harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
W(f) = \sum_{m, r, n}a_{f}(m, r, n)q_{1}^{m}q_{2}^{n}, W'(f) = \frac{1}{2}\sum_{m, r, n}r a_{f}(m, r, n)q_{1}^{m}q_{2}^{n}, W''(f) = \frac{1}{2}\sum_{m, r, n}r^{2}a_{f}(m, r, n)q_{1}^{m}q_{2}^{n}.
-(\vec{\Theta}_\alpha\cdot\vec{e}\,)\gamma^t_{\alpha\beta}E_sM_{st}^{\parallel}\psi_0=-\frac{\i}{2}\gamma^s_{\alpha\beta}E_s[\vec{\Theta}_\alpha\cdot\vec{e}\,,M_{ut}^{\parallel}M_{ut}^{\parallel}]\psi_0-\i\frac{d^2-d}{8}(\vec{\Theta}_\alpha\cdot\vec{e}\,)\gamma^s_{\alpha\beta}E_s\psi_0\;,
\widehat{\varphi}\circ\widehat{f}=\widehat{\varphi}\left(\widehat{f}_{n}+\widehat{g}_{n}\right)=\widehat{\varphi}\circ\widehat{f}_{n}+\sum_{i\geq1}\frac{\widehat{\varphi}^{(i)}\circ\widehat{f}_{n}}{i!}\widehat{g}_{n}^{i},
\begin{array}{l}\partial X(z) \sigma_{\vartheta}(w,\bar{w}) \sim(z-w)^{-(1-\vartheta)}\tau_{\vartheta}(w,\bar{w}), \partial \bar{X}(z) \sigma_{\vartheta}(w,\bar{w}) \sim(z-w)^{-\vartheta}\tau'_{\vartheta}(w,\bar{w}),\end{array}
i \, \Gamma_{0 , 0}^{\, \infty , \, \alpha_{0}}( 0 ) : = - c_{1}^{\, R}\quad \mathrm{a n d}\quad \Gamma_{l , 2 l}^{\, \infty , \, \alpha_{0}}( 0 ) = 0 \, , \, \Gamma_{l , 2 l - 1}^{\, \infty , \, \alpha_{0}}( 0 ) = 0 \, , \, \partial_{\mu}\, \partial_{\nu}\, \Gamma_{l , 2 l - 1}^{\, \infty , \, \alpha_{0}}( 0 ) = 0 \quad \mathrm{f o r}\quad l > 0 \ ,
\left(\frac{1}{T}+D^\pm\right)f^\pm=\pm\mu^\pm_e\hat{v}_1\left(\partial_x\phi+\frac{1}{T}b\right)\pm\mu^\pm_p\hat{v}_1\partial_x\psi\pm\frac{1}{T}\left(\mu^\pm_e\hat{v}_2+\mu^\pm_p\right)\psi
\begin{array}{l}\beta''-(n-1)(h')^{-1}h'''=\pm \bar{\lambda}, \noalign{\smallskip}(h')^2\bar{\lambda}=\pm h'h''\beta'\mp (n-2)(h'')^2\mp h'''+\bar{c}(n-2), \noalign{\smallskip}Ric_N=\bar{c}(n-2)g_N, \end{array}
(3d)= -\frac{\lambda^2}{2^2.3}i^5 \int{\cal D}k_1{\cal D}k_2{\cal D}k_3{\cal D}k_4 \frac{1}{k_1^2 k_2^2 k_3^2 k_4^2 (p+k_1 +k_2 -k_3 -k_4)^2}\; .
\begin{array}{l}{\rm SINR}_{v,j}^q= 2y_{v,j}^q \sqrt{p_{v,j}^{q}{\left({{R_{\rm{PD}}}{G_{v,j}^{q}}}\right)}^2}-y_{v,j}^q \left({\sum\limits_{j'\ne j}{\sum\limits_{v'\ne v}p_{v',j'}^{q}{\left({{R_{\rm{PD}}}{G_{v',j}^{q}}}\right)}^2}+ N_{\rm VLC}B_{\rm VLC}}\right).\end{array}
\mbox{\mathversion{bold}$v$}_{0ss}=(-(q^{0}_{1})^{2}-(q^{0}_{2})^{2})\mbox{\mathversion{bold}$v$}_{0}+ q^{0}_{1s}\mbox{\mathversion{bold}$e$}^{0}+ q^{0}_{2s}\mbox{\mathversion{bold}$w$}^{0},
\mathbb{E}_{\rho_0 \otimes \Phi}\left[X_t c_{f_t}| \mathcal{Y}_t \right] = \mathbb{E}_{\rho_0 \otimes \Phi}\left[\mathbb{E}_{\rho_0 \otimes \Phi}\left[X_t | \mathcal{Y}_t\right]c_{f_t}\right], \pi_t \left(X_t c_{f_t}\right) = \pi_t \left(\hat{X}_t c_{f_t}\right).
m_{Q}( T , \mu ) \equiv m_{Q}= - \frac{1}{2}g_{Q Q}^{1 / 2}\sum_{Q^{\prime}= U , D}g_{Q^{\prime}Q^{\prime}}^{1 / 2}{\langle ( \bar{Q^{\prime}}Q^{\prime})^{( 1 )}\rangle}_{T},
\begin{aligned}\bar H^6-3\bar H^4S+3\bar H^2S^2-3S^2+2S &\geq \dfrac{(\bar H^2-S)^2}4\bigl[(5-\dfrac{3}{32})S-(4-\dfrac3{8})\bar H^2+\dfrac9{16}\bigl ].\end{aligned}
{\cal Z}= \sum_{\alpha}\mathrm{T r}( q^{H_{\overline{{\alpha V}}}^{L}}{\overline{q}}^{H_{\overline{{\alpha V}}}^{R}}e^{2 \pi i \alpha s}\xi ({\overline{{\alpha V}}},{\bf n}) \eta ({\overline{{\alpha V}}},{\cal N}_{\overline{{\alpha V}}},{\bf n}) ) .
f(x)\cdot (g_1(x),g_2(x)) =(f(x)g_1(x)\mod x^{n_1}-a_1, f(x)g_2(x)\mod x^{n_2}-a_2) =(r(x)g_1(x)\mod x^{n_1}-a_1, r(x)g_2(x)\mod x^{n_2}-a_2) =r(x)\cdot (g_1(x),g_2(x)).
