harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
w^{-2\delta}\langleP(\chi,\chi;-z^{\ast})|V(\psi,\rho)|P\left(\overline{\chi},\overline{\chi};-{\frac{1}{w}}\right)\rangle=z^{-2\delta}\langleP(\chi,\chi;-w^{\ast})|V(\psi,\rho)|P\left(\overline{\chi},\overline{\chi};-{\frac{1}{z}}\right)\rangle\,,
\left.{\calF}^{(N_{F})}(\mathrm{v};\left\{m_{i}\right\},\Lambda_{N_{F}})\right|_{n=2}=-\frac{i}{\pi}\mathrm{v}^{2}\left(\frac{\Lambda_{N_{F}}}{\mathrm{v}}\right)^{2(4-N_{F})}\sum_{\delta=0}^{N_{F}}{f}_{\delta}^{(N_{F})}\left(\frac{M_{\delta}^{(N_{F})}}{\mathrm{v}^{2\delta}}\right)
c^{2}\Phi^{2}={\frac{1}{\mu}}{\frac{\partial{\calL}}{\partial\mu}}\,\qquadc^{2}\Phi^{2}{\calK}=-{\frac{1}{s}}{\frac{\partial{\calL}}{\partials}}\,\qquadc^{2}\Phi^{2}{\calA}=-{\frac{1}{2y}}{\frac{\partial{\calL}}{\partialy}}\.
{\calL}[\phi,\sigma]=\frac{Z}{2}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{\mu^{\varepsilon}Z_{2}\phi^{2}\sigma}{2\sqrt{\lambda}Z_{\zeta}\zeta}+\frac{\lambda\mu^{2\varepsilon}}{4!}\left(Z_{4}+\frac{3Z_{2}^{2}}{\lambdaZ_{\zeta}\zeta}\right)\phi^{4}+\frac{\sigma^{2}}{2\lambdaZ_{\zeta}\zeta}-\frac{\sigmaJ}{\sqrt{\lambda}}
=\frac{eg}{4\piA}\hat{e}_{z}\int_{string}\frac{xz}{(x^{2}+z^{2})^{\frac{3}{2}}}dV=\frac{egz}{4\pi}\int_{-\infty}^{0}\frac{xdx}{(x^{2}+z^{2})^{\frac{3}{2}}}\hat{e}_{z}
S_{2}=\intd^{2}\xi\left[\sqrt{g}\left(g^{\alpha\beta}\partial_{\alpha}\psi^{a}\partial_{\beta}\psi^{a}+2g^{\alpha\beta}w_{\alpha}^{[ab]}\partial_{\beta}\psi^{a}\psi^{b}+2\psi^{a}\psi^{a}\right)+O(t^{2}\psi^{2})\right]\:.
\tilde{\Pi}_{i}^{a}({\bfr})=\Pi_{i}^{a}({\bfr})+\sum_{m=0}\sum_{n=0}\sum_{r=0}{\textstyle\frac{g^{m+n+r}}{m!n!}}(-1)^{m+n+r}f_{(m)}^{\vec{\mu}ac}f_{(n)}^{\vec{\nu}cd}f_{(r)}^{\vec{\delta}dh}{\calR}_{(m)}^{\vec{\mu}}({\bfr})\,{\calM}_{(n)}^{\vec{\nu}}({\bfr}){\textstyle\frac{\partial_{i}}{\partial^{2}}}\left({\calT}_{(r)}^{\vec{\delta}}({\bfr})j_{0}^{h}({\bfr})\right)\,,
\psi_{\sigma,N_{r},m_{\sigma}}=\frac{C_{N_{r},2|m_{\sigma}|}}{2{\sqrt{N_{r}+|m_{\sigma}|+\frac12}}}(\lambda_{N_{r},m_{\sigma}}|w|)^{|m_{\sigma}|}\mathrm{e}^{-\frac{\lambda_{N_{r},m_{\sigma}}|w|}{2}}L_{N_{r}}^{2|m_{\sigma}|}(\lambda_{N_{r},m_{\sigma}}|w|)\mathrm{e}^{i(m_{\sigma}-\sigma)\mathrm{arg}w}/{\sqrt2\pi}.
\tilde{D}_{-}\tilde{D}_{+}\tilde{W}=e^{2\tilde{W}}\tilde{\Psi}_{L}^{+}\tilde{\Psi}_{R}^{-},~~~\tilde{D}_{\pm}=\tilde{\partial}_{\pm}+i\tilde{\eta}^{\pm}\tilde{\partial}_{\pm\pm},
U(x,\lambda)=\frac{1}{2}\frac{L_{0}(x)+L_{1}(x)}{1-\lambda}-\,\frac{1}{2}\frac{L_{0}(x)-L_{1}(x)}{1+\lambda}=\,\frac{\lambdaL_{0}(x)+L_{1}(x)}{1-{\lambda}^{2}}
\beta_{o}^{\mu}=\sum_{e\neq0}\beta_{e}^{\mu}U_{-e,o}\,\,,\quad\beta_{o}^{b}=\sum_{e\neq0}\beta_{e}^{b}U_{-e,o}\,\,,\quad\beta_{o}^{c}=\sum_{e\neq0}U_{o,-e}^{-1}\beta_{e}^{c}\,\,.
