harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{x_u(z)}= \frac{1}{N}\sum\limits_{n = 1}^N{\frac{{q{{\left|{{\lambda_{u,n}}}\right|}^2}}}{{\frac{1}{N}\sum\nolimits_{i = 1}^U{\frac{q}{{1 +{x_i(z)}}}{{\left|{{\lambda_{i,n}}}\right|}^2}}-{z}}}}.
\begin{array}{c}C_2=\frac 12L_{AB}L^{AB}=E_o(E_o+3)+s(s+1) C_4=W_AW^A=-s(s+1)(E_o-1)(E_o-2) W_A=\frac 18\epsilon_{ABCDE}L^{BC}L^{DE}\end{array}
C_{m_1,\ldots,m_n}^{k_1,\ldots,k_n}(t):=\sum_{l_1=0}^{k_1}\cdots \sum_{l_n=0}^{k_n}\prod_{j=1}^n\binom{k_j}{l_j}(-1)^{l_j+k_j+1}(l_1+k_1t+\cdots + l_j+k_jt)^{m_j-1}.
\mathsf{M}_{x_0}^{\sigma^s}(\Gamma_X\times \Gamma_A)=\int_{\Gamma_X}\sigma^s(\Gamma_A|x)\mathsf{M}_{x_0}^{\sigma^s}(dx\times \textbf{A}),~\Gamma_X\in{\cal B}(\textbf{X}),~\Gamma_A\in{\cal B}(\textbf{A})
{{\cal L}^{1}}^{\mathrm{e f f}}( H , \mu , T ) = \frac{e H}{2 \pi^{2}}\sum_{n = 0}^{\infty}b_{n}\int_{- \infty}^{\infty}d \! p_{z}{}~ \frac{p_{z}^{2}}{\varepsilon_{n}( p_{z})}~ ( f_{+}( T ) + f_{-}( T ) ) .
P_s\big[e^{-\frac12a|\cdot|^2 + \langle b, \cdot\rangle}\big](x)=a_s^{-\frac{n}{2}}{\rm exp}\, \bigg[-\frac1{2a_s}\big(ae^{-2s}|x|^2 - 2e^{-s}\langle b,x\rangle-(1-e^{-2s}) |b|^2\big)\bigg].
\varphi_j \left( A_1, A_2 \right) = g_j \left(A_1 \cdot i, A_2 \cdot i \right) \frac{(c_1 i + d_1)^2 (c_2 i + d_2)^2}{\det A_1 \, \det A_2}, (A_1, A_2) = \left( (\begin{smallmatrix}a_1 b_1 c_1 d_1\end{smallmatrix}), \, (\begin{smallmatrix}a_2 b_2 c_2 d_2\end{smallmatrix}) \right).
+ ~ \theta ~ \Biggl ( \tilde{A}_{\mu}( x ) + 4 \widetilde B_{\mu}( x ) ~ +{\textstyle \frac 1 2}\Bigl [ A_{\mu}( x ) + 2 B_{\mu}( x ) ~ , ~ \int_{- \infty}^{x}\kern - 1 . 2 e m d y ( A_{0}( y ) + 2 B_{0}( y ) ) \Bigr ] \Biggr ) ~ + ~{\cal O}( \theta^{2}) ~ .
