harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
D_{\mu}^{A C}J^{\mu C}= \partial_{\mu}J^{\mu A}-{\frac{i g}{2}}d^{A B C}[ A_{\mu}^{B}, J^{\mu C}]_{\mathrm{M}}+{\frac{g}{2}}f^{A B C}\{A_{\mu}^{B}, J^{\mu C}\}_{\mathrm{M}}= 0
x_{\ell}=\sum_{((\ell_1,\ldots, \ell_j))}\frac{1}{\prod_{m=1}^{\kappa(\ell_1, \ldots, \ell_j)}g_m!}\prod_{k=1}^j{\ell-\sum_{i=1}^{k-1}\ell_i \choose \ell_k}^2 \frac{\left( \ell_k ! \right)^2}{2 \ell_k}.
\frac{1+tH(x)}{1-t^{2}E(x)H(x)}=\sum_{\lambda\;\mathrm{odd}}\frac{p_{\lambda}}{z_{\lambda}}2^{l(\lambda)}\frac{A_{l(\lambda)}(t^{2})}{(1-t^{2})^{l(\lambda)+1}}x^{\left|\lambda\right|}+t\sum_{\lambda}\frac{p_{\lambda}}{z_{\lambda}}\frac{B_{o(\lambda)}(t^{2})}{(1-t^{2})^{o(\lambda)+1}}x^{\left|\lambda\right|}.
\frac{d}{dt}E_{\sigma}(\phi_n^t(u)) =DE_\sigma(\phi_n^t(u))\cdot X_n(\phi_n^t(u)) &\leq -\psi_n(\phi_n^t(u))DE_{\sigma_n}(\phi_n^t(u))\cdot X_n^0(\phi_n^t(u))+C(\phi_n^t(u))(\sigma_n^2-\sigma^2)\Vert{X_n^0(\phi_n^t(u))}\Vert_{\mathrm{W}^{2,2}_{\phi_n^t(u)}(S^1,TM)}&\leq -2\psi_n(\phi_n^t(u))\delta+2C(\phi_n^t(u))(\sigma_n^2-\sigma^2)
\mathfrak{a}_2^2 \mathfrak{a}_3^3 = (1 \otimes y_2^2+ y_2 \otimes 1) (1 \otimes y_3^3- y_3 \otimes y_3^2+ y_3^2 \otimes y_3- y_3^3 \otimes 1) &= (y_1y_2+ y_2y_3) \otimes y_1y_2y_3- y_1y_2y_3 \otimes (y_1y_2+ y_2y_3).
N_{\ell,0}^p(\Gamma)^\circ =\bigoplus_{\substack{\ell'=\max(2, \ell-2p) \ell'\equiv\ell\bmod2}}^\ell\;\bigoplus_{\substack{m'=\max(0, 2(\ell - \ell' -p)) m'\equiv0\bmod2}}^{\ell-\ell'}\;D_+^{\frac{\ell-\ell'-m'}2}\,{U}^{\frac{m'}{2}}\,S_{\ell',m'}(\Gamma)\;\oplus\;N_{\ell,0}^p(\Gamma)^\circ_1,
[z]\hat\otimes_A [\lambda^{(d)}] = (-1)^{d-1}[z]\hat\otimes_A \big([\mathrm{ext}] \hat\otimes_{C}[\lambda^{(d-1)}] \big) = (-1)^{d-1}\big( [z]\hat\otimes_A [\mathrm{ext}] \big) \hat\otimes_{C}[\lambda^{(d-1)}] = (-1)^{d-1}\partial[z] \hat\otimes_{C}[\lambda^{(d-1)}]
\lambda_{1+2j}\left(\begin{bmatrix}A X X^* B\end{bmatrix}\right) \le \inf_{\dim{\mathcal{S}}=n-j}\lambda_{1}\left( A_{\mathcal{S}}+B_{\mathcal{S}}\right) + \delta_2(X) &=\lambda_{1+j}(A+B) + \delta_2(X)
w h a t^{\prime}s t h e f a c t o r ? ? ? S_{g}={m^{2}}\int d^{4}x \, \sqrt{- g}\, ( D_{\mu}g )_{a \bar{b}}\, \, g^{\bar{b}c}\, ( D^{\mu}g )_{c \bar{d}}\, g^{\bar{d}a}
\hat{M}_{0}( z ) = \displaystyle{\left( \begin{array}{c c c}{{\hat{w}( z ) + \textstyle{\frac{1}{2}}\hat{\lambda}( z )}}&{{0}}&{{0}}\\{{i b{\cdot \gamma}}}&{{\hat{w}( z ) - \frac{1}{2}\hat{\lambda}( z )}}&{{\sqrt{2}\hat{\rho}{}^{b}( z )}}\\{{- \sqrt{2}\hat{\bar{\rho}}_{a}( z )}}&{{0}}&{{\hat{t}{}_{a}{}^{b}( z )}}\\ \end{array}\right)}\, .
g_{e}(A;\sigma) = g_{e'_{m}}(A;\sigma)^{s_{m}}...g_{e'_{1}}(A;\sigma)^{s_{1}}, P^{e}_{X}(A,E;\sigma) = c_{e'_{m}}(l,l')P^{e'^{s_{m}}_{m}}_{X}(A,E;\sigma)+...+c_{e'_{1}}(l,l')P^{e'^{s_{1}}_{1}}_{\textup{Ad}_{(g_{e'_{m}}(A;\sigma)^{s_{m}}...g_{e'_{2}}(A;\sigma)^{s_{2}})^{-1}}(X)}(A,E;\sigma), \sum_{n=1}^{m}c_{e'_{n}}(l,l') =1,
H(\mathsf{m})[X+1] =H[\mathsf{m}X+\mathsf{m}] \ref{l-Hps}=H[\mathsf{m}X]H[\mathsf{m}] \ref{l-Hps}=\frac{H(\mathsf{m})}{1-\mathsf{m}}, \ref{l-HE}
\begin{array}{c}{{v_{1}= \left( 0 , 1 , 0 , 0 , 0 , 0 \, ; \, 1 , 0 \right) \}}\\{{v_{2}= \left( 0 , 0 , 1 , 0 , 0 , 0 \, ; \, 0 , 1 \right) .}}\\ \end{array}
\left| \begin{array}{c c c c}{f^{1}}&{\cdots}&{f^{N}}&{f}\\{f^{( 1 ) 1}}&{\cdots}&{f^{( 1 ) N}}&{f^{( 1 )}}\\{\vdots}&{\cdots}&{\vdots}\\{f^{( N ) 1}}&{\cdots}&{f^{( N ) N}}&{f^{( N )}}\\ \end{array}\right| = 0 .