{\calL}=-\frac{1}{12}(\partial_{\mu}A_{\nu\lambda}+\partial_{\lambda}A_{\mu\nu}+\partial_{\nu}A_{\lambda\mu})^{2}+\frac{m^{2}}{4}A_{\mu\nu}^{2}-\overline{{\psi}}(\gamma_{\mu}\partial^{\mu}+M)\psi-\frac{1}{2}g\varepsilon_{\mu\nu\lambda\varrho}\overline{{\psi}}\gamma^{\nu}\psi\partial^{\mu}A^{\lambda\varrho}.
i\frac{\partial}{\partialt}\langle\vec{q}\,^{\prime},t\mid\vec{q},0\rangle=-\frac{1}{2}\sum_{i=1}^{3}\frac{\partial^{2}}{\partialq_{i}^{2}}\langle\vec{q}\,^{\prime},t\mid\vec{q},0\rangle+V(\vec{q}\,^{\prime})\langle\vec{q}\,^{\prime},t\mid\vec{q},0\rangle
x^{a}\partial^{\gamma}(f\starg)=\sum_{b\leqa}\sum_{\epsilon\leq\gamma}(-i)^{\vertb\vert}\left(\begin{array}{c}{a}\\{b}\\\end{array}\right)\left(\begin{array}{c}{\gamma}\\{\epsilon}\\\end{array}\right)x^{a-b}\partial^{\gamma-\epsilon}f\star\hat{\partial}^{b}\partial^{\epsilon}g
\dot{x}^{\sigma}\frac{\partialg_{\mu\nu}}{\partialx^{\sigma}}=\frac{1}{2}\left[2\Gamma_{\mu\nu\sigma}\dot{x}^{\sigma}+2\Gamma_{\nu\mu\sigma}\dot{x}^{\sigma}+\rho_{\mu\nu}+\rho_{\nu\mu}\right].
{\calL}_{\mathrm{fermion}}={\overline{{\tilde{\psi}}}}\big({\tilde{X}}_{i},t\big)\left(i\gamma^{0}\partial_{t}+i\gamma^{i}\left(\frac{1+{\tilde{R}}^{2}}{2}\partial_{{\tilde{X}}_{i}}-{\tilde{X}}_{i}\right)\right){\tilde{\psi}}\big({\tilde{X}}_{i},t\big),
F(g_{i}({{{(h^{1})}^{-1}}g^{1}{h^{1}}})_{k};({gh{g^{-1}}})_{n}h_{m}^{1})=F[({P^{-1}}(h^{1}))_{k}^{k^{\prime}}g_{i}g_{k^{\prime}}^{1};P(g)_{n}^{n^{\prime}}h_{n^{\prime}}h_{m}^{1}]
E_{ik}=[i,k]E_{ik}^{*},\;\;E_{\bar{i}\bar{k}}=(\hat{\bar{i}},\hat{\bar{k}})E_{\bar{i}\bar{k}},\;\;E_{i\bar{k}}=u[1,i]v(1,\hat{\bar{k}})E_{i\bar{k}}^{*},
A_{1}^{(1)}(x)\;=\;\frac{1}{e}\;\frac{-x_{2}}{x^{2}+\lambda^{2}}\;\;\;\;,\;\;\;\;A_{2}^{(1)}(x)\;=\;\frac{1}{e}\;\frac{x_{1}}{x^{2}+\lambda^{2}}\;.
H_{p}=1+\sum_{i}{\frac{Q_{i}}{[|\vec{x}-\vec{x}_{0\,i}|^{2}+{\frac{4Q}{(p-2)^{2}}}|\vec{z}-\vec{z}_{0}|^{p-2}]^{\frac{(p+2)}{2(p-2)}}}},\\\H_{p+2}={\frac{Q}{|\vec{z}-\vec{z}_{0}|^{4-p}}}.
S[t,\varrho,\varphi,x]=\int_{0}^{t}d\tau\int_{0}^{\pi}d\sigma\left[{\frac{\alpha}{2}}\left(\dot{x}^{2}-x^{2}\right)+{\frac{\beta}{2}}\left(\dot{\varphi}^{2}-\varphi^{2}+2\dot{\varrho}\dot{\varphi}-2\varrho^{\prime}\varphi^{\prime}\right)\right]\;\;\;,
U(t,\vec{x})=\left(Pe^{-ig\int_{0}^{t}dt^{\prime}\,A_{0}(t^{\prime},\vec{x})}\right)\left(Pe^{ig\int_{0}^{\beta}dt^{\prime}\,A_{0}(t^{\prime},\vec{x})}\right)^{{\frac{t}{\beta}}}
H={\frac{1}{16\piG}}\intd^{2}x\left[e_{t}^{a}\varepsilon^{ij}\left(R_{aij}+{\frac{1}{l^{2}}}\varepsilon_{abc}e_{i}^{b}e_{j}^{c}\right)+\omega_{t}^{a}\left(D_{i}e_{aj}-D_{j}e_{ai}\right)\right]\,,
\left(\begin{array}{c}{D^{+}}\\{D^{-}}\\\end{array}\right)\left|j_{3},s=-1\right\rangle=\left(\begin{array}{c}{\left|j_{3},s=-1;\,l+1,-\frac{1}{2}\right\rangle}\\{\left|j_{3},s=-1;\,l,+\frac{1}{2}\right\rangle}\\\end{array}\right).
\hat{R}_{\mu}{}^{\nu}(z;g)=\Omega(z;g)^{-1}R_{\mu}{}^{\nu}(z;g)=\textstyle{\frac{1}{2}}\mathrm{tr}(\gamma_{\mu}L(z;g)\gamma^{\nu}L^{-1}(z;g))\in\mathrm{SO}(1,2)\,.
\Pi_{o}^{(2)}(0)=\frac{g_{r}^{2}}{M_{r}}(1+M_{r}\frac{\partial}{\partialM_{r}})J(M_{r},T)=-\frac{g_{r}^{2}}{2\pi}\left(1+\frac{1}{e^{x}-1}+\frac{1}{x}\mathrm{ln}(1-e^{-x})\right)\;.
