/2\mu)}=\left[2^{L+1}\Gamma(L+1)\right]^{2}\sqrt{\frac{n^{\prime}!N^{\prime}}{2\pi\Gamma(N^{\prime}+L+1)}}\sqrt{\frac{n^{\prime\prime}!N^{\prime\prime}}{2\pi\Gamma(N^{\prime\prime}+L+1)}}B_{L,N^{\prime},N^{\prime\prime}}^{(c/2\mu)}
\Omega_{ij}\left(\tilde{g}_{2}=-{\frac{1}{3}},\tilde{g}_{3}={\frac{1}{27}}\right)=\left(\begin{array}{cc}{\frac{11}{10}}&{\frac{9}{5}}\\{-\frac{7}{60}}&{\frac{3}{20}}\\\end{array}\right).
\sum_{s+t=n-1}(i^{2})^{s+t+1}(k^{2})^{s}(k^{2})^{t}=(k^{2})^{n-1}\sum_{s+t=n-1}(-1)^{n}(k^{2})^{s-n+1}(k^{2})^{t}=\frac{(-k^{2})^{n}-(-k^{2})^{n}}{k^{2}-k^{2}}
{\calC}\Gamma_{a}=\left(\begin{matrix}{c\,\tilde{\gamma}_{a}}&{0}\\{0}&{c^{T}\gamma_{a}}\\\end{matrix}\right)\otimes\Omega\;\qquad{\calC}\Gamma_{a^{\prime}}=\left(\begin{matrix}{0}&{-c}\\{c^{T}}&{0}\\\end{matrix}\right)\otimes\Omega\gamma_{a^{\prime}}\.
\{\,g_{ab}(z,\bar{z}),\,g_{cd}(y,\bar{y})\,\}=\frac{\gamma^{2}}{4}\,\,\Theta\,[2g_{ad}(z,\bar{y})\,g_{cb}(y,\bar{z})-g_{ab}(z,\bar{z})\,g_{cd}(y,\bar{y})],
V\left(z\right)=-\left(\alpha\left(0\right)-\beta\left(0\right)\right)\frac{\pi^{3}}{a^{4}}G\left(\frac{\piz}{a}\right)+\left(\alpha\left(0\right)+\beta\left(0\right)\right)\left(-\frac{7}{8}\right)\frac{\pi^{3}}{360a^{4}}.
\lambda_{2N}(\alpha)={\frac{1+\sum_{m=0}^{N-1}\left({\frac{i\alpha}{2}}\right)^{2m+1}{\frac{(2N+2m)!}{(2m+1)!(2N-2m-2)!}}}{\sum_{m=0}^{N-1}\left({\frac{i\alpha}{2}}\right)^{2m}{\frac{(2N+2m-1)!}{(2m)!(2N-2m-1)!}}-i\alpha(4N+1)}}\times\lambda_{0}(\alpha)
\left\{\prod_{l_{i}^{(k)}}\left(\prod_{i=0}^{l_{i}^{(k)}-1}(\Theta_{j}-i)\right)-\prod_{i=1}^{|l_{0}^{(k)}|}(i-|l_{0}^{(k)}|-\Theta_{0})\prod_{l_{i}^{(k)},j\neq0}\left(\prod_{i=0}^{|l_{i}^{(k)}|-1}(\Theta_{j}+|l_{i}^{(k)}|-i)\right)x_{k}\right\}\Pi(x)=0
\begin{aligned}{}&{{}U_{1}:}&{}&{{}\qquadx_{1}(t)=0,}&{}&{{}\qquady_{1}(t)=t,}&{}&{{}\qquadu_{1}(t)=\mu,}&{}&{{}\qquadv_{1}(t)=0}\\{}&{{}U_{3}:}&{}&{{}\qquadx_{3}(t^{\prime})=t^{\prime},}&{}&{{}\qquady_{3}(t^{\prime})=0,}&{}&{{}\qquadu_{3}(t^{\prime})={\frac{\mu}{t^{\prime}}},}&{}&{{}\qquadv_{3}(t^{\prime})=0.}\\\end{aligned}
S_{WZW2}[g,\tilde{A}_{z}]={\frac{1}{4\pi}}\int_{\partialM}Tr(g^{-1}\partial_{z}gg^{-1}\partial_{\overline{{z}}}g-2g^{-1}\partial_{\overline{{z}}}g\tilde{A}_{z})+{\frac{1}{12\pi}}\int_{M}Tr(g^{-1}dg)^{3}
d{}^{*}{\hat{F}}^{(4)}+dA^{(3)}\wedge{\hat{F}}^{(4)}=H^{(3)}\wedge{}^{*}J^{(6)},\qquadd{}^{*}{\hat{F}}^{(7)}\equiv-d{\hat{F}}^{(4)}={~}^{*}J^{(6)}.