\sum_{y \in V_x}u^\rho_{xy}= \sum_{r = 1}^k \sum_{y \in V_X}\delta_{x\sigma^r(y)}p_r + \sum_{s = 1}^k \sum_{y \in V_X}\delta_{x\tau^s(y)}q_s - 1 = \sum_{r = 1}^k p_r + \sum_{s = 1}^k q_s - 1 = 1
B_{t}=\frac{1}{\sqrt{C(H)}}\left\{\int_{-\infty}^{0}\left[(t-s)^{H-\frac{1}{2}}-(-s)^{H-\frac{1}{2}}\right]dW_{s}+\int_{0}^{t}(t-s)^{H-\frac{1}{2}}dW_{s}\right\},
F(s_{1},s_{2},...,s_{n})=\int_{0}^{\infty}\cdot\cdot\cdot\int_{0}^{\infty}f(t_{1},t_{2},...,t_{n})e^{-s_{n}t_{n}-s_{n-1}t_{n-1}\cdot\cdot\cdot\cdot\cdot\cdot s_{1}t_{1}}dt_{1}\cdot\cdot\cdot dt_{n}
\left( \begin{smallmatrix}c 0 0 c' \end{smallmatrix}\right) A \left( \begin{smallmatrix}\lambda \kappa \end{smallmatrix}\right)= B \left( \begin{smallmatrix}\lambda \kappa \end{smallmatrix}\right)\ \{\rm mod}\left( \begin{smallmatrix}Y Y' \end{smallmatrix}\right)\,.
[ p \cdot \hat{H}, \widetilde{M}]_{\mu \nu}= \left[ p \cdot \hat{H}, \,{( i / 2 )}\sum_{\rho \in \Delta_{+}}g_{| \rho |}\,{| \rho |^{2}}\, y ( \rho \cdot q ) \, \hat{s}_{\rho}\right]_{\mu \nu}, \quad \mu , \nu \in{\cal R}
\psi(W'_0) =_H w_0a'_1a_3t_1a_3t_1 a^{\prime \,-1}_1 w_1^{-1}w_3 &=_H w_0a_1a_2w_1a_1a_2w_1a'_1 a^{\prime \,-1}_1 w_1^{-1}w_3 & =_H w_0a_1a_2w_1a_1a_2 w_3 &=_H \theta(W).
a_{24}^{2413}=a_{24}^{2431}=a_{24}^{2143}=a_{24}^{2134}=a_{24}^{2341}=a_{24}^{2314}= a_{24}^{1243}=a_{24}^{1234}=a_{24}^{1324}=a_{24}^{3241}=a_{24}^{3214}=a_{24}^{3124}
\Bigl\|\sum_{i=1}^\infty \|P^{\tilde{\textsf{H}^*}}_{[n_{i-1}, n_i)}\tilde{y^*}\|_\sim e^*_{n_{i-1}}\Bigr\|_{S^*}\leqslant \|\tilde{y^*}_0\|_\sim + \Bigl\|\sum_{i=1}^\infty a_{n_{i-1}-1}e_{n_{i-1}}\Bigr\|_{S^*}+ \Bigl\|\sum_{i=1}^\infty \bigl\|\sum_{j=n_{i-1}}^{n_i-1}a_j\tilde{y^*_j}\bigr\|_\sim e_{n_{i-1}}\Bigr\|_{S^*},
D\Bigl(\frac{g_1(\lambda)g_2(\lambda)}{H_\lambda}\Bigr) =g_1(\lambda)D\Bigl(\frac{g_2(\lambda)}{H_\lambda}\Bigr)+g_2(\lambda)D\Bigl(\frac{g_1(\lambda)}{H_\lambda}\Bigr) & +\sum_{\lambda^{+}}\frac 1{H_{\lambda^+}}\bigl(g_1(\lambda^{+})-g_1(\lambda)\bigr)\, \bigl(g_2(\lambda^{+})-g_2(\lambda)\bigr)
\lim_{\left\vert k\right\vert \rightarrow\infty}\left\Vert a\right\Vert_{L^{p}\left( L^{p^{\prime}}\right)}=\lim_{\left\vert k\right\vert\rightarrow\infty}\left\Vert a^{\ast}\right\Vert_{L^{p^{\prime}}\left(L^{p}\right)}=0.
\mathsf{L}F=\binom{-\rho_\infty\chi_1\rho_g-\frac{1}{\rho_\infty}f}{-\frac{1}{\rho_\infty}\chi_2\rho_f-\rho_\infty g}\,, \quad\mbox{with}F = \binom{f}{g}\,,
\sum_{n_{i}= 0}^{\infty}\left[ \begin{array}{c}{n_{i}+ \tilde{n}_{i}+ \tilde{V}_{i , r}}\\{n_{i}}\\ \end{array}\right] \tilde{x}^{n_{i}}= \frac{1}{( 1 - \tilde{x})^{\tilde{n}_{i}+ 1 + \tilde{V}_{i}, r}}.
\mathrm{tr}(\log (E_x-\frac{1}{\lambda}X ))= \sum_{j=1}^\infty \frac{(-1)^{j-1}}{j}(-\frac{1}{\lambda})^j \mathrm{tr}(X^j) = \sum_{j=1}^\infty \frac{(-1)^{j-1}}{j}(-\frac{1}{\lambda})^j x^j \Phi(Q) = \Phi(Q) \log(1-\frac{x}{\lambda}).