\hat{\tau}^{(2)}_g(C_2)=\rho_g(C_2)\cdot \hat{\tau}^{(2)}(C_2)=\rho_g(C_2)\cdot\left(\begin{array}{c c c c c c}-1 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 -1 \end{array}\right),
\mathcal{L}_{1}=-\mathcal{L}_{1}^{\ast}=P_{\sigma\tau}^{\ast}Q^{\tau\sigma}=\frac{1}{2}\left( P_{\sigma\tau}^{\ast}Q^{\tau\sigma}-P_{\sigma\tau}\overset{\ast}{\left. Q^{\tau\sigma}\right.}\right) & =\frac{i}{2}e^{\sigma\tau\kappa\rho}P_{\sigma\tau}P_{\kappa\rho}^{\ast}=4i\left( \mathbf{E}\cdot\mathbf{B-H}\cdot\mathbf{D}\right) .
\langle M_{ap}^{2}\rangle(\theta)=\int_{0}^{2\theta}\frac{\phi d\phi}{\theta^{2}}[\xi_{++}(\phi)+\xi_{\times\times}(\phi)]T_{+}(\frac{\phi}{\theta})=\int_{0}^{2\theta}\frac{\phi d\phi}{\theta^{2}}[\xi_{++}(\phi)-\xi_{\times\times}(\phi)]T_{-}(\frac{\phi}{\theta})
\mu_0(k) = (c_-(k) - 3k^2) \frac{4 K^2(k)}{\pi^2}, \mu_1(k) = -3k^2 \frac{4 K^2(k)}{\pi^2}, \mu_2(k)= 0, & \mu_3(k)= (3 - 3k^2) \frac{4 K^2(k)}{\pi^2}, \mu_4(k) = (c_+(k) - 3k^2) \frac{4 K^2(k)}{\pi^2},
\begin{array}{l}A^{\prime}(z) = -\sqrt{\frac{2}{k}}(A^{+3}+ A^{-3}+ i W)(z) \ \bar{A}^{\prime}(z) = \sqrt{\frac{2}{k}}(A^{+3}+ A^{-3}- i W)(z) \ Q^{\prime}(z) = \frac{2}{\sqrt{k}}Q^{+K}(z) \ \bar{Q}^{\prime}(z) = \frac{2}{\sqrt{k}}Q^{-K}(z)\end{array}
F_l^-= \{(x,0)\,|\, 0\le x\le r\}\cup\{(0,y)\,|\, 0\le y\le r\}, F_r^-= \{(x,0)\,|\, R\le x\le R+1\}\cup\{(0,y)\,|\, R\le y\le R+1\}& \cup\{(x,R+1)\,|\, 0\le x\le R+1\}\cup\{(R+1,y)\,|\, 0\le y\le R+1\}, F_b^+= \{(x,0)\,|\, r\le x\le R\}, F_t^+= \{(0,y)\,|\,r\le y\le R\}.
\mathbf{\Phi}_{{\nu}}^{\prime}(z)=\frac{(2{\nu}+1)\left( \frac{z}{2}\right)^{2{\nu}}}{\Gamma \left({\nu}+1\right) \Gamma \left({\nu}+2\right)}\prod_{n\geq1}\left( 1-\frac{z^{4}}{\gamma_{{\nu},n}^{\prime 4}}\right) ,
\begin{array}{l}\;n=0,1,\ldots \left\lfloor\begin{array}{l}\varepsilon\leq\gamma_n\leq(2-\varepsilon)/\beta y_n=x_n-\gamma_n Bx_n x_{n+1}=J_{\gamma_n A}y_n.\end{array}\right. begin{align*}2mm]\end{array}
\langle A\delta_L^u,v\rangle=\langle\delta_L^u,A^tv\rangle=\langle u,(A^tv)|_L\rangle=\langle u,(A^t)|_L(v|_L)\rangle=\langle A'u,v|_L\rangle=\langle\delta_L^{A'u},v\rangle\;.