{\cal L}_{\mathrm{g a u g e}}\sim{\frac{( r M_{h})^{6}}{g_{h}^{2}}}\; \mathrm{t r}F^{2}\ \ \mathrm{a n d}\ \ \{\cal L}_{\mathrm{g r a v}}\sim{\frac{r^{6}M_{h}^{8}}{g_{h}^{2}}}\;{\cal R}\ ,
F_\alpha^k=\biggl(x^\alpha-\sum_{x^{\eta}\in \mathcal N(J^{(k)})_{\vert\alpha\vert}}C_{\alpha\eta k k}x^\eta \biggr)e_k-\mathop{\sum_{l\neq k,x^\eta e_l\in \mathcal N(J^{(l)})e_l}}_{\deg(x^\eta e_l)=\deg(x^\alpha e_k)}C_{\alpha\eta k l}x^\eta e_l
(Hf)(z_1,\ldots,z_N) = \frac{d}{ds}(e^{sH}f)(z_1,\ldots,z_N)|_{s=0}= \frac{d}{ds}\big[(w\chi)(e^{sH})\cdot f\big((w\alpha_1)(e^{-sH})z_1,\ldots, (w\alpha_N)(e^{-sH})z_N \big)\big]|_{s=0}=d(w\chi)(H)f(z_1,\ldots,z_N)-\sum_{r=1}^N \sum_{s=1}^{\ell}d(w\alpha_r)(H)z_{rs}\frac{\partial f}{\partial z_{rs}}(z_1,\ldots,z_N),
p_{i,j}^{(t+1)}\overset{}{=}\left[ P^{t+1}\right]_{i,j}^{}= \begin{dcases}(1-p)^{t+1}\mbox{if}\; i=j=0; q_{j}^{}p_{i,j}^{(t)}\,+\, p_{j-1}^{}p_{i,j-1}^{(t)}\mbox{if}\; 0 < i \le j \le n j < t; 0 \mbox{otherwise}; \end{dcases}
\alpha_2= \frac{\alpha(1+\bar{\alpha}-\kappa)(-2+\alpha-\bar{\alpha}+2 \kappa)}{2\kappa^3}, \alpha_3= \frac{\alpha(1+\bar{\alpha}-\kappa)\left(6+\alpha^2+\bar{\alpha}^2-3 \alpha(2+\bar{\alpha}-2 \kappa)-5 \bar{\alpha}(\kappa-1)-11 \kappa+6 \kappa^2\right)}{6\kappa^4},
(\partial_{t}^+e_h^{u^n},e_h^{u^n})_{\mathcal T_h}=\frac{\|e_h^{u^n}\|^2_{\mathcal{T}_h}-\|e_h^{u^{n-1}}\|^2_{\mathcal{T}_h}+(\Delta t)^2\|\partial_{t}^+ e_h^{u^n}\|^2_{\mathcal{T}_h}}{2\Delta t}.
K\left(e,\bar{e},\bar{e},n\right)-K\left(n,\bar{n},\bar{e},n\right)=W\left(e,\bar{e},\bar{e},n\right)-W\left(n,\bar{n},\bar{e},n\right)=2W\left(e,\bar{e},\bar{e},n\right),~~~~~~~~~~~~~~
\mathfrak{W}_{t}=\left\{i|f_{i}(\mathcal{X}^{t})\geq\theta\mathop{\max}_{j=1,2,\cdots,q}f_{j}(\mathcal{X}^{t})+(1-\theta)\mathbf{E}_{j\sim \mathbf{p}}[f_{j}(\mathcal{X}^{t})]\right\},
1 - \frac{v^2M^{\theta}_{2,1/v}}{2n}+ \frac{v^4M_{4,1/v}^{\theta}}{48 n}- \frac{2N^{\theta}_{1/v}}{n}\leq \sum_{i=1}^n \frac{\cos(v\theta_i)}{n}\leq 1 \ ,
\begin{cases}\partial_{t}V_{2}= D \partial_{xx}^{2}V_{2}+ \beta_{1}(m^{\ast}+ \epsilon_{0}) V_{2}+ \frac{\beta_{2}V_{4}}{K}(m^{\ast}+ \epsilon_{0}) - V_{2}(d+\gamma), \partial_{t}V_{4}= D_{4}\partial_{xx}^{2}V_{4}- U \partial_{x}V_{4}+ \xi V_{2}+ V_{4}(-\delta) + g V_{4},\end{cases}
U_t =d_j \Delta U+\beta(u_j+1) ( x,t)\in \Omega\times(0,T_{max}) V_t =\tilde d_i\Delta_M V+K_f(u_j+v_i+1)^l (x,t)\in M\times(0,T_{max}) d_j\frac{\partial U}{\partial \eta}=K_g(u_j+v_i+1) (x,t)\in M\times(0,T_{max}) U =U_0 x\in\Omega ,t=0 V =V_0 x\in M ,t=0
\Phi^\ell(\tau ,\bar\tau,\phi) = \sum_{n=0}^{2\ell}a_n t^\ell_{0n}(\tau ,\bar\tau,\phi) = \frac{e^{-i\ell\phi}}{(1+|\tau|^2)^\ell}\sum_{n=0}^{2\ell}b_n \tau^n \equiv e^{-i\ell\phi}\varphi(\tau ,\bar\tau)\,.