\sum_{\ell=0}^k \binom{k}{\ell}(-1)^{\ell+k}\biggl[ \frac{1}{n^{k-1}}\biggr] \sum_{m=1}^{k-1}(-1)^{m+1}\frac{1}{m}\bigl( F_{(i+\ell)}- 1 \bigr)^m = \frac{(k-2)!}{r^{k-1}}
< n > = \frac{1}{e^{\hbar \omega / k T}- 1}- \lambda^{2}\frac{\hbar \omega}{k T}\frac{e^{3 \hbar \omega / k T}+ 4 e^{2 \hbar \omega / k T}+ e^{\hbar \omega / k T}}{( e^{\hbar \omega / k T}- 1 )^{4}}.
\psi_{1}= \left( \begin{array}{c}{{| 0 \rangle \! \rangle \langle \! \langle 0 |}}\\{{| 0 \rangle \! \rangle \langle \! \langle 1 |}}\\{{\vdots}}\\{{| 0 \rangle \! \rangle \langle \! \langle k - 1 |}}\\ \end{array}\right) \, , \quad \psi_{2}= \left( \begin{array}{c}{{0}}\\{{0}}\\{{\vdots}}\\{{0}}\\ \end{array}\right) \, , \quad \xi = S_{k}\, ,
\theta_{c A}^{a b}= 2 \gamma^{[ a}\rho_{A}^{b ] c}+ \gamma^{c}\rho_{A}^{a b}; \quad \theta_{c}^{a b \A}= 2 \gamma^{[ a}\rho^{b ] c \vert A}+ \gamma^{c}\rho^{a b \vert A}
\frac{d}{dz}[z^{-b_{1}}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|z)]=-z^{-1-b_{1}}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{1}+1,b_{2},...,b_{q}\end{matrix}|z),m\ge1,
S_{\xi}= \int d^{4}x \sqrt{- g}\left[ -{\frac{1}{2}}| \xi | \hat{\phi}^{2}\tilde{R}- \frac{| \xi |}{\xi}\left({\frac{1}{2}}{\cal D}_{\mu}\hat{\phi}{\cal D}^{\mu}\hat{\phi}+ V ( \hat{\phi}) \right) -{\frac{1}{4}}H_{\mu \nu}H^{\mu \nu}\right] ,
\mathrm{d e t}_{3}[ \; 1 \; - \; K ( \not{\! \! B}) \; ] \; : = \; \exp \; \left( \; - \sum_{n = 3}^{\infty}\; \frac{1}{n}\; \mathrm{T r}[ K ( \not{\! \! B})^{n}] \; \right) \; .
k_j = N_j\frac{1-\delta_{\{-j\}}}{1-\delta_{\{1,\dots,K\}}}+ \left(N_j\frac{1-\delta_{\{-j\}}}{1-\delta_{\{1,\dots,K\}}}\right)^{3/4}, \mbox{and\quad}n_1 = \max_{j}\frac{k_j+k_j^{3/4}}{p_j}.
0= -\lambda h(x_0)=Mh(x_0)=\sum_{y\sim x_0}M_{x_0y}h(y) = \sum_{\substack{y\sim x_0 y\in D_j\setminus D_i}}M_{x_0y}h(y)=\left(1-\frac{a_j}{a_i}\right)\sum_{\substack{y\sim x_0 y\in D_j\setminus D_i}}M_{x_0y}f(y).
\Psi(G^{\{v \}})= \sqrt{-1}\sqrt{2|k+h^\vee|}:a \Phi_{[e,v]}:+\sum_{\alpha \in S_{1/2}}: [v,u_{\alpha}]^\natural\Phi^{\alpha}: -2(k+1) T \Phi_{[e,v]}& +\frac{1}{3}\sum_{\alpha ,\beta \in S_{1/2}}: \Phi^{\alpha}\Phi^{\beta}\Phi_{[u_{\beta},[u_{\alpha},v]]}: .
\begin{array}{r c l}N=\sum_{a=1}^3{\alpha^a}^\dagger\alpha^a\ \ \ , \ \ \N^\dagger=N\ \ \ , B^\dagger=\sum_{a=1}^3{\alpha^a}^\dagger{\alpha^a}^\dagger\ \ \ , \ \ \B=\sum_{a=1}^3\alpha^a\alpha^a\ \ \ ,\end{array}
|\nabla\chi_m^i(x)|\leq \frac{2}{\sqrt{\mu_m^i\mu_m^{i-1}}}{\rm and}|\nabla^2\chi_m^i(x)|\leq \frac{4}{\mu_m^i\mu_m^{i-1}},\quad\mbox{for \any}\i=1,\cdots,k.
0 =s_k^2\left(\lambda\mathcal{L}_0(W_k)+\frac{1}{2}\mathcal{S}_{0,2}(U_k,U_k)\right) &+u_k\mathcal{L}_0(W_k)+\lambda s_k^3\mathcal{S}_{0,2}(U_k,W_k)+\frac{s_k^3}{6}\mathcal{S}_{0,3}(U_k,U_k,U_k)+o(s_k^3) &=\lambda s_k^2\mathcal{L}_0(E_k)+u_k\mathcal{L}_0(W_k) &+\frac{s_k^3}{6}(6\lambda \mathcal{S}_{0,2}(U_k,W_k)+\mathcal{S}_{0,3}(U_k,U_k,U_k)) +o(s_k^3).
\sum_{j=1}^{2N-1}\eta_{j,N}\le \sum_{h=1}^{\infty}\frac{d^{2h}}{h! (h-1)!}\left\{\frac 1N \sum_{j=1}^{2N-1}(1- x_j^2)^{h-1}\left(1+x_j \right) \right\}\,.