\begin{array}{r c l}{d \left( \hat{G}_{( 5 )}+ \hat{H}\hat{C}_{( 2 )}\right)}&{=}&{0 ,}\\{}&{}&{}\\{d \left( \hat{\overline{{G}}}_{( 5 )}+ \hat{H}\hat{\overline{{C}}}_{( 2 )}\right)}&{=}&{0 ,}\\ \end{array}
0 \geq \ (2-2q-\alpha q)KH + 2q(1+\alpha)u^{q-1}& + \frac{2\lambda_2|\nabla_1h_{11}|^2}{(H^2-4K)[2K-\alpha\lambda_1(\lambda_1-\lambda_2)]^2}B_1 + \frac{2\lambda_1 |\nabla_2h_{22}|^2}{(H^2-4K)[2K+\alpha\lambda_2(\lambda_1-\lambda_2)]^2}B_2,
\epsilon = s_p(hs) \epsilon(hs,\alpha_m) = \begin{cases}(c,-d)_p(-d,p^m)_p cd \neq 0, \mathrm{ord}_p(d) , (-d,p^m)_p cd \neq 0, \mathrm{ord}_p(d) , (c,p^m)_p d = 0, (-d,p^m)_p c = 0. \end{cases}
\frac{d}{dt}\Big( - 2dr \wedge \Im(\Omega) - \frac{1}{4}\frac{1}{2}\omega^2 \Big) = 2 \cal{L}_{\nabla (\log |\Omega|_\omega)}\Big( - 2dr \wedge \Im(\Omega) + \frac{1}{4}\frac{1}{2}\omega^2 \Big)
\int_a^{r_3}(r_2-a)^{1+\zeta}(r_3-r_2)^\zeta dr_2= \int_0^{r_3-a}r_2^{1+\zeta}(r_3-a-r_2)^\zeta dr_2 = (r_3-a)^{2(1+\zeta)}B\Big((1+\zeta)+1, \zeta+1\Big),
\vec{A}=\frac{\sqrt{2}a\int C(\frac{u^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u)sin(\omega u)du}{C(\frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u)}\partial_{x}
\begin{array}{l l}{{\begin{array}{l l}{{\varphi ( r ) = \eta}}&{{r \rightarrow \infty}}\\{{\varphi ( r ) = 0}}&{{r = 0}}\\ \end{array}}}&{{\begin{array}{l l}{{P ( r ) = 0}}&{{r \rightarrow \infty}}\\{{P ( r ) = 1}}&{{r = 0 .}}\\ \end{array}}}\\ \end{array}
I(u) =\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\mathcal{P}\Phi(u(x)-u(y))K(x,y)dx dy -\frac{1}{p_{s}^{\ast}}\int_\Omega |u|^{p_{s}^{\ast}}-\lambda \int_{\Omega}F(x,u)dx &\geq \frac{\Lambda^{-2}}{p}[u]_{s,p}^{p}-\frac{1}{p^{\ast}_{s}}\int_{\Omega}|u|^{p_{s}^{\ast}}-\frac{1}{c_{2}}\lambda\int_{\Omega}f(x,u)u dx
P_{n-k}^{\left( k\right)}=P_{r}^{\left( k\right)}P_{n-\left( k+r\right)}^{\left( k+r\right)}-q_{1,r-1}P_{n-\left( k+r+1\right)}^{\left(k+r+1\right)}-...-q_{d,r-1}P_{n-\left( k+r+d\right)}^{\left( k+r+d\right)},
\begin{array}{ccl}Aw_{\tilde{l}_{m-1}+s,\tilde{l}_{n-1}+t}= \mu_1^{(mn)}w_{\tilde{l}_{m-1}+s,\tilde{l}_{n-1}+t}+ b_{mn}\u_{\tilde{l}_{m-1}+s,\tilde{l}_{n-1}+t}.\end{array}
{\cal L}(J) = \frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ +{\cal D}_\mu \phi{\cal D}^\mu \phi^{*}+ \frac{\lambda}{4}(\phi \phi^*)^2+ J(\phi \phi^*)- \zeta/2 J^2+{\cal L}^{\rm CT}+{\cal L}^{\rm gauge}
\bigl( [c, x_1][x_2, x_3] + [c, x_2 ] [x_1, x_3] \bigr) [x_4, x_5] + \bigl( [c, x_1][x_2, x_4] + [c, x_2 ] [x_1, x_4] \bigr) [x_3, x_5] = \ [c, x_1] \bigl( [x_2, x_3] [x_4, x_5] + [x_2, x_4] [x_3, x_5] \bigr) + [c, x_2 ] \bigl( [x_1, x_3] [x_4, x_5] + [x_1, x_4] [x_3, x_5] \bigr) \in I^{(n)},
\varphi_{1}(tw_{0}) \geq \Lambda^{-2}t^{p}[w_{0}]_{s,p}^{p}-t^{p_{s}^{\ast}}\int_\Omega |w_{0}|^{p^{\ast}_{s}}dx-\lambda c_{4}\int_{\Omega}|tw_{0}|^{q}dx &\geq A_{1}t^{p}-Bt^{p_{s}^{\ast}}-E_{1}t^{q}
\hat{L}_{h_{+},h_{-}}=\frac{\sum_{i=1}^{n}(g^{*}(x_{i})-f^{*}(x_{i}))y_{i}}{\sum_{i=1}^{n}(g_{+}^{*}(x_{i})-f_{+}^{*}(x_{i}))}= \frac{\sum_{i=1}^{n}k_{+}(x_{i}/h_{+})y_{i}}{\sum_{i=1}^{n}k_{+}(x_{i}/h_{+})}- \frac{\sum_{i=1}^{n}k_{-}(x_{i}/h_{-})y_{i}}{\sum_{i=1}^{n}k_{-}(x_{i}/h_{-})},
\Bigg(1- Q_{T}Q_{R}\sum_{i=0}^{Q_{T}Q_{R}}\binom{Q_{T}Q_{R}-1}{i}\frac{(-1)^i}{(i+1)}\Bigg( \frac{1}{1+\frac{(i+1)\overline{\gamma}_{W}}{\overline{\gamma}_{M}}}- \frac{e^{\frac{-(2^{R_{s}}-1)(i+1)}{\overline{\gamma}_{M}}}}{1+\frac{(i+1) 2^{R_{s}}\overline{\gamma}_{W}}{\overline{\gamma}_{M}}}\Bigg)\Bigg).