\begin{array}{lcr}{U^{+{\frac{n(n+1)}{2}}}=uM^{+{\frac{n(n+1)}{2}}}}\\{V^{+{\frac{n(n+1)}{2}}}=vN^{+{\frac{n(n+1)}{2}}}}\\{Z^{+{(n+1)}}=zS^{+{(n+1)}};}\\\end{array}
\bar{I}^{(2l-2\varepsilon)}=\sum_{j=0}^{l}\left(\frac{m^{2}}{4m^{2}-q^{2}}\right)^{j-\varepsilon}\frac{\Gamma(2j-2\varepsilon-3)\Gamma(2-j+\varepsilon)}{i^{2j-2\varepsilon}\Gamma(j-1-\varepsilon)}+C_{\varepsilon}^{l},
F_{(e)}^{*\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\alpha\beta}F_{(e)\alpha\beta}=\left(\begin{array}{cccc}{0}&{-B_{x}}&{-B_{y}}&{-B_{z}}\\{B_{x}}&{0}&{E_{z}}&{-E_{y}}\\{B_{y}}&{-E_{z}}&{0}&{E_{x}}\\{B_{z}}&{E_{y}}&{-E_{x}}&{0}\\\end{array}\right)
E={\frac{1}{n!}}\intd\Omega\left[E_{1}\left|Y_{a_{1}\cdotsa_{n}}^{(l)}\right|^{2}+E_{2}\left|Y_{a_{2}\cdotsa_{n}}^{(l)}\right|^{2}+E_{3}\left|Y_{a_{3}\cdotsa_{n}}^{(l)}\right|^{2}\right].
\begin{array}{lllll}{\phi_{1}=p_{y}}&{}&{}&{\psi_{1}=p_{\beta}}\\{\phi_{2}=x}&{}&{}&{\psi_{2}=\alpha}\\{\phi_{3}=p_{z}}&{}&{}&{\psi_{3}=p_{\alpha}}\\{}&{}&{}&{\psi_{4}=\beta}\\\end{array}
E=\kappa\,c^{2}F\left(\frac{\calE}{\kappac^{2}},\frac{\vec{\calP}^{2}}{\kappa^{2}c^{2}}\right)\,,\qquad\vec{P}=\vec{\calP}\,G\left(\frac{\calE}{\kappac^{2}},\frac{\vec{\calP}^{2}}{\kappa^{2}c^{2}}\right)\,.
+\frac{(ig)^{3}}{2}\int_{0}^{1}d\tau_{1}\int_{0}^{\tau_{1}}d\tau_{2}\int_{0}^{\tau_{2}}d\tau_{3}A_{b}(X(\tau_{1}))[A_{c}(X(\tau_{2})),A_{d}(X(\tau_{3}))]\partial_{\tau_{1}}X^{b}\partial_{\tau_{2}}X^{c}\partial_{\tau_{3}}X^{d}+{\calO}(e^{4}),
g_{\mu\nu}{\frac{dx^{\mu}}{d\tilde{\lambda}}}{\frac{dx^{\nu}}{d\tilde{\lambda}}}=-{\calW}^{-1}\left[\epsilon_{D}\left({\frac{d\lambda}{d\tilde{\lambda}}}\right)^{2}+\left({\frac{dy}{d\tilde{\lambda}}}\right)^{2}\right].
T_{1}(p,q)={\frac{C_{1}(q)}{\alpha^{2}}}\left({\frac{p}{\alpha}}\right)^{-2}+{\frac{A_{1}(q)}{\alpha^{2}}}\left({\frac{p}{\alpha}}\right)^{-(3/2)+(1/2)\eta}+{\frac{B_{1}(q)}{\alpha^{2}}}\left({\frac{p}{\alpha}}\right)^{-(3/2)-(1/2)\eta},
\langle\psi_{1}^{I_{1}}(x_{1})\cdots\psi_{n}^{I_{n}}(x_{n})\rangle=e^{i(\kappa_{1}+\cdots+\kappa_{n})\Omega}\langle\psi_{1}^{I_{1}}(x_{1})\cdots\psi_{n}^{I_{n}}(x_{n})\rangle\,,
m_{network}^{2}={\frac{1}{(2\pi)^{4}\alpha^{3}}}A\left(\begin{matrix}{p_{1}}&{q_{1}}&{p_{2}}&{q_{2}}\\\end{matrix}\right)(M\pmL)\left(\begin{matrix}{p_{1}}\\{q_{1}}\\{p_{2}}\\{q_{2}}\\\end{matrix}\right),\
V=G_{N}{\frac{m_{1}m_{2}}{r}}+\int_{0}^{\infty}{\frac{dm}{k}}G_{N}{\frac{m}{k}}{\frac{m_{1}m_{2}e^{-mr}}{r}}=G_{N}{\frac{m_{1}m_{2}}{r}}\left(1+{\frac{1}{k^{2}r^{2}}}\right)\;.
T^{\mu\nu}(t,{\bfv})=\sum\frac{P_{i}^{\mu}P_{i}^{\nu}}{E_{i}}\delta^{D-1}({\bfx}-{\bfv}t)\Theta(-t)+\frac{{P^{\prime}}_{i}^{\mu}{P^{\prime}}_{i}^{\nu}}{E_{i}^{\prime}}\delta^{D-1}({\bfx^{\prime}}-{\bfv^{\prime}}t)\Theta(t)\,,
<\alpha_{p}(t)>~={}~(-i)^{p/{\bar{n}}}{\frac{\lambda(p)}{\lambda(p)-1}}({\frac{C_{\bar{n}}^{\frac{1}{2}}}{{C_{\bar{n}}^{\frac{1}{2}}}^{*}}})^{{\frac{p}{\bar{n}}}}~J_{p/{\bar{n}}}(\Phi({\bar{n}},p))
{\calS}_{5D}=\intd^{4}xdy\,\sqrt{G}\,[2M_{5}^{3}R-\Lambda(y)]-\sum_{i}\intd^{4}x\sqrt{\hat{g}}V_{i}+\intd^{4}xdy\sqrt{\hat{g}}{\calL}_{m}(\Phi_{i},\hat{g})
\begin{array}{c}{x^{\mu}(\sigma,\tau)=x_{R}^{\mu}(\sigma,\tau)+x_{L}^{\mu}(\sigma,\tau)}\\{x_{R,L}^{\mu}(\sigma,\tau)=\frac12x_{0}^{\mu}+\frac12l^{2}p^{\mu}(\tau\mp\sigma)+\frac{il}2{\sum}\frac1n\alpha_{n,\pm}^{\mu}e^{-2in(\tau\mp\sigma)}}\\\end{array}
S_{b}={\frac{-1}{16\piG}}\intd^{4}x\sqrt{-\hat{g}}\left\{\hat{\calR}+\{\frac{6}{1+2q}}\{\frac{(\hat{\partial}\varphi)^{2}}{\varphi^{2}}}+\{\frac{12q}{1+2q}}\(\hat{\partial}\psi)^{2}\\right\}.