\min_{y,u}\mathcal{J}(y,u):= \frac{1}{2}\int_0^T \int_{\Omega}\left(y(\boldsymbol{x}, t) - y_d(\boldsymbol{x},t) \right)^2 d\boldsymbol{x}\,dt+ \frac{\lambda}{2}\int_0^T \int_{\Omega}\left( u(\boldsymbol{x},t) \right)^2 d\boldsymbol{x}\,dt
I_2 = \int_{B_{|x|/2}(x)}\frac{v(y)^q}{|x-y|^{n-\alpha}|y|^{\sigma_1}}\,dy \leq{}\frac{C}{(1 + |x|^2)^{q\theta_2 + \sigma_{1}/2}}\int_{B_{|x|/2}(x)}\frac{1}{|x-y|^{n-\alpha}}\,dy \leq{}\frac{C}{(1 + |x|^2)^{q\theta_2 +\sigma_{1}/2 -\alpha/2}};
\left[ T_{(A,i)},T_{(B,j)}\right] =\left( K_{ij}{}^{k}-K_{ij}{}^{k+n}\right) C_{AB}{}^{C}T_{(C,k)},\quad A,B,C =1,\ldots,\dim\mathfrak{g},i,j,k=0,\cdot\cdot\cdot,n-1.
\pi(s_{\mu,\nu})\xi_m=\rho(a_{\mu,\nu})\xi_m+\sum_{j=1}^{r_{\mu,\nu}}t(x^{(\mu,\nu)}_{j,1}) t(x^{(\mu,\nu)}_{j,2})\cdots t(x^{(\mu,\nu)}_{j,N_j})\xi_m=\tau(s_{\mu,\nu})\xi_m
[ \delta_{\chi}^{( 1 )}, \delta_{h}^{( 1 )}] \, m_{A}^{\phantom{A}B}\, = \, i \left( \Psi_{A}( \chi^{\top}h^{\top}C )^{B}- ( h \chi )_{A}( \Psi^{\top}C )^{B}\right) \; ,
( - m_{\pi}^{2}+ \partial_{\bot}^{2}) \, \omega_{\pi}= \frac{\lambda}{2 L}\int_{- L}^{L}d x^{-}( \varphi_{\pi}^{3}+ \varphi_{\pi}\varphi_{\sigma}^{2}+ 2 v \varphi_{\pi}\varphi_{\sigma}) ,
{\cal E}= \frac{1}{2}\partial^{a}\left( \frac{1}{e}e_{\mu}^{m}( \partial_{m}( e e_{a}^{\mu}) ) \right) + \frac{1}{4}\left( \frac{1}{e}e_{\mu}^{m}( \partial_{m}( e e_{a}^{\mu}) ) \right)^{2}
v_l = J_{-N - 1}( \vert \delta_{- N - 1}\rangle \langle \delta_{-N}\vert + \vert \delta_{- N}\rangle \langle \delta_{-N-1}\vert ), v_r = J_{N}( \vert \delta_{N +1}\rangle \langle \delta_{N}\vert + \vert \delta_{N}\rangle \langle \delta_{N+1}\vert ).
L_{\phi}= - \frac{3}{4}c \int_{- R}^{R}e^{- 2 \sigma ( y )}\sqrt{- \gamma}\partial_{\mu}\phi \partial^{\mu}\phi d y = - \frac{3}{2}\frac{k R^{2}}{e^{2 k R}- 1}\eta^{\mu \nu}\partial_{\mu}\phi \partial^{\nu}\phi .
f_{\infty}^{(\alpha)}(0) = \Omega_{\alpha}^2\,\frac{2^{\alpha}}{\pi}\int_0^1 \xi^{\frac{\alpha-1}{2}}(1-\xi)^{-\frac{1}{2}}{\rm d}\xi = \Omega_{\alpha}^2\, \frac{2^{\alpha}}{\pi}\frac{(\frac{\alpha-1}{2})!(-\frac{1}{2})!}{\frac{\alpha}{2}!}=\Omega_{\alpha}^2\,\frac{\alpha !}{\frac{\alpha}{2}!\frac{\alpha}{2}!}>0
\Omega^{(p)}\big(\sigma(\underline{e}^{\alpha t^a}\wedge \underline{e}^{\beta t^b}\wedge \underline{e}^{\gamma t^c})\big)-\Omega^{(p)}(\underline{e}^{\alpha t^a}\wedge \underline{e}^{\beta t^b}\wedge \underline{e}^{\gamma t^c})= d(\sum_{i+j+k=q \atop{0 \leq i,j,k}}\frac{x^q}{q}S(a,b,c;i,j,k)\alpha^{(i)}\beta^{(j)}\gamma^{(k)}),
CC_8 =\Pi(8)\left((-1)^1\frac{(-1)^2}{L_2}+(-1)^4\frac{(-1)^4}{L_1^4}\right) &=(T^3-T)^2\left(-\frac{1}{(T^3-T)(T^9-T)}+\frac{1}{(T^3-T)^4}\right) &=\frac{1}{(T^3-T)(T^9-T)}\,.
8 \pi i \frac{d}{d \theta}F ( \theta ) = \int_{C}\frac{d z}{\sinh^{2}{\frac{1}{2}( z - \theta )}}\ln F ( z ) = \int_{\infty}^{\infty}\frac{d z}{\sinh^{2}{\frac{1}{2}( z - \theta )}}\ln{\frac{F ( z )}{F ( z + 2 \pi i )}}\ .
1 \mapsto \begin{pmatrix}1 0 0 1\end{pmatrix}, c_2 \mapsto \begin{pmatrix}0 -\sqrt{-1}\sqrt{-1}0\end{pmatrix}, c_1 \mapsto \begin{pmatrix}0 1 1 0\end{pmatrix}, c_1c_2 \mapsto \begin{pmatrix}\sqrt{-1}0 0 -\sqrt{-1}\end{pmatrix}.