\lim_{\lambda\to\infty}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{(E_\Phi^r)_t(\mathbb{R}^n)}=\left[\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{(E_\Phi^r)_t(\mathbb{R}^n)};
U_{j}=\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]p_{j}q^{\beta-2}_{j}&V_{j}=\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]q^{\beta-1}_{j}
=\frac{\frac{(\begin{matrix}n+1\\n\end{matrix})(\begin{matrix}m\\m\end{matrix})}{(\begin{matrix}n+m+1\\n+m\end{matrix})}\frac{1}{2}}{\frac{(\begin{matrix}n+1\\n\end{matrix})(\begin{matrix}m\\m\end{matrix})}{(\begin{matrix}n+m+1\\n+m\end{matrix})}\frac{1}{2}+\frac{(\begin{matrix}n\\n\end{matrix})(\begin{matrix}m+1\\m\end{matrix})}{(\begin{matrix}n+m+1\\n+m\end{matrix})}\frac{1}{2}}
\left(\frac{t}{\log\left(1+t\right)}\right)^{r}\left(1+t\right)^{x}=\sum_{n=0}^{\infty}C_{n}^{\left(r\right)}\left(x\right)\frac{t^{n}}{n!},\quad\left(\mbox{see \cite{key-3,key-9}}\right).
\square \zeta = e^{-2\beta}\left(\zeta_{rr}+ \frac{\zeta_r}{r}+ (\alpha_r -\beta_r)\zeta_r \right) &= e^{-2\beta}r^{a-3}\frac{(1-a)^2}{2}\left(1-a + r^2 \kappa e^{2\beta}\, \mathbf{f}\right).
\int_{SO(N)}h(g)^P\,dg=\frac{1}{i^{N-1}}\int_{[-1,1]^{\frac{(N-1)(N-2)}{2}}}\int_{(S^1)^{N-1}}\left[\widetilde{h_{SO(N)}}(x_1,\ldots,z_{n-1})\right]^P &\cdot J_{SO(N)}(x_1,\ldots,x_{\frac{(N-1)(N-2)}{2}})\frac{dz_1}{z_1}\ldots\frac{dz_{N-1}}{z_{N-1}}dx_1\ldots dx_{\frac{(N-1)(N-2)}{2}}.
r_{circular}=\frac{\sum_{i=1}^{n}sin(x_{i}-\overline{x})sin(y_{i}-\overline{y})}{\sqrt{\sum_{i=1}^{n}sin(x_{i}-\overline{x})^{2}}\sqrt{\sum_{i=1}^{n}sin(y_{i}-\overline{y})^{2}}}
y_{\mathtt{D}_{i}}=\mathbf{h}_{\mathtt{D}_{i}}^{H}\mathbf{w}_{\mathtt{D}_{i}}s_{\mathtt{D}_{i}}+\underbrace{\sum_{k\neq i}^{K_{\mathtt{D}}}\mathbf{h}_{\mathtt{D}_{i}}^{H}\mathbf{w}_{\mathtt{D}_{k}}s_{\mathtt{D}_{k}}}_{\textrm{MUI}}+\underbrace{\sum_{j=1}^{K_{\mathtt{U}}}g_{ji}s_{\mathtt{U}_{j}}}_{\textrm{CCI}}+n_{\mathtt{D}_{i}}
\mu(\mathbf{x}_{0}+\varepsilon \mathbf{x})= \varepsilon^{\alpha-n}L(\varepsilon)\upsilon(\mathbf{x})+o(\varepsilon^{\alpha-n}L(\varepsilon)) \ \ \ \mbox{as}\varepsilon\to0^{+}, \ \ \ \mbox{in}\mathcal{D}'(\mathbb{R}^{n}),
\int_{\Omega}\frac{\gamma}{\varepsilon}|\beta_{n}(\varphi_{n})|^{2}+ \beta_{n}'(\varphi_{n}) |\nabla \varphi_{n}|^{2}\, dx & = \int_{\Omega}\left ( -\frac{\gamma}{\varepsilon}\Lambda'(\varphi_{n}) + (\mu_{n}+ \chi \sigma_{n}) \right ) (\beta_{n}(\varphi_{n}) - \beta_{n}(-1)) + \frac{\gamma}{\varepsilon}\beta_{n}(\varphi_{n}) \beta_{n}(-1) \, dx.