<\bar{0}|\phi(x)\phi(x^{\prime})|\bar{0}>=-{\frac{A^{\alpha}(x)A^{\alpha}(x^{\prime})}{4\pi^{2}}}\left\{[v(x)-v(x^{\prime})]^{2}+(y-y^{\prime})^{2}+(z-z^{\prime})^{2}-(t-t^{\prime})^{2}\right\}
K(t,{\bfx},s\,\vert\,t,{\bfx},0)={\frac{1}{4\pis}}\int{\frac{dp_{y}}{2\pi}}\;\langlex\vert\,e^{-i{\hat{G}}s}\,\vertx\rangle\qquad\mathrm{where}\qquad{\hat{G}}\equiv-d_{x}^{2}+(p_{y}-qA)^{2}.
\psi_{\mp}^{\mu}|p,\pm\rangle_{RR}^{0}=\psi_{\pm}^{i}|p,\pm\rangle_{RR}^{0}=0\,,\qquad\psi_{\pm}^{\alpha}\equiv{\frac{1}{\sqrt{2}}}\left(\psi_{0}^{\alpha}\pmi\tilde{\psi}_{0}^{\alpha}\right)\,.
\varepsilon_{p}^{D}(L)=\frac{L^{p-D}}{2^{D+1}}\sum_{q=0}^{p-1}(-1)^{q+1}C_{p}^{q}\left(\sqrt{\pi}\right)^{q-D}\Gamma\left(\frac{D-q}{2}\right)A\left(1,\cdots,1;\frac{D-q}{2}\right)
{\frac{{\calW}^{(2)}\pi^{2\omega}e^{2i\pi\omega}}{C_{F}C_{A}(2T)^{4-4\omega}(LT)^{2}}}={\frac{1}{(\omega-1)^{2}}}+{\frac{9-4\gamma}{2(\omega-1)}}+{\frac{39}{2}}-9\gamma+2\gamma^{2}+{\frac{\pi^{2}}{6}}+{\calO}(\omega-1)\,
\int_{0}^{\beta_{T}}d\tau\int_{-\infty}^{\infty}dx\,v^{(\perp)}(\tau,x)\hat{{\calG}}_{\perp}^{(1)}v^{(\perp)}(\tau,x)=\sum_{n,m}c_{nm}^{(\perp)}\left[\left(\frac{2\pin}{\beta_{T}}\right)^{2}+\left(\omega_{m}^{(\perp)}\right)^{2}\right]\left(c_{nm}^{(\perp)}\right)^{*},
\bar{P}_{\pi}^{(\pm)}=2(1-x^{2})(1-{\frac{\omega}{v}})+\omega^{2}x^{2}(1-v^{2}x^{2})\mps\sqrt{1-v^{2}x^{2}}\;(2-2x^{2}-\omegavx^{2}){\frac{M\omega}{p_{\perp}}}\,.
S_{\mathrm{topol}}={\frac{3\gamma}{2}}\int_{0}^{\infty}dr{\frac{\partial}{\partialr}}\left(R^{\prime}(r)-{\frac{1}{3}}R^{3}(r)\right)={\frac{3\gamma}{2}}[R^{\prime}(r)-{\frac{1}{3}}R^{3}(r)]_{r\rightarrow\infty}=\gamma\.
\langleT_{i}^{k}\rangle_{N}^{(b)}\left(\xi_{1},\xi_{2},\xi\right)=\langleT_{i}^{k}\rangle_{N}^{(1b)}\left(\xi_{2},\xi\right)-A_{d}\delta_{i}^{k}\int_{0}^{\infty}dk\,k^{d}\int_{0}^{\infty}d\omega\frac{I_{\omega}^{\prime}(k\xi_{1})}{I_{\omega}^{\prime}(k\xi_{2})}\frac{F^{(i)}[N_{\omega}(k\xi,k\xi_{2})]}{N_{\omega}^{\prime}(k\xi_{1},k\xi_{2})}.
w={\frac{1}{(2\pi)^{D-1}}}\sum_{S}{\frac{\beta_{1}+\beta_{2}}{\pi\epsilon}}\sum_{k=1}^{\infty}(-1)^{(k+1)(a_{S}+1)}(|\epsilon|/k)^{D/2}e^{-\pikM_{S}^{2}/|\epsilon|}.
D_{\mathrm{G}}^{2}[U,U^{\prime}]=\sum_{x,i}\sum_{\alpha=0}^{3}\left[((u_{\alpha})_{x,i}-(u_{\alpha}^{\prime})_{x,i})^{2}+((\dot{u}_{\alpha})_{x,i}-(\dot{u}_{\alpha}^{\prime})_{x,i})^{2}\right]\,,
1-\frac14\omega_{0}^{2}r_{0}^{2}+\frac{1}{64}\omega_{0}^{4}r_{0}^{4}-\frac{1}{384}\omega_{0}^{6}r_{0}^{6}-\frac{7}{64}\kappa^{2}r_{0}^{2}-\frac{15}{128}\omega_{0}^{2}\kappa^{2}r_{0}^{4}=\frac{c_{1}}{\alpha_{0}}K_{0}(k_{0}r_{0}),
X=\left(\begin{matrix}{z}\\{x^{+}}\\{x}\\{x^{-}}\\\end{matrix}\right)\quad,\quadg_{o}\cdotX=\left(\begin{matrix}{z+t\,R}\\{x^{+}}\\{x+tx^{+}}\\{x^{-}+tx+\frac{1}{2}t^{2}x^{+}}\\\end{matrix}\right)~.
\Pi_{-}^{AB}|0\rangle_{D}=\sum_{C=-\infty}^{N-1}\prod_{D=-\infty}^{C-1}\Pi_{0}^{DD}\left[\Pi_{-}^{AB}\,,\Pi_{0}^{CC}\right]\prod_{G=C+1}^{N-1}\Pi_{0}^{GG}|0\rangle.