\left(\begin{array}{l}{\Psi_{1}}\\{\Psi_{2}}\\\end{array}\right)=\sqrt{\frac{1+x_{5}}{2}}\left(\begin{array}{l}{u_{1}}\\{u_{2}}\\\end{array}\right),\quad\left(\begin{array}{l}{\Psi_{3}}\\{\Psi_{4}}\\\end{array}\right)=\sqrt{\frac{1}{2(1+x_{5})}}(x_{4}-ix_{i}\sigma_{i})\left(\begin{array}{l}{u_{1}}\\{u_{2}}\\\end{array}\right).
\begin{array}{cc}{AQ^{3}\leftrightarrowaq^{3}}&{Q^{6}\leftrightarrowq^{6}}\\{A\bar{Q}^{3}\leftrightarrowa\bar{q}^{3}}&{\bar{Q}^{6}\leftrightarrow\bar{q}^{6}}\\{A^{3}Q^{3}\leftrightarrowa^{3}q^{3}}&{Q\bar{Q}\leftrightarrowM_{0}}\\{A^{3}\bar{Q}^{3}\leftrightarrowa^{3}\bar{q}^{3}}&{QA^{2}\bar{Q}\leftrightarrowM_{2}}\\{A^{4}\leftrightarrowa^{4}}&{}\\\end{array}
G^{\prime}(\sigma_{i}^{a},\sigma_{j}^{a})=\sum_{n_{1}=-\infty}^{\infty}\sum_{n_{2}=0}^{\infty}{\frac{4\pi\alpha^{\prime}}{\omega_{n_{1}n_{2}}C_{n_{1}n_{2}}}}{\bar{\Psi}}_{-n_{1}n_{2}}(\sigma_{i}^{a})\Psi_{n_{1}n_{2}}(\sigma_{j}^{a})\quad.
\left(\begin{array}{c}{m^{\prime}}\\{N^{\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{a}&{b}\\{c}&{d}\\\end{array}\right)\left(\begin{array}{c}{m}\\{N}\\\end{array}\right),\qquad\Theta^{\prime}=\frac{c+d\Theta}{a+b\Theta},
\deltaf_{2}^{9}\sim-\mathrm{Tr}(\tilde{B}_{1}^{9})^{2}\sum_{i,j}^{3}(2\phi_{i,j,1}-\phi_{i,j,2}-\phi_{i,j,3})+\sqrt{3}\mathrm{Tr}(\tilde{B}_{1}^{9})^{2}(\rho_{1,1,2}-\rho_{1,1,3})\;.
\dot{\varrho}+3{\frac{\dot{a}_{0}}{a_{0}}}\left(\varrho+{\frac{\wp}{c^{2}}}\right)=-{\frac{\dot{G}_{4}}{G_{4}}}\varrho+{\frac{3kc\dot{c}}{4\piG_{4}a_{0}^{2}}}-{\frac{3{\calC}c\dot{c}}{4\piG_{4}a_{0}^{4}}}.
\begin{array}{c}{D_{TM}^{(-2)}(t)=\displaystyle{\frac{{n_{0}}^{2}\,R^{2}\,t\,\left(10-\frac{10\,{n_{0}}^{2}\,R^{2}\,t^{2}}{a^{2}}+\frac{{n_{0}}^{4}\,R^{4}\,t^{4}}{a^{4}}\right)}{8\,a^{2}\,{\left(1+\frac{{n_{0}}^{2}\,R^{2}\,t^{2}}{a^{2}}\right)}^{4}}+\frac{Z^{2}}{2\,t^{3}}}}\\\end{array}
\left(\sigma_{\mathrm{abs}}^{6D}\right)_{l=0,K=0}=\left(\tilde{\sigma}_{\mathrm{abs}}^{6D}\right)_{l=0,K=0}\simeq{\calA}_{H}^{6D}\left\vert{\frac{\Gamma(1-i{\frac{\omega}{4\piT_{L}}})\Gamma(1-i{\frac{\omega}{4\piT_{R}}})}{\Gamma(1-i{\frac{\omega}{2\piT_{H}}})}}\right\vert^{2}.