\left. \begin{array}{l l l}{a_{1}}&{=}&{( 0 . 7 7 , 1 . 9 5 ) , \quad a_{2}= ( 3 . 2 5 , 1 . 9 5 ) ;}\\{b_{1}}&{=}&{( 1 . 3 2 , 1 . 9 5 ) , \quad b_{2}= ( 2 . 7 0 , 1 . 9 5 ) .}\\ \end{array}\right.
\int_{0}^a (e^{iu^2})'g(u\epsilon)u^{-1}du=e^{ia^2}\epsilon \left(\frac{g(a\epsilon)}{a\epsilon}\right)-\epsilon g'(0)-\epsilon\int_{0}^a e^{iu^2}\left(\frac{g(u\epsilon)}{\epsilon u}\right)'du\,.
\Phi_j(f_0,f_1,f_2,\ldots,f_{N-1}) =\exp_K (-t_1f_0)\ldots \exp_K(-(t_j-t_{j-1})f_{j-1})f_j\exp_K(t_j-t_{j-1})f_{j-1})\ldots\exp_K t_1f_0.
e^{-iN\theta}+ \cdots + e^{-i\theta}= e^{-i\frac{(N+1)}{2}\theta}\left(e^{-i\frac{N-1}{2}\theta}+ \cdots + e^{i\frac{N-1}{2}\theta}\right) &=e^{-i\frac{(N+1)}{2}\theta}\left(\frac{\sin \frac{N\theta}{2}}{\sin \frac{\theta}{2}}\right)\;
\frac{1}{|(q+\xi_1)^2 - E - i \eta |}& = \frac{1}{| (q+\xi_2)^2 - E - i 2^{-1/2}\eta | + |(q+\xi_1)^2 - E - i \eta |- | (q+\xi_2)^2 - E - i 2^{-1/2}\eta |}& \leq \frac{1}{|(q+\xi_2)^2 - E - i 2^{-1/2}\eta |}.
\delta_{\lambda^{\dagger}}\widehat \Lambda^{\dagger}( \lambda^{\dagger}, A^{\dagger}) - \delta_{\eta^{\dagger}}\widehat \Lambda^{\dagger}( \lambda^{\dagger}, A^{\dagger}) - i [ \widehat \Lambda^{\dagger}( \lambda^{\dagger}, A^{\dagger}) \stackrel{*}{,}\widehat \Lambda^{\dagger}( \eta^{\dagger}, A^{\dagger}) ] = \widehat \Lambda^{\dagger}( - i [ \lambda^{\dagger}, \eta^{\dagger}] , A^{\dagger}) .
0\leq u\leq \theta_1<1 \quad\hbox{on}\partial D_2\cap\{x_1=k/2\},\quad\hbox{(by Step \ref{step:C_1_base}applied with $r=k$)}0\leq u\leq 1 \quad\hbox{on}D_2,\quad\hbox{(by \eqref{eq:A_u_bounds})}Au = 0 \quad\hbox{on}D_2,\quad\hbox{(by \eqref{eq:A_u})}.
I = \frac{1}{2}\frac{\mu}{g^{2}}t r \int d^{3}x [ - \mu ( v_{m}- U^{- 1}\partial_{m}U ) ( v^{m}- U^{- 1}\partial^{m}U ) + \epsilon^{m n p}v_{m}( \partial_{n}v_{p}+ \frac{2}{3}v_{n}v_{p}) ] ,
F^{\mu \lambda_{2}\dots \lambda_{p + 2}}F_{\nu \lambda_{2}\dots \lambda_{p + 2}}= \bar{\delta}_{\nu}^{\mu}( p + 1 ) ! \F_{t y_{1}\dots y_{p}r}F^{t y_{1}\dots y_{p}r}= - \bar{\delta}_{\nu}^{\mu}( p + 1 ) ! \e^{- 2 a \phi}{\frac{Q^{2}}{( G r )^{2 ( d - 1 )}}},
\chi_1 = \chi_{\left\{p>\frac{3n-2}{2n-1}\right\}}= \begin{cases}1, \mbox{if}p> \frac{3n-2}{2n-1}, 0, \mbox{if}p \le \frac{3n-2}{2n-1}, \end{cases}\chi_2 = 1 - \chi_1 = \chi_{\left\{p\le \frac{3n-2}{2n-1}\right\}}.
Z_{i}= \frac{2 \pi M_{1 1}^{3}V_{7}}{V_{3 , i}^{2}}\ \ ( i \leq N_{3}) \ , \quad \frac{1}{2 \kappa_{4}^{2}}= Z_{N_{3}+ 1}={2 \pi M_{1 1}^{9}V_{7}}\ .
e V^{\mathrm{Q E D_{3}}}( \vec{r}) = \frac{e^{2}}{2 \pi}\left( 1 + \frac{e^{2}}{8 \pi m}\right) K_{0}\left( \frac{e^{2}}{8 \pi}r \right) + \frac{8}{m r^{2}}\left[ 1 - \frac{e^{2}}{8 \pi}r K_{1}\left( \frac{e^{2}}{8 \pi}r \right) \right] L .