\partial^{\mu}H_{\mu\mu_{1}\ldots\mu_{p-2}}^{*}=\delta\left(\frac{\partial^{\mu}}{\Box+M^{2}}\left(M\eta_{\mu\mu_{1}\ldots\mu_{p-2}}^{*}-\frac1{p-1}\partial_{\left[\mu\right.}C_{\left.\mu_{1}\ldots\mu_{p-2}\right]}^{*}\right)\right),
H_{\mathrm{cl}}=\frac{m^{2}}{2\lambda}\int\left(\phi_{0}^{\prime}(x)^{2}+U(\phi_{0})^{2}\right)dx=\mp\frac{m^{2}}{\lambda}\intU(\phi_{0}(x))\phi_{0}^{\prime}(x)=\mpZ_{\mathrm{cl}},
2\frac{d^{2}\mu}{dr^{2}}+\left(\frac{d\mu}{dr}\right)^{2}+\frac{2}{r}\left(\frac{d\mu}{dr}-\frac{d\nu}{dr}\right)-\frac{d\mu}{dr}\frac{d\nu}{dr}-e^{\nu-\mu}\left(2\frac{d^{2}\nu}{dt^{2}}+\left(\frac{d\nu}{dt}\right)^{2}-\frac{d\nu}{dt}\frac{d\mu}{dt}\right)=0.
\sum_{b}\sum_{\alpha,\gamma}\;\sum_{\beta,\beta^{\prime}{\mathrm{\scriptsize{~with~}}}\beta\neq\beta^{\prime}}P_{(b\beta)}^{(a\alpha)}\:P_{(c\gamma)}^{(b\beta)}\;=\;\delta^{2}\:Q_{c}^{a}(x,y)\;.
{\calZ}(G,\lambdaA,N)=\sum_{\nu\in\Sigma_{G}}\frac{(-1)^{\tilde{t}}}{|S_{\nu}|}e^{-\frac{n\lambdaA}{2}}\frac{(\lambdaA)^{(i+t+\tilde{t}+h)}}{i!t!\tilde{t}!h!}N^{n(2-2G)-2(t+\tilde{t}+h)-i}\left[1+{\calO}\left({\frac{1}{N}}\right)\right],
F_{-}=T(\phi)F_{+}=\left(\begin{array}{llll}{e^{-2iq_{1}\phi}}&{0}&{\cdots}&{0}\\{0}&{e^{-2iq_{2}\phi}}&{}&{0}\\{\vdots}&{}&{\ddots}&{\vdots}\\{0}&{0}&{\cdots}&{e^{-2iq_{n}\phi}}\\\end{array}\right)F_{+}
\langle\bar{P}\rangle_{0}=\kappa{\frac{\partial}{\partial\bar{t}}}\,\bar{F}_{0}={\frac{\partial}{\partial\bar{t}}}\,\Bigl(\bar{t}t_{s}+F_{0}(t_{s})\Bigr)=t_{s}=\bar{t}^{\,1-\bar{\gamma}}\,
S={\textstyle\frac{1}{16\pi}}\intd^{4}x\sqrt{|g|}\,\left\{R+2(\partial\phi)^{2}+{\textstyle\frac{1}{2}}e^{4\phi}(\partiala)^{2}-e^{-2\phi}\sum_{n=1}^{N}F^{(n)}F^{(n)}+a\sum_{n=1}^{N}F^{(n)}\^{\star}F^{(n)}\right\}\,.
U(\phi_{1},\phi_{2})=\frac{1}{2}\left(\frac{\partialW}{\partial\phi_{1}}\right)^{2}+\frac{1}{2}\left(\frac{\partialW}{\partial\phi_{2}}\right)^{2}\qquad;\qquadW(\phi)=4\sqrt{2}\left(\frac{1}{3}\phi_{1}^{3}-\frac{1}{4}\phi_{1}+\frac{\sigma}{2}\phi_{1}\phi_{2}^{2}\right)
\intd^{n}x\,J_{a}(x)\,\frac{\delta^{L}Z_{J,K,\xi}}{\deltaK_{a}(x)}-\sum_{r\geq2}\frac1{r!}\,\frac{\partial^{R}Z_{J,K,\xi}}{\partial\xi^{B}}\,f_{A_{1}\cdotsA_{r}}^{B}\xi^{A_{r}}\cdots\xi^{A_{1}}=0.
\Theta_{\mu}(\vec{x},t)\,=\,{\frac{1}{(2\pi)^{n-1}}}\int_{-\infty}^{\infty}{\frac{d\vec{K}}{2w(\vec{K})}}\,\lbrace{{\bfb}(\vec{K})\}e^{-iw(\vec{K})t+i\vec{K}\cdot\vec{x}}\,\,+\,\,c.c.\rbrace\,,
\begin{array}{rll}{P_{i}P_{j}\,}&{=\,\delta_{ij}P_{j}}&{}\\{P_{i}N_{j}^{\pm}\,}&{=\delta_{ij}N_{j}^{\pm}}&{}\\{N_{i}^{\pm}N_{j}^{\pm}\,}&{=N_{i}^{\pm}N_{j}^{\mp}}&{=0\qquad.}\\\end{array}
R_{\alpha\dot{\alpha}}=R_{\alpha\dot{\alpha}}^{(0)}-i\overline{\theta}_{\dot{\beta}}\bigl(\overline{J}^{\dot{\beta}}{}_{\dot{\alpha}\alpha}-{\frac{2}{3}}\delta_{\dot{\alpha}}^{\dot{\beta}}\overline{J}^{\dot{\gamma}}{}_{\dot{\gamma}\alpha}\bigr)+\cdots,
{\calI}=\frac{1}{4\pi}\int\left(\frac{1+|z|^{2}}{1+|R|^{2}}\left|\frac{\mathrm{d}R}{\mathrm{d}z}\right|\right)^{4}\frac{2i\mathrm{d}z\mathrm{d}{\bar{z}}}{(1+|z|^{2})^{2}}.