\begin{array}{rcl}{\hat{\lambda}_{0}}&{=}&{\hat{a}_{0}+ie^{-\hat{\phi}_{0}}\,,}\\{}&{}&{}\\{\Upsilon}&{=}&{{\calQ}_{a}-i{\calQ}_{d}\,,}\\{}&{}&{}\\{\chi}&{=}&{{\calF}-i{\calW}\,,}\\{}&{}&{}\\{\Theta}&{=}&{{\calQ}_{e}^{I}{\calQ}_{m}^{I}-i{\calZ}\,.}\\\end{array}
4\pi\beta_{\alpha_{\lambda}}=24\alpha_{\lambda}^{2}+12\alpha_{\lambda}\alpha_{t}-\frac{9}{5}\alpha_{\lambda}\alpha_{1}-9\alpha_{\lambda}\alpha_{2}+\frac{27}{200}\alpha_{1}^{2}+\frac{9}{20}\alpha_{1}\alpha_{2}+\frac{9}{8}\alpha_{2}^{2}-6\alpha_{t}^{2}
{\frac{\partial}{\partialz_{\mu}}}\bigg\lbrackF_{\mu}(z-y;z-x)\bigg\rbrack_{aa^{\prime}}^{bb^{\prime}}=i\bigg\lbrack\delta(y-z)-\delta(x-z)\bigg\rbrack\delta_{a}^{b^{\prime}}\delta_{a^{\prime}}^{b}\;\;.
\left[\frac{\partial}{\partials}-\beta_{g}(g^{2},\alpha)\frac{\partial}{\partialg^{2}}-\beta_{\alpha}(g^{2},\alpha)\frac{\partial}{\partial\alpha}+n\gamma(g^{2},\alpha)+(n-4)\right]\Gamma^{(n)}(\{e^{s}\tilde{p}_{i}\},g^{2},\alpha,m)=0.
N_{n}=\left(\begin{array}{cc}{O(\psi^{4})}&{O(\psi^{3})}\\{0}&{O(\psi^{4})}\\\end{array}\right),\,\,Y_{n}=\left(\begin{array}{cc}{O(\psi^{2})+A_{i}O(\psi^{4})}&{O(\psi)+A_{i}O(\psi^{3})}\\{O(\psi^{3})}&{O(\psi^{2})+A_{i}O(\psi^{4})}\\\end{array}\right).
\lbrackR_{\mu},R_{\nu}\rbrack\approx\Big(\frac{1}{4}P^{2}-\frac{1}{2}(m_{1}^{2}+m_{2}^{2})+\frac{(m_{1}^{2}-m_{2}^{2})^{2}}{4P^{2}}\Bigr)\Pi_{\mu}^{\lambda}\Pi_{\nu}^{\sigma}J_{\lambda\sigma},
R_{q}^{a}=-\sum_{\frac{[p,t;q]}{[w,t;q]}}\mu_{w}\Gamma_{qt}^{pq}\Gamma_{wq}^{qt}-\sum_{\frac{(p,t;q)}{[w,t;q]}}\mu_{w}\Gamma_{wq}^{qt}-\sum_{\frac{[p,t;q]}{(w,t;q)}}\mu_{w}\frac{\mu_{q}}{\mu_{t}}\Gamma_{qt}^{pq}-\sum_{(p,q;w)}\mu_{w}\Gamma_{wq}^{qw}.
0\stackrel{}{\rightarrow}{}^{s}E_{k}\stackrel{}{\longrightarrow}\bigoplus_{a_{1}<a_{2}<\cdots<a_{s}}{\calO}(km+n_{a_{1}}+\cdots+n_{a_{s}})\stackrel{f}{\longrightarrow}{}^{s-1}E_{k+1}\stackrel{}{\rightarrow}0.
\left([\delta_{\varepsilon_{1}}\,,\,[\delta_{\varepsilon_{2}}\,,\,\delta_{\varepsilon_{3}}]]+[\delta_{\varepsilon_{3}}\,,\,[\delta_{\varepsilon_{1}}\,,\,\delta_{\varepsilon_{2}}]]+[\delta_{\varepsilon_{2}}\,,\,[\delta_{\varepsilon_{3}}\,,\,\delta_{\varepsilon_{1}}]]\right)q^{r}=0,
{X}^{\underline{{m}}}=\hat{X}^{\underline{{m}}}(\xi)\equiv\hat{X}^{\underline{{m}}}(\tau,\sigma),\quad{\Theta}^{1\underline{{\mu}}}=\hat{\Theta}^{1\underline{{\mu}}}(\xi),\quad{\Theta}_{\underline{{\mu}}}^{2}=\hat{\Theta}_{\underline{{\mu}}}^{2}(\xi).\quad
H_{\varphi}\left(a\right)|\vec{M},\vec{M}^{\prime}\rangle=\frac{\pi}{L}\sum_{i<j}\left[\left(M_{ij}-\frac{gL}{2\pi}(a_{i}-a_{j})\right)^{2}+\left(M_{ij}^{\prime}-\frac{gL}{2\pi}(a_{i}-a_{j})\right)^{2}\,\right]|\vec{M},\vec{M}^{\prime}\rangle
\left\{\mathrm{Tr}\Big[H^{(1)}H^{(l_{1})}H^{(l_{\pi(2)})}...H^{(l_{\pi(n)})}\Big]-\mathrm{Tr}\Big[H^{(1)}H^{(l_{\pi(2)})}...H^{(l_{\pi(n)})}H^{(l_{1})}\Big]\right\}\;.