N_{i,1}^s = -\eta d T\big( \frac{1}{d T}\sum_{t=0}^{\lfloor sT \rfloor-1}\sum_{j=1}^d f(\frac{t}{T})A(\frac{i}{d},\frac{j}{d})\Theta(\frac{t}{T},\frac{j}{d})+o(1) \big) = -\eta d T \big( \int_0^{\frac{\lfloor sT\rfloor}{T}}\int_0^1 f(u)A(\frac{i}{d},y)\Theta(u,y)\mathrm{d}y\mathrm{d}u+o(1)\big).
g_{1}([X,Y],V) =g_{1}(\varphi V,\mathcal{V}(\nabla^{^{M_1}}_{X}\mathcal{B}Y-\nabla^{^{M_1}}_{Y}\mathcal{B}X))+g_{2}(\pi_{\ast}(\mathcal{C}Y),(\nabla \pi_{\ast})(X, \varphi V)) &-g_{2}(\pi_{\ast}(\mathcal{C}X),(\nabla\pi_{\ast})(Y,\varphi V)),
\Omega_{7 5}\equiv 8 \pi \overline{{{G}}}\, \rho_{0}^{\mathrm{m a t t e r}}/ [ 3 ( 7 5 \ \mathrm{k m ~ s}^{- 1}\mathrm{M p c}^{- 1})^{2}] = \rho_{0}^{\mathrm{m a t t e r}}/ 1 . 0 5 6 8 \times 1 0^{- 2 9}\mathrm{g ~ c m}^{- 3}\ .
\mathrm{Re}\langle Aa,v\rangle_{X,X^\ast}\le\frac{1}{2}\left|\sum_{k,m\in\mathbb{Z}}i(k-m)c_{k-m}a_m\overline{v_k}\right| &=\frac{1}{2}\left|\langle(Bc)*a,v\rangle_{X,X^\ast}\right| &\le\frac{1}{2}\left\|Bc\right\|_1\|a\|_X^2.
\frac{u_t(t)}{u(t)}= -\frac{1}{2(-t)}-\frac{3}{8}(-t)^{-4}-\frac{111}{32}(-t)^{-7}-\frac{1509}{16}(-t)^{-10}\hphantom{\frac{u_t(t)}{u(t)}=}{}-\frac{2617599}{512}(-t)^{-13}-\frac{944695983}{2048}(-t)^{-16}-\frac{127756233309}{2048}(-t)^{-19}+\cdots.
\int_{\mathbb{G}}\left(\sum_{j=1}^{\ell}|\mathcal{R}_{j}^{\frac{a_{j}}{\nu_{j}}}u(x)|^{p-2}\mathcal{R}_{j}^{\frac{a_{j}}{\nu_{j}}}u(x)\overline{\mathcal{R}_{j}^{\frac{a_{j}}{\nu_{j}}}\psi(x)}-|u(x)|^{q-2}u(x)\overline{\psi(x)}\right)dx=0
\delta_{i_1 i_2}[\delta_{k i_1}D_{i_2 \pi(i_3 \cdots i_{n})}] = \delta_{k 1}D_{1 \pi(i_3 \cdots i_{n}}+ \delta_{k 2}D_{2 \pi(i_3 \cdots i_{n})}+ \delta_{k 3}D_{3 \pi(i_3 \cdots i_{n})}= D_{k \pi(i_3 \cdots i_{n})}
J_{n-1}=J_jJ_{n-j}+2J_{j-1}J_{n-j-1}\sum_{j=1}^n2^jJ_{n-j}=\frac{2}{3}(nJ_{n-1}+(n-1)J_n). \sum_{j=1}^n(-1)^{n+j}J_j =\frac{1}{3}(nJ_{n-1}-(n-2)J_n)
\nu_{\Phi^\alpha_{ij}}(z) \geqslant -\sum_{u=1}^3 \nu^{\infty}_{F_u^{ij}}(z)+\sum_{u=1}^3\nu^0_{F_u^{ij}}(z)+d_2+d_3-1 &\geqslant -\sum_{u=1}^3 \nu^{\infty}_{F_u^{ij}}(z)+\max\{0,-d_1\}+\max\{d_2,0\}+\max\{d_3,0\}-1\geqslant P(z).
T_0= A (y^0\frac{\partial}{\partial y^4}-y^4\frac{\partial}{\partial y^0})- B_i( y^0\frac{\partial}{\partial y^i}+y^i\frac{\partial}{\partial y^0})- C_j (y^4\frac{\partial}{\partial y^j}+y^j\frac{\partial}{\partial y^4})+ D_p \epsilon_{pqr}y^q\frac{\partial}{\partial y^r}
0\!=\!-\bar{E}^T\!v_c^{(l)}\!-\!\rho\bar{E}^T\!\!\left(\!\!\Delta w^{(k+1)}\!-\!\bar{E}\Delta x^{(k+1)}\!+r_c^{(l)}\!+\frac{1}{\rho}\Delta v_c^{(k)}\!\!\right)\!=\!-\bar{E}^T\!v_c^{(l)}\!-\!\bar{E}^T\!\Delta v_c^{(k+1)}.
(2\ell\!+\!1)(w\!-\!2)\Bar{q}(w)\!-\!w\Bar{q}(w)\!-\!2w(w\!-\!2)\Bar{q}'(w) =\binom{-\frac{1}{2}}{\ell}(-2)^{\ell+1}(2\ell+1) &=\frac{1\cdot3\cdots(2\ell\!-\!1)(2\ell\!+\!1)}{\ell!}(-2).
\lim_{n\rightarrow+\infty}\frac 1n \operatorname{\widehat{\mu}_{\min}^{\mathrm{asy}}}(\overline L^{\otimes n}\otimes(A,\varphi_1))=\lim_{n\rightarrow+\infty}\frac 1n\operatorname{\widehat{\mu}_{\min}^{\mathrm{asy}}}(\overline L^{\otimes n}\otimes(A,\varphi_2)).