V(\sigma)=\frac{N\sigma^{2}}{2G}-{\frac{i}{2}}\int_{0}^{\infty}\frac{ds}{s}\int\frac{d^{3}p}{(2\pi)^{3}}tre^{is[P_{\mu}P^{\mu}-{\frac{g}{4}}\sigma^{\mu\nu}F_{\mu\nu}^{a}\tau^{a}-\sigma^{2}]}.
\frac{d}{d\tau}(g_{\mu\nu}{\dot{x}}^{\nu})-\frac{1}{2}\partial_{\mu}g_{\nu\lambda}{\dot{x}}^{\nu}{\dot{x}}^{\lambda}=(kR\tilde{d}_{\tau}\tilde{F}_{\mu\nu}+\frac{\tilde{m}k\sqrt{\alpha^{\prime}}}{R}F_{\mu\nu}){\dot{x}}^{\nu}.
\varepsilon_{\Omega}-\varepsilon_{\Lambda}=\varepsilon_{\Omega^{*}}-\varepsilon_{\Lambda^{*}}=\sum_{k}\left(({\Omega_{k}^{*}}^{2}-{\Lambda_{k}^{*}}^{2})-2\,k\,\beta\,(\Omega_{k}^{*}-\Lambda_{k}^{*})\right)\,,\qquad|\Omega|=|\Lambda|\,,
\int_{G}t_{mn}^{(\rho)}(g)\,dg=\sum_{j=1}^{\ell}\bigl<v_{m};P^{(j)}v_{n}\bigr>=\sum_{j=1}^{\ell}\bigl<P^{(j)}v_{m};P^{(j)}v_{n}\bigr>=\sum_{j=1}^{\ell}P_{m}^{(j)}P_{n}^{(j)},
{\calL}=\frac{1}{2}\left\{\partial_{\mu}\varphi_{i}\,\partial^{\mu}\varphi_{i}+i\bar{\psi}_{i}\gamma^{\mu}\partial_{\mu}\psi_{i}+f_{i}f_{i}+2f_{i}\,\frac{\partial{W}}{\partial\varphi_{i}}-\frac{\partial^{2}{W}}{\partial\varphi_{i}\partial\varphi_{j}}\,\bar{\psi}_{i}\psi_{j}\right\}\;.
\chi_{h_{1}}(q)=q^{-c/24+h_{1}}\sum_{\ell=0}^{\infty}d_{\ell}^{h_{1}}\,q^{\ell},\qquad\chi_{h_{2}}(q)=q^{-c/24+h_{2}}\sum_{\ell=0}^{\infty}d_{\ell}^{h_{2}}\,q^{\ell},\qquadd_{\ell}^{h_{1}},d_{\ell}^{h_{2}}\ge0
ds_{11}^{2}={\frac{1}{({\frac{1}{6}}C_{abc}H^{a}H^{b}H^{c})^{\frac{1}{3}}}}\left[dudv+H_{0}du^{2}+{\frac{1}{6}}C_{abc}H^{a}H^{b}H^{c}d\vec{x}^{2}+H^{a}\omega_{a}\right],
W_{\mu}^{\alpha^{\prime}}M_{\alpha^{\prime}\beta^{\prime}}W^{\beta^{\prime}\mu}=\sum_{(i)}\bar{\phi}_{V}^{(i)}\{-g_{2}W_{\mu}^{\alpha}\frac{\tau_{\alpha}}{2}-g_{1}\frac{n}{2}B_{\mu}\}^{2}\phi_{V}^{(i)},
{\hat{U}}_{P}({\hat{J}}^{0}(0),{\hat{\bfJ}}(0)){\hat{U}}_{P}^{-1}={\hat{U}}_{T}({\hat{J}}^{0}(0),{\hat{\bfJ}}(0)){\hat{U}}_{T}^{-1}=({\hat{J}}^{0}(0),-{\hat{\bfJ}}(0))
\begin{array}{lcr}{x_{0}=X_{0}[R_{1}+|Y_{0}|^{2}+|Y_{1}|^{2}]^{-{\frac{1}{2}}}}\\{x_{1}=X_{1}[R_{1}+|Y_{0}|^{2}+|Y_{1}|^{2}]^{-{\frac{1}{2}}},}\\\end{array}
L(X,F,g_{m}^{2},l_{s})=\sum_{I,L}c_{I,L}(N,l_{s}X){\frac{F^{2}}{g_{m}^{2}}}\left({\frac{g_{m}^{2}F^{2}}{X^{7}}}\right)^{L}\left({\frac{F}{X^{2}}}\right)^{I-2L-2}.
lim_{Q\toq}[{{\frac{Q^{\frac{kN_{B}}{2}}{(B^{-})}^{k}}{([k]!)^{\frac{1}{2}}}},{{\frac{{(B^{+})}^{k}Q^{{\frac{kN_{B}}{2}}}}{([k]!)^{\frac{1}{2}}}}}}]~=~1
B=\frac{1}{e}Q=-\frac{i}{e}\int_{-\infty}^{+\infty}dx~J_{2}^{EM}(x,0)=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{+\infty}dx\sum_{i}\chi_{i}^{\prime}=\frac{1}{\sqrt{\pi}}\sum_{i}[\chi_{i}(\infty,0)-\chi_{i}(-\infty,0)].