\bar{\mathsf h}_{ij}\to \cfrac{\rho_{ij}^*}{(\rho_{ij}^* + (t^*)^2)^2 -(t^*)^2(\zeta_j)^2}(\bar{w}^*_{ij}(\rho_{ij}^* + (t^*)^2) - \bar{w}^*_{ji}t^* \zeta_j)= \cfrac{t^*\zeta_i - \zeta_j^2}{\zeta_i^2 - \zeta_j^2}\bar{w}^*_{ij}- \cfrac{\zeta_j(\zeta_i - t^*)}{\zeta_i^2 - \zeta_j^2}\bar{w}^*_{ji},
\sup_{\varphi \in F_q}\Bigg|\int_{\mathcal{D}}u(x)\partial^{\alpha}\varphi(x)dx\Bigg| = \sup_{n \in \mathbb{N}}\Bigg|\int_{\mathcal{D}}u(x)\partial^{\alpha}f_n^q(x)dx \Bigg| < + \infty
\Pi_{\xi}:= \begin{bmatrix}\Phi_1 \Phi_2\end{bmatrix}, \Phi_k \in B\big(\mathbb{C}^{m_k}, \, L^2_{m_2}(0, \, \xi)\big); & \big(\Phi_1 g_1\big)(x)=\Phi_1(x)g_1, \Phi_2 g_2=I_{m_2}g_2\equiv g_2,
A_{w,1}f(v_{\ell,\ell,0})=(q^3+q^2)f(v_{\ell,\ell,1})+(q+1)f(v_{\ell+1,\ell,0}) &A_{w,2}f(v_{\ell,\ell,0})=q^4f(v_{\ell-1,\ell-1,0})+q(q+1)^2f(v_{\ell+1,\ell,1})+f(v_{\ell+1,\ell+1,0}) &A_{w,3}f(v_{\ell,\ell,0})=(q^3+q^2)f(v_{\ell,\ell-1,0})+(q+1)f(v_{\ell+1,\ell+1,1}).
jf_1 g_{i+j-1}= g'_jf_1f_i - f_1f'_i g_j = g'_j f_1 f_i - ((i-1)f_i - f'_1 f_i)g_j &= (g'_j f_1 - g_j f'_1)f_i - (i-1) f_i = (j-i+1)f_i g_j
H ={\frac{1}{1 6 \pi G}}\int d^{2}x \left[ e_{t}^{a}\varepsilon^{i j}\left( R_{a i j}+{\frac{1}{l^{2}}}\varepsilon_{a b c}e_{i}^{b}e_{j}^{c}\right) + \omega_{t}^{a}\left( D_{i}e_{a j}- D_{j}e_{a i}\right) \right] ,
(\overline{A}_1)_j^{j'}(l)=\begin{cases}-\dfrac{2}{3}\,c_2\,\left(\dfrac{j-j'}{j'}\right) \sqrt{\lvert j-j'\rvert \xi_{j-j'}}-2\,c_3\,\dfrac{1}{j' (j'-j)}\,\sqrt{\lvert j-j'\rvert \xi_{j-j'}}\mbox{if}\,\,j-j'\in S, begin{align*}3mm]0 \qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\mbox{otherwise},\end{cases}
\log \, Z \left( \beta \right) = - \frac{\beta}{2 r}\sum_{\ell = 0}^{\infty}D_{\ell}M_{\ell}\left( \mu , \chi \right) - \sum_{\ell = 0}^{\infty}D_{\ell}\log \left( 1 - e^{- \beta M_{\ell}\left( \mu , \chi \right) / r}\right) ,
A = \langle x_1, x_2, x_3, x_4 \rangle/(r_1, r_2, r_3, r_4, r_5, r_6) \ r_1 = x_1x_4 + x_4x_1 - x_2^2, r_2 = x_2x_4 + x_4x_2 -x_3^2-x_4^2, r_3 = x_3x_4 + x_4x_3 - x_1^2, r_4 = x_1x_3 + x_3x_1 - x_4^2, r_5 = x_2x_3 + x_3x_2 - x_1^2, r_6 = x_1x_2 + x_2x_1 - x_3^2.