{\Pr}_r(S_1, K_1) = \frac{|Z(S_1, K_1)|}{|S_1|}\underset{r \in [s_1, K_1]}{\underset{s_1 + Z(S_1, K_1)\in\frac{S_1}{Z(S_1, K_1)}}\sum}\frac{1}{|[s_1,K_1]|}.
e_{\alpha}e_{\beta}^{-1}\sum_{i =1}^{k_{\alpha}}\sum_{j=1}^{k_{\beta}}( q^{j -i + 1}- q^{j-i})+ \sum_{i =1}^{k_{\alpha}}e_{\alpha}e_{\beta}^{-1}q^{-i+1}= e_{\alpha}e_{\beta}^{-1}\sum_{\ell=k_{\beta}- k_{\alpha}+1}^{k_{\beta}}q^{\ell}.
E_{g . s .}= \sum_{\alpha = 1}^{\frac{N}{2}}\epsilon_{\alpha}= N \int_{- \infty}^{\infty}\epsilon ( \lambda ) \rho ( \lambda ) d \lambda = - \frac{J N}{4}\int_{- \infty}^{\infty}d \lambda \frac{1}{\left( \lambda^{2}+ \frac{1}{4}\right) \cosh \pi \lambda}= - J N \ln 2
g(x)= 4 \kappa_0 - 2 \kappa_0^2 + 6 \kappa_0 \lambda_0 + 4 \kappa_0 \lambda_0^2 + 6 \kappa_0 \eta + 4 \kappa_0 \lambda_0 \eta - 3 \eta^2 - 2 \kappa_0 \eta^2 &- 4 \kappa_0 \lambda_0 \eta^2 + 4 \eta^3 - 8 \kappa_0 \eta^3 + 2 \eta^4 + 4 \kappa_0 \eta^4 - 2 \eta^5 - 4 \eta^6 + 4 \eta^7 - \eta^8.
\hspace{- 2 0 m m}c ( n ) \equiv \Gamma ( - n ) e^{-{\frac{n ( n + 1 ) T}{2}}}\prod_{i = 1}^{n}\int d^{2}z_{i}\, \prod_{i < j}| z_{i}- z_{j}|^{2}\prod_{i , j}{\frac{1}{| 1 - z_{i}\bar{z}_{j}e^{- T}|^{{\frac{2}{Q^{2}}}+ 1}}}.
\left(\sum_{j=2}^{\infty}M_j\big(\frac{M-M_1}{1-M_1}\big)^{2-j}\right)\bigg(1-M_1\bigg)^{-1}\geq\frac{M(\sum_{j=1}^{\infty}M_jM^{1-j}-M_1)}{1-M_1}>1.
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( 1- y^m z^n \right)^{\frac{m^4}{n^5}}= \left\{\frac{\left(1-yz\right)^{y^5 - 4 y^4 + 6y^3 +y}}{\left(1-z\right)^{4}}\right\}^{\frac{1}{(1-y)^5}}
\varepsilon = \frac{1}{R^{4}}\left\{\left[ - \frac{\pi^{2}}{9 0}+ \frac{( m R )^{2}}{2 4}- \frac{( m R )^{3}}{1 2 \pi}+ \ldots \right] + \lambda \left[ \frac{1}{1 1 5 2}- \frac{m R}{1 9 2 \pi}+ \ldots \right] \right\}+ \ldots
\hspace{-.5cm}\hat{T}_m(F) = - \frac{\delta F}{\delta \phi^I}\Big(\frac{\delta}{\delta \bar{\phi}_I}- \hbox{$\frac{1}{2}$}m^2 (P_-)_{Jc}^{Ic}\frac{\delta}{\delta \eta_J}\Big) -\hbox{$\frac{1}{2}$}(\hbar/i) \epsilon_{ab}\frac{\delta}{\delta \phi_{I a}^*}\frac{\delta^2 F}{\delta \phi^I \delta \phi^J}\frac{\delta}{\delta \phi_{J b}^*}+ (i/\hbar) m^2 F.
a\Big(1+\frac{\tilde{\gamma}_2-1}{n}\Big)+(1-a)\Big(\frac{1}{q}+\frac{\tilde{\gamma}_3}{n}\Big)=\frac{n}{\gamma_1 s+n}\Big(a\Big(1+\frac{\gamma_2-1}{n}\Big)+(1-a)\Big(\frac{1}{q}+\frac{\gamma_3}{n}\Big)\Big)=\frac{1}{s}.
f(t) =\vert \kappa\vert \sum\limits_{i<j}\left\vert \mathrm{i}\frac{g_{j}^{\prime}(t) e^{\mathrm{i}g_{j}(t)}-g_{i}^{\prime}(t)e^{\mathrm{i}g_{i}(t)}}{e^{\mathrm{i}g_{j}(t)}-e^{\mathrm{i}g_{i}(t)}}\right\vert
F_{i}^{-n}F_{j}=q^{-n(\alpha_{i},\alpha_{j})}F_{j}F_{i}^{-n}+\sum_{j=1}^{\infty}q^{(-n-j)(\alpha_{i},\alpha_{j})}\left[ \begin{array}{c}-n j\end{array}\right]_{(\alpha_{i},\alpha_{i})/2}(ad_{q}F_{i})^{j}(F_{j})F_{i}^{-n-j},
\left(P(B)\right)^{N+1}(k^d\lambda^k) =\sum_{r=0}^d P(\lambda)^{N+r-d}A_{d,N,r}\sum_{s=0}^r Q_{r,s}(\lambda) \left(k^s\lambda^k\right) &=\sum_{s=0}^d \sum_{r=s}^d P(\lambda)^{N+r-d}A_{d,N,r}Q_{r,s}(\lambda)\left(k^s\lambda^k\right).
[ \delta ( \epsilon_{1}) , \delta ( \epsilon_{2}) ] \zeta^{A}= \frac 1 2 \partial_{a}\zeta^{A}\bar{\epsilon}_{2}\gamma^{a}\epsilon_{1}+ \frac 1 4 \Gamma^{A}\bar{\epsilon}_{2}\epsilon_{1}- \frac 1 4 \gamma_{a}\Gamma^{A}\bar{\epsilon}_{2}\gamma^{a}\epsilon_{1}\, .