\frac{dX}{dt}=\frac{1}{i\hbar}\left(\left(\begin{array}{cc}{0}&{q}\\{1}&{0}\\\end{array}\right)X\left(\begin{array}{cc}{q^{\frac{1}{2}}}&{0}\\{0}&{q^{-\frac{1}{2}}}\\\end{array}\right)-\left(\begin{array}{cc}{q^{\frac{1}{2}}}&{0}\\{0}&{q^{-\frac{1}{2}}}\\\end{array}\right)X\left(\begin{array}{cc}{0}&{1}\\{q}&{0}\\\end{array}\right)\right),
\frac{1}{\sqrt{\pi}}\partial_{\mu}\phi_{a,j}=\varepsilon_{\nu\mu}J_{a,j}^{\nu}=\bar{\psi}_{a,j}\gamma_{\mu}\gamma^{5}\psi_{a,j},\qquad\Lambda_{ab}^{jk}\neq0,\quad\forall\,b,k\,\,.\,
W\to\sum\lambda_{ijk}\,e^{-i\varphi_{T}^{(1)}}\,e^{-i\varphi_{U}^{(1)}}\alpha_{i}^{(1)}\alpha_{j}^{(2)}\alpha_{k}^{(3)}=e^{-i\varphi_{T}^{(1)}-i\varphi_{U}^{(1)}}W,
\begin{array}{rcl}{\eta_{12}}&{=}&{-\left(P_{12}{}^{t_{2}}\left((r_{12}{}^{t_{2}})^{-1}\right)\right)^{t_{2}}}\\{}&{=}&{-\mathrm{tr}_{3}\left(P_{23}(r_{23}{}^{t_{3}})^{-1}\right)P_{12}}\\{}&{=}&{-D_{2}P_{12}.}\\\end{array}
S=\intd^{2}\sigmad\theta_{1+}d\theta_{1-}g_{ij}D_{1+}\varphi^{i}D_{1-}\varphi^{j}+\intd^{2}\sigmadtd\theta_{1+}d\theta_{1-}H_{ijk}\partial_{t}\varphi^{i}D_{1+}\varphi^{j}D_{1-}\varphi^{k}
\Phi(\{x_{i}\})=\frac{1}{\sqrt{N!}}\left|\begin{array}{ccc}{\chi_{1}(x_{1})}&{\cdots}&{\chi_{N}(x_{1})}\\{\vdots}&{~}&{\vdots}\\{\chi_{1}(x_{N})}&{\cdots}&{\chi_{N}(x_{N})}\\\end{array}\right|
S_{1}^{~\bullet}=\beta\left.{\frac{\partial{\tilde{W}_{1}^{~\bullet}(\beta,\alpha,y,\ldots)}}{\partial\beta}}\right|_{\alpha,y,\ldots}+\alpha\left.{\frac{\partial{\tilde{W}_{1}^{~\bullet}(\beta,\alpha,y,\ldots)}}{\partial\alpha}}\right|_{\beta,y,\ldots}-W_{1}^{\bullet}~~~.
S=\frac{1}{16\piG_{5}}\intd^{5}x\sqrt{g}\lbraceR-\frac{1}{3}(\partial\phi)^{2}-\frac{1}{4}e^{\frac{4\phi}{3}}\bar{F}^{2}-\frac{1}{4}e^{-\frac{2\phi}{3}}F^{2}\rbrace
{\calL}_{R}=-\frac14G_{\mu\nu}^{i}G^{i\mu\nu}+\frac14G_{\mu\nu}^{i}B^{j\mu}B^{k\nu}\epsilon_{ijk}-\frac18g^{2}B_{\mu}^{j}B_{\nu}^{k}(B^{j\mu}B^{k\nu}-B^{k\mu}B^{j\nu}),
\psi_{-2\dot{q}}^{1,2-}=-\frac{i}{2}\nabla_{-}\chi_{\dot{q}}^{1,2},\quad\psi_{+2\dot{q}}^{1,2-}=\frac{i}{4}\Omega_{+}^{-2i}\tilde{\gamma}_{\dot{q}p}^{i}\chi_{p}^{1,2},
\|(t_{0},t_{1},t_{2},t_{3})\|_{c}^{2}=c\sum_{0}^{3}\|t_{i}(+\infty)\|^{2}+\int_{0}^{+\infty}\sum_{0}^{3}\left(\|t_{i}(s)\|^{2}-\|t_{i}(+\infty)\|^{2}\right)ds.
S=\int\mathrm{d}^{4}x\sqrt{-G}\mathrm{e}^{-\Phi}[{\calR}_{G}+\partial_{\mu}\Phi\partial^{\mu}\Phi-F_{\mu\nu}^{a}(LML)_{ab}(F^{b})^{\mu\nu}+{\frac{1}{8}}\mathrm{Tr}(\partial_{\nu}ML\partial^{\mu}ML)]\,
E_{7}:\;[3\cdot{\bf56}]\stackrel{\langle{\bf56}\rangle}{\rightarrow}E_{6}:\;[2\cdot{\bf27}+2\cdot{\bf\overline{{27}}}]\stackrel{\langle{\bf27}\rangle}{\rightarrow}F_{4}:[3\cdot{\bf26}]\stackrel{\langle{\bf26}\rangle}{\rightarrow}SO(9)\;[2\cdot{\bf16}+3\cdot{\bf9}]
\begin{aligned}{\forall\,m,n\neq0,~~~\left[a_{m}^{r},a_{n}^{s}\right]=\mathrm{sgn}(n)\delta_{m+n,0}\,\delta^{rs}~~~\mathrm{and}~~~\{\theta_{m}^{1a},\theta_{n}^{1b}\}=\frac{1}{4}(\Gamma^{+})^{ab}\,\delta_{m+n,0}}\\\end{aligned}
G(\xi_{*})=\int_{0}^{\infty}ds\left\{\left[\vartheta_{3}\left({\frac{\pi}{2}},q\right)\right]^{d_{\perp}}-1\right\}=\int_{0}^{\infty}ds\left\{\left[\vartheta_{4}\left(0,q\right)\right]^{d_{\perp}}-1\right\}.
=S_{01}(\Phi)+{\frac{\deltaS_{0}(\Phi)}{\delta\Phi^{A}}}\Phi_{1}^{A}(\xi)-\varphi_{A}^{*}{\frac{\partial\tilde{\varphi}^{A}}{\partial\Phi_{B}}}({\frac{\partial\Phi_{1}^{B}(\xi)}{\partial\alpha}}-{\frac{\partial\Phi_{1}^{B}(\xi)}{\partial\varphi^{C}}}{\frac{\partial\tilde{\varphi}^{C}}{\partial\Phi^{D}}}{\frac{\partial\Phi^{D}(\xi)}{\partial\alpha}})c+X_{p}^{(1)}(\xi).