\langle S_{I}[ \phi_{<}, \phi_{>}]_{2}\rangle_{0}= \int a ( t ) d t \int d^{3}\vec{k}( - \frac{1}{2}) \phi_{<}( \vec{k}, t ) \ \delta \tilde{m}^{2}( t , s )_{1}p h i_{<}( \vec{k}, t )
0 = \big( z_1 f(\alpha) \big) \varphi = \sum\nolimits_{j=0}^d f_j \, (z_1 \alpha^j) \varphi = \sum\nolimits_{j=0}^d f_j \, (z_1 \lambda_j) = z_1 \big( \sum\nolimits_{j=0}^d f_j \lambda_j \big) = z_1 f(\lambda_1)
\Delta f -b_1 f_x -b_2 f_y=\dfrac{2F^{-2}\rho(2M+\Delta \psi)\psi_y-2F^{-4}\rho^2}{|\nabla \psi|^2}- (2M+\Delta \psi)M+\dfrac{1}{F^2}\rho_p\psi_y,
n'=|T|+ |(U\cup V)\setminus (U_4\cup V_4)| \leq 30h^2\varepsilon n+160 h^3\varepsilon n^{\frac{3}{2}}+3\cdot(6h)^{s+3}\varepsilon^{\frac{s+1}{2}}n^{\frac{s+2}{2}}.
\int \frac{d^4q}{(2 \pi)^4}\:\hat{V}(q) \:e^{-i \:\frac{q}{2}\:(x+y)}\:\int_{-\infty}^\infty d\tau\;\epsilon(q^2) \:\delta^{(\alpha)}(z)\:H^{(\beta)}_z(x,y) \:{\mathcal{P}}(\tau)\;e^{-iq \tau (y-x)}
Q^{N,aV}_{t}\left(\max_{1\le k\le d}d(\bar M^N_k)\ge N\delta\right) &\le e^{CN\log N}\biggl[ \int_{\max_{1\le k\le d}d(\bar M^N_k)\ge N\delta}e^{-\frac{\beta}{5}\sum_{k=1}^d d(\bar M^N_k)^2-N^2\sum_k\int f_k(x)dL^{N}_k(x)}\prod d\lambda_i^k +e^{-cN^2}\biggr] &\le e^{CN\log N}(e^{-\frac{\beta}{6}N^2 \delta^2}+e^{-cN^2})
m_{2,t}(x, m_{1 ,t}(y,z)) + m_{1,t}(x, m_{2 ,t}(y,z)) = m_{2,t}(m_{1,t}(x, y),z) + m_{1,t}(m_{2,t}(x, y),z)- m_{2,t}(m_{1,t}(x, z),y)- m_{1,t}(m_{2,t}(x, z),y)
\begin{array}{c}L=\left[ 1+\frac 12h\right] i\bar \psi \gamma^a\left[ \eta_a^\mu +h_a^\mu\right] \left[ \partial_\mu +\Omega_\mu \right] \psi ++\left[ 1+\frac12h\right] m\bar \psi \psi\end{array}
\widetilde{m}^{\varepsilon}_{\mathbb{II},\mathbb{I}}(\bar{\xi}_{1},\bar{\xi}_{2}):=\sum_{Q'\in\mathbb{Q}'_{\mathbb{II}},Q''\in\mathbb{Q}''_{\mathbb{I}}}\widetilde{m}^{\varepsilon}(\bar{\xi}_{1},\bar{\xi}_{2})\phi_{Q'}(\xi^{1}_{1},\xi^{1}_{2})\phi_{Q''}(\xi^{2}_{1},\xi^{2}_{2}).
f_0^{\lambda_i(x,\lambda_j(y,z))}= -f_i^{\lambda_0(x,\lambda_j(y,z))}-\sum_{m+n=i,~m,n>0}f_m^{\lambda_n(x,\lambda_j(y,z))}+\sum_{m+n= i,~m,n>0}[f_m^x,f_n^{\lambda_j(y,z)}] +[f_0^x,f_i^{\lambda_j(y,z)}]+[f_i^x,f_0^{\lambda_j(y,z)}].
[x, y, z] = -(-1)^{\bar x \bar y}[y, x, z], [x, y, z] = -(-1)^{\bar y \bar z}[x, z, y], [x, y, [z, u, v]] = [[x, y, z], u, v] + (-1)^{\bar z(\bar x+\bar y)}[z, [x, y, u], v] + (-1)^{(\bar z+\bar u)(\bar x+\bar y)}[z, u, [x, y, v]],
\varphi^{(p,q)}(\zeta)=\frac{(-1)^{q+\nu+1}}{q}(1-\zeta\bar{\zeta})\zeta^{m+\nu-q+1}\bar{\zeta}^{m+\nu-p+1}\sum_{j=0}^{\nu}(-1)^j\frac{(j+\nu+m+1)!}{j!(j+m)!(\nu-j)!}(\zeta\bar{\zeta})^j,
\begin{aligned}\mathcal{F}(t,h) \equiv \int h^2 \varrho\, m_1 +\alpha_1 t \int \left|\nabla_\xi h\right|^2 \varrho\, m_0 + \alpha_2 t^2 \int \nabla_x h\cdot\nabla_{\xi}h\,\varrho \,m_0 &+ \alpha_3 t^3 \int \left|\nabla_x h\right|^2 \varrho\, m_0,\end{aligned}
d_K(p,q) = \left( \left[ \sum_{j=1}^n (x_j-x_j')^2 + (y_j-y_j')^2 \right]^2 + \left[ t-t' + 2 \sum_{j=1}^n (x_j'y_j - x_j y_j') \right]^2 \right)^{1/4}&\approx \left[ \sum_{j=1}^n (x_j-x_j')^2 + (y_j-y_j')^2 \right]^{1/2}+ \left| t-t' + 2 \sum_{j=1}^n (x_j'y_j - x_j y_j') \right|^{1/2}
\begin{array}{rl}\mathcal{L}u_{11}= -F^{ij,kl}D_1w_{ij}D_1w_{kl}+F^{ij}(D_{11}A_{ij}+2(D_{1p_k}A_{ij})u_{k1}) +F^{ij}D^2_{p_kp_l}A_{ij}u_{k1}u_{l1}+D_{11}\tilde{B}+2(D_{1p_k}\tilde{B})u_{k1}+(D^2_{p_kp_l}\tilde{B})u_{k1}u_{l1}, \end{array}