\Theta^{[2]}: \begin{pmatrix}\rho \kappa_u \kappa_s \end{pmatrix}\mapsto\begin{pmatrix*}[l]A_c Dk_c + Dg_c(K{}{}) \kappa - Dk_c\left(R\right) P_\rho -A_u^{-1}Dg_u(K{}{}) \kappa + A_u^{-1}\kappa_u(R) P_\rho A_s \kappa_s(T) Q_\rho + Dg_s(K{}{}\circ T) \kappa(T) Q_\rho\end{pmatrix*}
g_t\,\mathbb{P}^{\delta,\mu}_0\, \stackrel{\mathcal{L}}{=}\,\min\{t,\,E_\mu\}\,\sup\{s\leq 1:X_s=0\}\,\mathbb{Q}_0^\delta &\stackrel{\mathcal{L}}{=}\,\min\{t,E_\mu\}\, g_1\,\mathbb{P}_0^\delta.
dq_{l-l_0}+t(-r_{l-l_0}-l_0,1) \overset{\textup{(\ref{align:d2})}}{\le}q_{l-l_0}t(-d,0)+t(-r_{l-l_0}-l_0,l) \overset{\textup{(\ref{align:d3})}}{\le}t(-dq_{l-l_0}-r_{l-l_0}-l_0,1)=t(-l,1).
\sum_{n \ge 1}\dim\big(3T^{(0)}(K_{n,n+m})^{ab}\big){t^{n}}= \sum_{k \ge 1}{k^{k+m-1}(k-1) ! t^{k}\over \prod\limits_{j=1}^{k-1}(1+k j t)}, \sum_{n \ge 1}{\rm Hilb}\big(3T^{(0)}(K_{n,m})^{ab},t\big) z^{n-1}=\sum_{k \ge 0}(1+k t)^{m-1}\prod_{j=1}^{k}{{z (1+j t) \over 1+j z}}.
S=\int d^{d-1}x\sqrt{|g|}\ \left[ R +{\textstyle\frac{1}{2}}\left(\partial\varphi \right)^{2}-{\textstyle\frac{1}{4}}e^{-a\varphi}F_{(2)}^{2}+{\textstyle\frac{1}{2}}\left({\cal D}\phi \right)^{2}-{\textstyle\frac{1}{2}}m^{2}e^{a\varphi}\right]\, ,
\left(x^2 \sqrt{l}\Phi_3^{'}\right)^{'}=\frac{l^{3/2}x^2 \Phi_3}{f}\left[\beta_2^2 \left(\Phi_1^2 + \Phi_3^2 + \Phi_4^2-1 \right) + \beta_3^2 \left(\frac{\Phi_3^2}{2}+ \frac{2}{5}\Phi_4^2 - \frac{1}{\sqrt{5}}\Phi_3 \Phi_4 -\frac{7}{30}\right) \right]
\sum_{1\le u\le n}\frac{[x^{n-\Sigma_{\mathrm{k}}- u}]S(x)}{[x^n]S(x)}[x^u]T_1(x)=\mathcal{O}\bigg( z_n^{\Sigma_{\mathrm{k}}}\sum_{u\ge 1}[x^u]T_1(x) z_n^u\bigg)=\mathcal{O}\big( z_n^{\Sigma_{\mathrm{k}}}T_1(z_n)\big)= o\big(z_n^{\Sigma_{\mathrm{k}}}\big).
\tilde{I}_{\rm DE}=\frac{\Omega_{\tilde{d}+1}}{16\pi}[(\tilde{d}+1)r_+^{\tilde{d}}-r_-^{\tilde{d}}]\beta_H -\frac{1}{4}\Sigma-\frac{\pi Q}{\tilde{d}\Omega_{\tilde{d}+1}r_+^{\tilde{d}}}Q\beta_H
\left\langle \frac{1}{2N}\mbox{Tr}\, H_S^2 \right\rangle -\left\langle \frac{1}{2N}\mbox{Tr}\, H_S \right\rangle^{2}=\left\langle \frac{1}{N}\mbox{Tr}\, H_0^2 \right\rangle -\left\langle \frac{1}{N}\mbox{Tr}\, H_0 \right\rangle^{2}+\frac{3}{8}a^4 = k^2 a^2 + \frac{9}{16}a^4
e^{iML\sinh \theta_{j}}\sum_{\mathbf{n}}T_{j}( \vec{\theta})^{\mathbf{m}}_{\mathbf{n}}\psi^{\mathbf{n}}( \vec{\theta}) =-\psi^{\mathbf{m}}( \vec{\theta}) \, ,\quad j=1,\ldots ,N\, ,
T_s(x\{i_{j-m+2}\}_m) =T_sT_{\omega}T_sT_{\omega}^{-1}T_sT_{\omega}(x\{i_{j-m+1}\}_{m-2}) &=T_{\omega}T_sT_{\omega}^{-1}T_sT_{\omega}T_s(x\{i_{j-m+1}\}_{m-2}) &=0
-\widetilde{\Delta}_sv=z_j(1-|w_j|^2)\left(\frac{-\widetilde{\Delta}_sv^\prime}{(1+\overline{w}_jv^\prime)^2}-\frac{-2\overline{w}_j\frac{(1-r_s^2)}{4}\left(\nabla_sv^\prime\cdot\nabla_sv^\prime\right)}{(1+\overline{w}_jv^\prime)^3}\right),
\begin{array}{ccccccccc}\frac{1}{l_M^3}& \mid \frac{R_1}{l_M^3}- \frac{R_1 R_2}{l_M^6}- \frac{R_1 R_2 R_3}{l_M^9}- \frac{R_1 R_2 R_3 R_4}{l_M^9}-\dots- \frac{1}{R_{d+1}}\end{array}
\dot{x}_k(t) = -4 \pi^2 s^2 (t-t_0) e^{-2\pi^2s^2(t-t_0)^2}, \mathcal{F}\{x_k(t)\}= \hat{x}_k(f) = \frac{1}{s \sqrt{2 \pi}}e^{\frac{-f^2}{2s^2}}.