\widetilde{\Phi}_{c}^{2}\simeq\frac{4!4N\widetilde{h}^{2}\widetilde{m}^{2}}{(4\pi)^{2}}\left(\frac{N\widetilde{h}^{4}}{(4\pi)^{2}}-\widetilde{\lambda}\right)^{-1}\simeq\frac{96\xi\widetilde{m}^{2}}{\widetilde{h}^{2}},\\\\\R_{c}\simeq\frac{2\widetilde{m}^{2}}{\widetilde{\xi}}.
q_{c}(a,b,r)=\frac{1}{8\pi}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}{dk\sum_{\gamma,\lambda}}\beta_{\lambdam}^{2}f_{\lambdam}^{(q)}(\gammaa,\gammar),\quadq_{c}=\varepsilon,\,p_{i},
\int_{0}^{\infty}\int_{0}^{\infty}dp\,dq\,p^{2}q^{2}f^{J}(p)f^{J}(q)Q_{J-1}(z_{1})=\pi\left(\frac{\alpha\mu}{n}\right)^{3}\left(\frac{2}{J}-\frac{1}{n}\right),
I(X)\sim\frac{G^{(0)}}{X}+\frac{G^{(1)}}{X^{2}}+\frac{G^{(2)}+f^{(3)}G^{(0)}}{X^{3}}+\,\frac{G^{(3)}+3f^{(3)}G^{(1)}}{X^{4}}+\cdots,\quad(X\to\infty).
\sum_{n=-\infty}^{\infty}\biggl(u_{\alphan}(\epsilon\gamma^{0}-(p-nl)\gamma^{1}-m{\bf1})\+\\frac{a}2\gamma^{0}(u_{\alphan+1}+u_{\alphan-1})\biggr)e^{i(p+nl)x-iEt}=0.
A_{n,m}=\sum_{L,L^{\prime}}\int\left(\prod_{r}dq_{r}d\overline{q}_{r}\right)\left[\left(\prod_{s}\partial_{\mu_{s}}\partial_{\bar{\mu}_{s}}\partial_{\nu_{s}}\partial_{\bar{\nu}_{s}}\right)Z_{L,L^{\prime}}^{(n)}K_{L,L^{\prime}}^{(n,m)}\right]_{\{k_{s},u_{s},v_{s}+\mu_{s}\nu_{s}\}}+S
|n_{i}^{\pm}>=\prod_{i}\frac{1}{\sqrt{n_{i}^{+}!\n_{i}^{-}!}}\,\left({\alpha_{i}^{+}}^{\dagger}\right)^{n_{i}^{+}}\left({\alpha_{i}^{-}}^{\dagger}\right)^{n_{i}^{-}}\,|0>\\\,
\mathrm{e}^{-2i\sqrt{\pi}{\eta}(0)}=\mathrm{exp}[\frac{1}{2}\int_{\kappa}^{\infty}\frac{dk_{+}}{k_{+}}]\mathrm{exp}[-2i\sqrt{\pi}{\eta}^{(-)}(0)]{\sigma}_{+}\mathrm{exp}[-2i\sqrt{\pi}{\eta}^{(+)}(0)].
A_{n}\stackrel{\epsilon\to0}{\longrightarrow}\frac{1}{n!}\left(\,\frac{3\alpha}{2\pi}\frac{1}{\epsilon}\,\right)^{n},s_{n}\stackrel{\epsilon\to0}{\longrightarrow}\frac{(-1)^{n}}{n!}\left(\,\frac{3\alpha}{2\pi}\frac{1}{\epsilon}\,\right)^{n}\;\;\;.
{}_{\sigma}R(u)=\left(\begin{array}{ccc|ccc}{\bar{a}}&{}&{}&{}&{}&{}\\{}&{\bar{b}}&{}&{\bar{d}}&{}&{}\\{}&{}&{\bar{c}}&{}&{\bar{d}}&{}\\\hline{}&{}&{}&{}&{}&{}\\{}&{\bar{d}}&{}&{\bar{c}}&{}&{}\\{}&{}&{\bar{d}}&{}&{\bar{b}}&{}\\{}&{}&{}&{}&{}&{\bar{a}}\\\end{array}\right),
A_{\mu}^{c}(x)=\frac{1}{2}(\oint_{C}d\tau\left(\frac{dy_{\mu}}{d\tau}\frac{1}{|x-y(\tau)|}-\psi_{\mu}(\tau)\psi_{\lambda}(\tau)\frac{\partial}{\partialy_{\lambda}}\left(\frac{1}{|x-y(\tau)|}\right)\right).
\widetilde\Sigma_{\psi}(p^{2})\=\g^{2}\,C_{2}^{\mathrm{fund}}\,(3+\xi)\int{\frac{dk^{2}}{16\pi^{2}}}\,{\frac{k^{2}}{{\calM}^{2}}}\,{\frac{A_{\psi}({\calM}^{2})}{A_{\psi}(m^{2})}}\,{\frac{\widetilde\Sigma_{\psi}(k^{2})}{k^{2}}}\,
\hat{K}_{\mathrm{c}}G(q,q^{\prime},p,p^{\prime};t,t^{\prime})={\frac{\partialG}{\partialt}}+{\frac{\partialH}{\partialp}}{\frac{\partialG}{\partialq}}-{\frac{\partialH}{\partialq}}{\frac{\partialG}{\partialp}}=\delta(t-t^{\prime})\;\delta(q-q^{\prime})\;\delta(p-p^{\prime})\;.
\psi(\tau)=\frac{1}{2}\int_{0}^{1}\varepsilon(\tau-\tau^{\prime})\omega(\tau^{\prime})d\tau^{\prime}+\frac{1}{2}\theta\;,\;\;\omega(\tau)=\dot{\psi}(\tau)\;,\;\varepsilon(\tau)=\mathrm{sign}\;\tau\;.