\kappa = \displaystyle \frac{2 \Gamma ( h / 2 )}{\pi \Gamma ( 1 - h / 2 )}\left( \displaystyle \frac{\sqrt{\pi}}{2 \Gamma \left( \displaystyle \frac{1}{2 - h}\right) \Gamma \left( \displaystyle \frac{h}{4 - 2 h}\right)}\right)^{2 - h}\, .
=\sup_{u,x\geq 0}\left\{\bigg{|}f(u,v)-f(x,y)\bigg{|};~~\bigg{|}u-x\bigg{|}\leq \delta_{1},\bigg{|}v-y\bigg{|}\leq \delta_{2},~~(u,v)\in \mathbb{R}_{+}^{2},~~(x,y)\in \mathbb{R}_{+}^{2}\right\}.
A=\begin{pmatrix}-2 1 0 0 0 0 0 0 \cdots 0 -3 1 1 0 0 0 0 \cdots 0 0 -4 0 1 1 1 0 \cdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \ddots\end{pmatrix}
\forall z \in C^2(S^{n-1}) \;\; \;\; \int_{S^{n-1}}z h_K dS_K = 0 \; \int_{S^{n-1}}\vec{\theta}\; z(\theta) dS_K(\theta) = \vec{0}\;\; \Rightarrow \int_{S^{n-1}}(-L_K z) z dV_K \geq \frac{n-p}{n-1}\int_{S^{n-1}}z^2 dV_K .
\binom{K-2}{s-1}D_{s}^{(K+1)}L_{s-1}^{(K)}+ \binom{K-1}{s-1}D_{s-1}^{(K)}L_{s}^{(K)}+\binom{K-2}{s-2}D_s^{(K)}L_s^{(K+1)}&= \binom{K-2}{s-1}D_{s-1}^{(K)}L_{s}^{(K+1)}+ \binom{K-2}{s-2}D_s^{(K+1)}L_{s}^{(K)}+\binom{K-1}{s-1}D_s^{(K)}L_{s-1}^{(K)}.
\Delta^{\alpha}_{2t-1}= \rho_{n-t}(v_1,u_2,\ldots,v_{n-1},u_n,v_n), \nabla^{\alpha}_{2t-1}= \rho_{n-t}(u_1,v_1,\ldots,u_{n-1},v_{n-1},u_n), \Delta^{\alpha}_{2t}= \rho_{n-1-t}(v_1,u_2,\ldots,v_{n-1},u_n), \nabla^{\alpha}_{2t}= \rho_{n-t}(u_1,v_1,\ldots,u_n,v_n).
\Bigl\langle \sup_{j_0 \geq s+k_0}\Bigl\lvert \sum_{j = j_0}^{\infty}T_{\mathcal N(j,s)}B_{j-s}\Bigr\rvert ,g \Bigr\rangle \lesssim \lvert I_0\rvert \langle f\rangle_{I_0}\langle g\rangle_{I_0}.
\lim_{{\varepsilon}\to 0}\sup_{x \in K}\left|{\varepsilon}\log{\mathbb{E}}\left(\exp(-{\varepsilon}^{-1}h(Y^{\varepsilon}_x)) \right) +\inf_{\varphi \in C([0,T]:E)}\{h(\varphi) + \tilde{I}_{x}(\varphi)\}\right|=0.
[\nabla_t,D^{k+1}]\psi =\sum_{l=0}^{k}D^l(R^{SM}(\partial_t,.))\star D^{k-l}\psi+\sum_{l=0}^{k}D^{k-l+1}\dot{g}\star D^{l+1}\psi &+\sum_{\sum l_i+\sum{m_j}=k}{}^{G}\nabla^{l_1}R^P\star\underbrace{D^{m_1+1}\phi\star\ldots\star D^{m_{l_1}+1}\phi}_{l_1-}\star D^{l_2}\nabla_t\phi\star D^{l_3+1}\phi\star D^{l_4}\psi.
y_{circular}(n_{1},n_{2})=\sum_{r_{1}}\sum_{r_{2}}[\sum_{m_{1}=0}^{Q_{1}-1}\sum_{m_{2}=0}^{Q_{2}-1}h(m_{1},m_{2})x(n_{1}-m_{1}-r_{1}N_{1},n_{2}-m_{2}-r_{2}N_{2})]
[\begin{matrix}-6+3i&-2+2i&2-2i&-3+6i\\ 6-1i&3+3i&5-5i&2+1i\\ 2-2i&2+2i&-3-1i&-5+3i\\ -2+1i&8+2i&7+1i&-2-4i\\ \end{matrix}].