\frac{1}{\sqrt{2}}\left(\begin{array}{cccccc}{}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{1}&{}\\{}&{}&{}&{}&{}&{}\\{}&{1}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}\\{}&{}&{}&{}&{}&{}\\\end{array}\right)\;,
f(x)=\sum_{i=0}^{\infty}c_{i}x^{i}=\int_{0}^{\infty}e^{-t}t^{b}F(xt)dt,\\\\\\nonumber\,F(y)=\sum_{i=0}^{\infty}{\frac{c_{i}}{(i+b)!}}y^{i}\\,
\frac{\partial\Phi(\rho,u)}{\partial\rho}=-2n(1-\rho^{2})^{n-1}\int_{vv^{\prime}=1}\cdots\intd\Omega_{v}\frac{\Phi_{0}(v)}{\left[(u-v)(u^{\prime}-v^{\prime})\right]^{n}}-{\calO}\left((1-\rho^{2})^{n+1}\right)~,~~~~\rho\longrightarrow1~.
{\frac{\dot{\bar{G}}_{4}}{\bar{G}_{4}}}=e^{2m_{0}(R_{0}-R)}{\frac{R_{\tau}\left[3(\Omega_{R}^{2}-1)R_{\tau\tau}+2m_{0}\Omega_{R}^{2}(c^{2}+R_{\tau}^{2})\right]}{(\Omega_{R}^{2}-1)(c^{2}+R_{\tau}^{2})}}.
{\cal{H}}(\mu^{2})={\frac{1}{8\pi}}\int_{-\Lambda}^{\Lambda}dk{\frac{2k^{2}+\mu^{2}}{(k^{2}+\mu^{2})^{1/2}}}-{\frac{\alpha_{0}}{\beta^{2}}}({\frac{\Lambda^{2}}{\mu^{2}}})^{-\beta^{2}/8\pi}
\rho_{1}=\Gamma_{11}\psi,\quad\rho_{3}={\frac{1}{6}}\psi^{3}+{\calF}\Gamma_{11}\psi,\quad\rho_{5}={\frac{1}{120}}\Gamma_{11}\psi^{5}+{\frac{1}{6}}{\calF}\psi^{3}+{\frac{1}{2}}{\calF}^{2}\Gamma_{11}\psi,\quad\mathrm{etc.},
\begin{array}{c}{Fun_{q}^{{\calR}}\left({\sfG}_{\tau}\right)\rightarrowA,}\\{L^{n}\mapstoG^{n-1}\widehat{L}^{n}G^{n^{-1}},}\\{M\mapstoG^{N}\widehat{M}G^{N^{-1}};}\\{\mathrm{here~}\widehat{M}=\widehat{L}^{1}\cdot\ldots\cdot\widehat{L}^{N}.}\\\end{array}
\left[\frac{a(v_{j})a^{\iota_{2}}(\bar{v}_{j})}{b(v_{j})b^{\iota_{2}}(\bar{v}_{j})}\right]^{N}=-\prod_{i\neqj}^{n}\frac{a(v_{j}-v_{i})}{a(v_{i}-v_{j})},\qquadj=1,...n.
\left[\left(\Box+\frac{e^{2}}{\pi}\right)\left(g_{\mu\nu}-\frac{\partial_{\mu}\partial_{\nu}}{\Box}\right)+\lambda\partial_{\mu}\partial_{\nu}\right]\bigtriangleup^{\nu\alpha}(x-y,e^{2}/\pi)=\delta^{(2)}(x-y)g_{\mu}^{\alpha}\;.
kW_{1}b_{2k}=\frac{-2\partial_{2}c}{(\partial_{1}c)^{2}+(\partial_{2}c)^{2}}\cdot[\sum_{s=r}^{k-1}s(\lambda_{s}\partial_{2}b_{2k-2s}+\omega_{s}b_{2k-2s})+\gammasb_{2s}b_{2k-2s}+\theta_{k}].
\mathrm{\frac{\kappa^{2}}{12}R^{-p-1}\frac{\partial}{\partialR}R^{p}\frac{\partial}{\partialR}=\frac{\kappa^{2}}{12}\left(\frac{1}{R}\frac{\partial^{2}}{\partialR^{2}}+\frac{p}{R^{2}}\frac{\partial}{\partialR}\right),}
A_{\infty}=\frac{1}{(1+q)Q}\left[\sqrt{4s^{2}-2(1+q)Qc}\frac{1+\left(\frac{k-1}{k+1}\right)^{\sqrt{4s^{2}-2(1+q)Qc}/D}}{1-\left(\frac{k-1}{k+1}\right)^{\sqrt{4s^{2}-2(1+q)Qc}/D}}-2s\right]