\gamma = \begin{cases}\|r_{\ell-1}^{(N)}\|_\infty, i = 1, \|r_{\ell}^{(i-1)}\|_\infty, i > 1, \end{cases}w^*_{\mathrm{add}}= \begin{cases}5, i = 1, 4, i > 1, \end{cases}
\begin{bmatrix}\Phi(f(\rho) A^*A) \Phi(f(\rho) A^*B) \Phi(f(\rho) g(\rho) A^*) 0 \Phi(f(\rho) B^*A) \Phi(f(\rho) B^*B) \Phi(f(\rho) g(\rho) B^*) 0 \Phi(f(\rho)g(\rho) A) \Phi(f(\rho) g(\rho) B) \Phi\left(f(\rho) g(\rho)^2\right) 0 0 0 0 \Phi\left(f(\rho) g(\rho)^2\right)\end{bmatrix}\geq 0.
M_{\varphi}[\hat{\eta}]:=-(M_{\varphi}[M_1])^{-1}\{M_{\varphi}[g_1]+M_{\varphi}[ M_2 g_2]+M_{\varphi}[M_3 g_3 ]-M_{\varphi}[ M_2(\partial_{\psi}\theta_0)^T]\,M_{\varphi}[ g_2 ]\}.
= \frac{1}{2}\times \frac{3}{2}\times \frac{2}{3}\times \frac{4}{3}\times \frac{3}{4}\times \frac{5}{4}\times \cdot \times \frac{2 0 1 2}{2 0 1 3}\times \frac{2 0 1 4}{2 0 1 3}
\overline{{{\theta}}}\Gamma^{N}\partial_{\nu}\theta = ( \theta_{R}+ \theta_{L})^{T}\Gamma^{0}\Gamma^{N}\partial_{\nu}( \theta_{R}+ \theta_{L}) = \theta_{R}^{T}\Gamma^{0}\Gamma^{N}\partial_{\nu}\theta_{R}+ \theta_{L}^{T}\Gamma^{0}\Gamma^{N}\partial_{\nu}\theta_{L}
G^{(n+1)}(x) = \frac{g^{(n+1)}(x)}{x}- \frac{(n+1)!}{x^{n+2}}\ \left[ (-1)^n\ (g(x)-1) + \sum_{j=1}^n (-1)^{n-j}\ \frac{x^j}{j!}\g^{(j)}(x) \right] = \frac{(n+1)!}{x^{n+2}}\ \left[ (-1)^{n+1}\ (g(x)-1) + \sum_{j=1}^n (-1)^{n+1-j}\ \frac{x^j}{j!}\g^{(j)}(x) + \frac{x^{n+1}}{(n+1)!}\g^{(n+1)}(x) \right]\,.
\sum_{m \geq 1}\frac{A(1, m)}{m}H_{m, p}= \frac{1}{\pi}\int_{-\infty}^\infty k(t) \Bigl( \sum_{m \geq 1}\frac{A(1, m)}{m}U(m^2 p, t) \Bigr) \tanh(\pi t) t \, d t.
H'(0) =-v\sum_{k=s-u}^{s-1}(2m+1-2k)\cosh(x_k')+u\sum_{k=s}^{s+v}(2m+1-2k)\cosh(x_k') &>-v\sum_{k=s-u}^{s-1}(2m+1-2k)\cosh(x_s')+u\sum_{k=s}^{s+v}(2m+1-2k)\cosh(x_s') &\geqslant(-uv(2m+1-2s)+uv(2m+1-2s))\cosh(x_s')=0
G_{ijkl}^{conn}=-\langle c_{i}\overline{c}_{j}c_{k}\overline{c}_{l}\rangle+\langle c_{i}\overline{c}_{j}\rangle\langle c_{k}\overline{c}_{l}\rangle-\langle c_{i}\overline{c}_{l}\rangle\langle c_{k}\overline{c}_{j}\rangle=\frac{\delta^{2}W[J]}{\delta J_{ji}\delta J_{lk}}|_{J=0}
\nabla_{\dot{A}a}\overline{{{W}}}^{2}= 2 W^{\dot{C}\dot{D}}\nabla_{\dot{A}a}W_{\dot{C}\dot{D}}= 4 i W^{\dot{C}\dot{D}}W_{\dot{C}\dot{D}\dot{A}a}- \frac{4}{3}W_{\dot{A}\dot{B}}\Lambda_{a}^{\dot{B}}
\beta=0 \gamma\in \{2m \mid m =2,3,4,7,12\}, \beta=1 \gamma\in \{2m \mid m =8,9,10,11,12,15\}, \beta=2 \gamma\in \{2m \mid m =4,6,7,8,9,10,11,12,13,14,15,51,52,53,54\}.