Y=(y_1,y_2,\dots,y_n)^t:=-B_{\infty}^{-1}R(x')\hat{Y}, B^{(k)}:=B_{\infty}^{-1}\left(R(x')\hat{B}^{(k)}R(x')^{-1}+{\partial R(x')\over \partial x_k}R(x')^{-1}\right)B_{\infty}, 1\le k\le n,
p_{S^m}(s^m)p_{X^{\rho m}|S^m}(x^{\rho m}|s^m)p_{Y^{\rho m}_1,Y^{\rho m}_2|X^{\rho m}}(y^{\rho m}_1,y^{\rho m}_2|x^{\rho m})\prod\limits_{i=1}^2p_{\hat{S}^m_i|Y^{\rho m}_i}(\hat{s}^m_i|y^{\rho m}_i),
{\boldsymbol\Lambda}_{\rm b}\left(\mu_{\rm b}\right){\boldsymbol\Psi}_{\rm b}+ \bar{\bf H}_{\rm b}^H\left({\bf Q}_{\rm s}- \gamma{\bf Q}_{\rm n}\right)\bar{\bf H}_{\rm b}{\boldsymbol\Psi}_{\rm b}={\bf 0}.
R_{j}(z) :=\sum_{n=0}^{\infty}\frac{R_{[n,j]}}{z^{n+1}}=\sum_{m=0}^{\infty}\frac{R_{[m(p+1)+j,j]}}{z^{m(p+1)+j+1}}, S_{j}(z) :=\sum_{n=0}^{\infty}\frac{S_{[n,j]}}{z^{n+1}}=\sum_{m=0}^{\infty}\frac{S_{[m(p+1)+j,j]}}{z^{m(p+1)+j+1}}, T_{j}(z) :=\sum_{n=0}^{\infty}\frac{T_{[n,j]}}{z^{n+1}}=\sum_{m=0}^{\infty}\frac{T_{[m(p+1)+j,j]}}{z^{m(p+1)+j+1}}.
[ \hat{q}^{i}, \hat{\Pi}_{j}]_{-}= \hbar \delta_{j}^{i}, \quad [ \hat{\psi}^{i}, \hat{\psi}^{j}]_{+}= \hbar \delta^{i j}, [ \hat{\varphi}_{m}, \hat{\varphi}_{n}]_{+}= \hbar \delta_{m n}
T_{\varepsilon}(\nu_1, \nu_2)(x) = T_{\varepsilon}(\nu_1, \nu_2)(x) - T_{\delta}(1_{B(x,2\varepsilon)^c}\nu_1, \nu_2)(w) &+ T_{\delta}(\nu_1, \nu_2)(w) - T_{\delta}(1_{B(x,2\varepsilon)}\nu_1, \nu_2)(w).
\begin{aligned}q_i(x_i)-y_i =0 \Leftrightarrow y_i = q_i(x_i) begin{align*}.2cm] \mbox{and}begin{align*}.2cm] \rho_jy_i[p_i(x_i)-d_i]+ \rho_iy_j[p_j(x_j)-d_j] = 0 \Leftrightarrow \rho_jy_i[p_i(x_i)-d_i]= -\rho_iy_j[p_j(x_j)-d_j] \end{aligned}
\begin{aligned}\langle (0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22), (0,23)(1,22)(2,11)(3,15)(4,17)(5,9)(6,19)(7,13)(8,20)(10,16)(12,21)(14,18), (2,16,9,6,8)(3,12,13,18,4)(7,17,10,11,22)(14,19,21,20,15) \rangle \end{aligned}
Y_{i,x}^{(1)}=Y_1\cap V(P_x^*), Y_{i,x}^{(2)}=\{y\in Y_1\setminus Y_{i,x}^{(1)}: z_{xy}\in V(P_x^*)\}, Y_{i,x}^{(3)}=Y_1\setminus(Y_{i,x}^{(1)}\cup Y_{i,x}^{(2)}).