U(x,t) = \frac{1}{\sqrt{4\,\pi\,K_{\alpha}\,t^{\alpha}}}\,H^{2,0}_{1,2}\left[ \frac{x^2}{4\,K_{\alpha}\,t^{\alpha}}\left|\begin{array}{ll}\displaystyle \left( 1-\frac{\alpha}{2},\alpha\right) [10pt](0,1) \left( \frac{1}{2},1\right) \end{array}\right.\right],
\delta W = \int_{M^{1 0}}\hat{I}_{1 0}^{1}= - \frac{1}{9 6 ( 2 \pi )^{5}}\int_{M^{1 0}}\hat{I}_{2}^{1}\wedge ( \frac{1}{4}( \hat{I}_{4})^{2}- X_{8})
\widehat{D}_{0}\pi_{a}+ i \varsigma{\frac{\partial E_{a}}{\partial \phi_{i}}}\tau_{i}- i \overline{{{p}}}^{\prime}{\frac{\partial \overline{{{J}}}^{a}}{\partial \overline{{{\phi}}}_{i}}}\overline{{{\tau}}}_{i}+ i E_{a}\beta - i \overline{{{J}}}^{a}\overline{{{\gamma}}}+{\cal O}( 1 / e^{2}) = 0
\mathbb{E}_0\left[ \left(U^{(12)}_h\right)^4\right] ={n \choose k}^{-4}\sum_{\mathbf{i}^1, \mathbf{i}^2, \mathbf{i}^3, \mathbf{i}^4 \in\mathcal{P}(n, k)}\mathbb{E}_0 \left[ \prod_{\omega = 1}^4h\left(\mathbf{R}^{(1 2)}_{\mathbf{i}^\omega, 1}, \dots,\mathbf{R}^{(1 2)}_{\mathbf{i}^\omega, k}\right) \right] .
\delta_i = 2\kappa - 1 \simeq \frac{\kappa - \frac{1}{2}}{\frac{1}{|\Sigma_{i-1}|}+ \frac{1}{2}}= \frac{|\Sigma_i| - \frac{1}{2}|\Sigma_{i-1}|}{1 + \frac{1}{2}|\Sigma_{i-1}|}\geq \frac{\mu_i - \mu_{i-1}}{\mu_{i-1}+ t_{i-1}}\,,
\langle F_{k,m},F_{k,m}\rangle = \frac{1}{\pi}\sum_{i=1}^3 \int_{\mathcal{F}}F_{k,m}|_{\alpha_i^{-1}}(z) \overline{F_{k,m}|_{\alpha_i^{-1}}(z)}y^{k-2}dx dy = \frac{1}{\pi}\left( I_1 + I_2 + I_3 \right).
E_{+}( X^{0}) = 2 N \int d \tilde{\sigma}\frac{e^{3 H ( X^{0}- \tilde{\sigma})}+ H^{2}L^{2}e^{- H ( X^{0}- \tilde{\sigma})}}{\sqrt{e^{4 H ( X^{0}- \tilde{\sigma})}+ e^{2 H ( X^{0}- \tilde{\sigma})}+ H^{2}L^{2}}}.
\sum_{k=2}^{\infty}\frac{\zeta\left( k\right)}{2^{k}}=\log2, \sum_{k=2}^{\infty}\frac{\zeta\left( k\right) k}{2^{k}}=\log2+\frac{3}{4}\zeta\left( 2\right) , \sum_{k=2}^{\infty}\frac{\zeta\left( k\right) k^{2}}{2^{k}}=\log2+\frac{9}{4}\zeta\left( 2\right) +\frac{14}{8}\zeta\left( 3\right) .
\sigma(E)=\frac{\pi e^{2}\hbar}{2mc\epsilon_{0}E}\sum_{n}^{N_{fs}}\frac{1}{N_{p}}\sum_{i}^{N_{p}}\Delta E_{0n}(x_{i})f_{0n}(x_{i})g(E-\Delta E_{0n}(x_{i}),\delta),
N_{M_1\cap T}\left(m\right) = \frac{1}{2}(2\pi)^{5/2}\Gamma(5/2)^{-1}\cdot t^{3/2}\cdot |2\cdot 3^4|^{-1/2}L(2,Dt_2^2)\zeta(4)^{-1}&\times\prod_{p = 2, 3}\frac{1 - \chi_D(p)p^{-2}}{1 - p^{-4}}\alpha_p\left(t,S_1\right),
L_n^{p,q}(f;x)=\frac{1}{\ell_n^{p,q}(x)}\sum_{k=0}^n f \left( \frac{p^{n-k+1}[k]_{p,q}}{[n-k+1]_{p,q}q^k}\right) p^{\frac{(n-k)(n-k-1)}{2}}q^{\frac{k(k-1)}{2}}\left[\begin{array}{c}n k\end{array}\right]_{p,q}x^k
\left[ \begin{array}{cccc}I I \dots I A_1 A_2 \dots A_{d_i+1}A_1^2 A_2^2 \dots A_{d_i+1}^2 \vdots \vdots \vdots \vdots A_1^{s_i-1}A_2^{s_i-1}\dots A_{d_i+1}^{s_i-1}\end{array}\right]\left[ \begin{array}{c}W_1C_1 W_2C_2 \vdots W_{d_i+1}C_{d_i+1}\end{array}\right]=0,
2 A ({\bf r^{\prime \prime}},{\bf r^{\prime}},{\bf r}) = ({\bf r}^{\prime}-{\bf r}) \times ({\bf r}-{\bf r}^{\prime \prime}) ={\bf r}^{\prime \prime}\times{\bf r}^{\prime}+{\bf r}^{\prime}\times{\bf r}+{\bf r}\times{\bf r}^{\prime \prime}
0\longrightarrow{\cal O}_{{\cal F}_I}\longrightarrow\oplus_{j\in I}{\cal O}_{{\cal G}_{(\{j\},\emptyset)}}\longrightarrow\oplus_{\stackrel{I'\subset I}{|I'|=2}}{\cal O}_{{\cal G}_{(I',\emptyset)}}\longrightarrow\ldots\longrightarrow{\cal O}_{{\cal G}_{(I,\emptyset)}}\longrightarrow 0
\lambda_\ell(N^j_k) =\prod\limits_{s=1}^j \left (1 - \prod^{k-s+j}_{i=\ell}\delta_i \right ) &= \prod\limits_{s=1}^j \left (1 - \prod^{k-s+j}_{i=\ell}\frac{(q^i-1)(q^{n-i+1}-1)}{(q^{i+1}-1)(q^{n-i}-1)}\right ).