\begin{array}{c}D(\beta ,R)-i\mu \gamma^0=e^{\mu \beta t}D(\beta ,R)\e^{-\mu \beta t}, K_{bc}(t,\vec{x};t^{\prime},\vec{x^{\prime}})=e^{\mu \beta t}k(t,\vec{x};t^{\prime},\vec{x^{\prime}})\e^{-\mu \beta t\prime}, \end{array}
\Gamma(\omega)= \omega^2\sigma\sigma^{-1}+ \omega^4\sum_{k=1}^\infty \frac{\bigl( \int_{Q_0}\sigma\phi^k\bigr)\otimes\bigl( \int_{Q_0}\sigma\phi^k\bigr)}{\alpha_k - \omega^2}=\sigma\Gamma(\omega)\sigma^{-1},\ \ \ \ \ \ \ \ \ \omega^2\notin\{0\}\cup\{\alpha_k\}_{k=1}^\infty,
( ( 1 2 ) ( 3 ) ( 1^{\prime}) ( 2^{\prime}) ( 3^{\prime}) ) , ( ( 1 ) ( 2 3 ) ( 1^{\prime}) ( 2^{\prime}) ( 3^{\prime}) ) , ( ( 1 3 ) ( 2 ) ( 1^{\prime}) ( 2^{\prime}) ( 3^{\prime}) ) ,
R_L(f)(\xi,y) = R(f\restriction_{[0,\alpha]\times \{y\}})(\xi) =T(f\restriction_{[0,\alpha] \times \{y\}}\otimes \chi_{\{y_0\}})(\xi,y_0)= \sum_{\eta \in \alpha}r(\eta,\xi)\cdot f(\eta,y).
L^{pzm}_{Weyl}= \int d^3{x}\left[ - \partial_i a_{-}\partial_{+}a_i + \frac 1 2 (\partial_{+}a_{-})^2 - \frac 1 4 (\partial_{i}a_j - \partial_{j}a_i)^2 + j^\perp a_\perp + j^{-}a_{-}\right]\;,
Q_{n}[ \overline{{{\psi}}}, \psi , A , h ] \; : = \; P [ \overline{{{\psi}}}, \psi , A , h ] \Big ( S_{M}[ \overline{{{\psi}}}, \psi ] \Big )^{n}\; \longrightarrow \; \tilde{Q_{n}}[ G , \frac{\delta}{\delta a}, A , h ] \; .
\sum_{n=0}^{\infty}b_{m}(2n)x^{n}=\left(\sum_{n=0}^{\infty}\binom{m+2n-1}{n}x^{2n}\right)\left(\sum_{n=0}^{\infty}b_{m}(n)x^{n}\right) & =\sum_{n=0}^{\infty}\left(\sum_{j=0}^{n}\binom{m+2(n-j)-1}{2(n-j)}b_{m}(j)\right)x^{n}.
I_\beta^{(\pm)}(\psi;\,k)=\pm\frac{1}{2\pi}\int_0^a d\eta\,\frac{\psi(i\eta)}{-k \pm i(\eta-\beta)}\pm\frac{1}{2\pi i}\int_0^\infty d\xi\,\frac{\psi(\xi \pm ia)}{\xi-k\pm i(a-\beta)}+\psi(k \pm i\beta).
({\cal H}_V)_{cl}=\sum_{{\vec n},{\vec m}}\sum_{s,s'}\kappa_{\vec m}\left[(a^{s\dagger}_{\vec n}\bar{u}^s_{\vec n}+b^{s}_{-{\vec n}}\bar{v}^s_{-{\vec n}})(a^{s'}_{\vec m}\gamma^0 u^{s'}_{\vec m}-b^{s'\dagger}_{-{\vec m}}\gamma^0 v^{s'}_{-{\vec m}})\right]_{cl}F({\vec n}-{\vec m},X),
\tilde{t}(v_0 \otimes v_0) = v_0 \otimes v_0 \tilde{t}(v_1 \otimes v_1) = v_1 \otimes v_1 \tilde{t}(v_0 \otimes v_1) = q v_1 \otimes v_0 +(1-q^2) v_0 \otimes v_1 \tilde{t}(v_1 \otimes v_0) = q v_0 \otimes v_1.
\real\left(\frac{z\mathtt{g}_{a, \nu}'(z)}{\mathtt{g}_{a, \nu}(z)}\right)\geq \frac{r\mathtt{g}_{a, \nu}'(r)}{\mathtt{g}_{a, \nu}(r)}= a (1-\nu)+ a^{a/2}\left(a\nu-a+1-\sum_{n=1}^\infty \frac{2 r^2}{\mathtt{j}_{\nu, n}^2-r^2}\right)
\left \lbrace \begin{array}{l}c+\left[\begin{array}{cc}I A^t\end{array}\right]\left[\begin{array}{c}u v\end{array}\right] =0 , Ax=b x\geq 0 u_i x_i=0 i=1,\ldots,n u_i \leq 0 v_j(a_j^t x-b_j)=0 j=1,\ldots,m\end{array}\right.
\sum_{n=1}^N\lambda_n\left(\frac{A_n}{\Lambda_n}\right)^{-p}- \left(\frac{p}{p+1}\right)\sum_{n=1}^N\lambda_na_n\left(\frac{A_n}{\Lambda_n}\right)^{-p-1}\geq \frac{\Lambda_n}{p+1}\left(\frac{A_n}{\Lambda_n}\right)^{-p}.
\rho^{2}\frac{\partial^{2}V}{\partial r^{2}}+\rho\sum\limits_{i=1}^{n}\gamma_{i}\frac{\partial^{2}V}{\partial r\partial\alpha_{i}}+\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{n}\gamma_{i}\gamma_{j}\frac{\partial^{2}V}{\partial\alpha_{i}\partial\alpha_{j}}\leq 0
I_{u}(r) = r^{1- n}\int_{b(x)= r}u^2|\nabla b| dx D_{u}(r) = r^{2- n}\int_{b(x)\leq r}|\nabla u|^2 dx \ , F_{u}(r)= r^{3- n}\int_{b(x)= r}\Big|\frac{\partial u}{\partial n}\Big|^2 |\nabla b| dx
\lim_{p\rightarrow\infty,p^{\prime}\rightarrow0^{+}}\tilde{\zeta}_{p,p^{\prime}}(s,c)=\sum_{m\in\mathbb{Z},n\neq0}\frac{1}{\left(m^{2}+c\,\left(n-\frac{1}{2}\right)^{2}\right)^{s}}.