d_{N}(\mathcal{M}) \geq (2N)^{-1/2}\cdot d_N\bigl( \{x \in\mathbb{R}^{2N}\ |\ \|x\|_1 \leq 1\}\bigr) &= (2N)^{-1/2}\cdot\left(\frac{2N - N}{2N}\right)^{1/2}= \frac{1}{2}N^{-1/2}.
\Phi^G_{1}= e+\zeta \pi_{0}^{-1}\;,\;\;\Phi^G_2 = \chi\;,\;\;\Phi^G_{3\mu}= b_\mu\;,\;\;\Phi^G_4 = x_{0}-\zeta \tau\;,\;\;\Phi^G_5 = \psi^0\;,\;\;
\widehat{\deg}(\overline V)=-\int_{\mathbb R}t\,\mathrm{d}(\dim_K(\mathcal F^t(\overline V))), \widehat{\deg}_+(\overline V)=-\int_0^{+\infty}t\,\mathrm{d}(\dim_K(\mathcal F^t(\overline V)))=\int_{0}^{+\infty}\dim_K(\mathcal F^t(\overline V))\,\mathrm{d}t.
[[x_1,\cdots, x_n],\alpha(y_1), \cdots, \alpha(y_{n-1})] =\sum_{i=1}^n [\alpha(x_1), \cdots, \alpha(x_{i-1}), [x_i, y_1, \cdots, y_{n-1}], \alpha(x_{i+1}), \cdots \alpha(x_n)],
p(x)+au_\infty^\perp\cdot x-p_\infty =\Big[\int_{\partial\Omega}\left\{T(\widetilde u,\widetilde p)+a\,\big(\widetilde u\otimes y^\perp-y^\perp\otimes\widetilde u\big)\right\}\nu\,d\sigma_y &+\int_\Omega f\,dy-\beta a x_0^\perp\Big]\cdot\frac{x}{2\pi |x|^2}+O(|x|^{-2})
\mathbf{D}_k=\begin{bmatrix}\mathbf{h}(1,k) \mathbf{h}(2,k) \cdots \mathbf{h}(R,k) \mathbf{h}(2,k) \mathbf{h}(3,k) \cdots \mathbf{h}(R+1,k) \vdots \vdots \ddots \vdots \mathbf{h}(S,k) \mathbf{h}(S+1,k) \cdots \mathbf{h}(Q,k) \end{bmatrix}
F(u) := (a(\cdot,\cdot,u,u_x) - a_0) u_{xx}+ f(\cdot,\cdot,u,u_x)\,, H(u) (\cdot,\pm\ell) := \pm h_0(\cdot,\pm\ell) u_x(\cdot,\pm\ell) \mp h (\cdot,\pm\ell,u(\cdot,\pm\ell),u_x(\cdot,\pm\ell)),
d\sigma_\Sigma:=(\overline{p}\omega_2-\overline{q}\omega_1)\wedge \omega,~~~~d\overline{\sigma_\Sigma}:=\frac{X_3u}{l}\omega_1\wedge \omega_2-\frac{(X_3u)^2}{2l^2}(\overline{p}\omega_2-\overline{q}\omega_1)\wedge \omega.
\int_0^t e^{-b(t-\tau)}|z(\tau)|^q \, d\tau= \int_0^{t_0}e^{-b(t-\tau)}|z(\tau)|^q \, d\tau +\int_{t_0}^t e^{-2^{-1}b(t-\tau)}\left(e^{-2^{-1}q^{-1}b(t-\tau)}|z(\tau)|\right)^q \, d\tau \leq C e^{-bt}\max_{\tau\in [0,\infty)}|z(\tau)|^q+ C |z(t)|^q \leq C |z(t)|^q.
V = \frac{1}{\mu c^{2}}C_{1}\frac{\alpha}{q}\hat{h}^{( 0 )}+ \frac{C_{2}- C_{1}}{\mu c^{2}}\frac{\alpha^{2}}{q^{2}}+ \frac{\hbar^{2}}{2}\frac{\alpha}{\mu^{2}c^{2}}C_{1}\left( \frac{{\bf q}}{q^{3}}\frac{\partial}{\partial{\bf q}}+ 4 \pi \lambda \delta ({\bf q}) \right) \, ,
B^{a ( 0 , p_{a})}, \H_{\mu_{1}\ldots \mu_{p + 1}}^{a ( p_{a}+ 1 , 0 )}\mathrm{a n d}\partial_{\alpha_{1}\ldots \alpha_{k}}H_{\mu_{1}\ldots \mu_{p + 1}}^{a ( p_{a}+ 1 , 0 )}.
{\mathcal N}= \left\{\left( \sum_{1 \le j \le n}[T_j,y_j],(y_j)_{1 \le j \le n}\right) \in{\mathcal C}_1 \times ({\mathcal I}^*)^n \mid \sum_{1 \le j \le n}[T_j,y_j] \in{\mathcal C}_1\right\},