\|u\|_{q+\alpha}^{q+\alpha}=\|u\|_{q+\alpha}^{p}\|u\|_{q+\alpha}^{q+\alpha-p}\le B_\alpha^{q+\alpha}\|\nabla u\|_p^p \Big(\frac{pq}{q-p}J(u_0)\Big)^{\frac{q+\alpha-p}{p}}.
L_{11}=\Delta^{-1}([d,b]d^{-1}+b+\Delta(c-db^{-1}a)^{-1})db^{-1}&=\Delta^{-1}(dbd^{-1}+\Delta(c-db^{-1}a)^{-1})db^{-1}&=\Delta^{-1}(d+\Delta(bd^{-1}c- a)^{-1}),
q\left( 0_{X}^{\ast}\right) =\inf_{S\in U,{{{\lambda}\in}}\mathbb{R}_{+}^{S}}\left\{{{{\sum\limits_{s\in S}{{\lambda_{s}b_{s}:}\sum\limits_{s\in S}{{\lambda_{s}a_{s}^{\ast}=}}}}-c^{\ast}}}\right\}=-\sup \mathrm{(ODP)}_{0_{X}^{\ast}}.
G_\delta(\alpha) = \oint_{S^1}\frac{\mathrm{d}w}{2 \pi i w}\frac{w^{-\delta}}{1 - 1/w + \alpha}= \begin{cases}\frac{1}{1+\alpha}\mathbf{1}_{|1+\alpha|>0}, - \mathbf{1}_{|1+\alpha| < 0}.\end{cases}
\delta_{B}( \left[ v^{3}\theta^{2}\right] ) = - i v^{2}\left( \left( \frac{d}{\phi d \phi}\right) f_{1}^{( 2 )}v^{i}\phi^{j}\phi^{k}+ f_{1}^{( 2 )}v^{i}\delta^{j k}\right) ( \epsilon \gamma^{k}\theta ) ( \theta \gamma^{i j}\theta ) .
O^{T}T O = \left( \begin{array}{cccc}0 1 0 0 -1 0 0 0 0 0 0 -1 0 0 1 0 \end{array}\right), ~ O^{T}Q O = \left( \begin{array}{cccc}0 -1 0 0 1 0 0 0 0 0 0 -1 0 0 1 0 \end{array}\right).
\Big| \int_0^t m e^{-m(t-s)}\big( \langle \psi(\cdot-ct), W_t \rangle - \langle \psi(\cdot-cs), W_s\rangle \big) ds \Big| & \leq M_{\beta}(T,\omega) \int_0^t m e^{-m(t-s)}(t-s)^{\beta}ds & \leq M_{\beta}(T,\omega) \frac{1}{m^{\beta}}\int_0^{\infty}e^{-r}r^{\beta}dr = M_{\beta}(T,\omega) \frac{1}{m^{\beta}}\Gamma(1+\beta),
\lim\limits_{v\rightarrow\infty}\sum_{k=0}^{v+3}\varphi (k) \bigl(1-D(v+3-k)\bigr) =\varphi(\infty)\lim\limits_{v\rightarrow\infty}\sum_{k=0}^{v+3}\bigl(1-D(k)\bigr) &=\varphi(\infty)\mathbb{E}S.
(F^{\mathrm{can}})^{\prime}_{\bar{H}^{\bar{l}^{\prime}}}= (\bar{\pi}^{(k+ \bar{l}^{\prime},\bar{l}^{\prime}+ 1)})_{*}\circ F_{(\bar{\pi}^{(k+\bar{l}^{\prime}+1, \bar{l}^{\prime}+1)})^{-1}(\bar{H}^{\bar{l}^{\prime}})}.
\begin{array}{rcl}I_1(\rho^2,0) = \displaystyle (4\pi)^{-2}(\pi\mu^2\rho^2)^{2-n/2}\Gamma(n/2-2) &= \displaystyle (4\pi)^{-2}\left[\Gamma(n/2-2)-\ln(\pi\mu^2\rho^2)+O(n/2-2)\right],\end{array}
\partial_\zeta\vec \psi(t;\zeta)=\mathcal{Q}(t;\zeta)\vec \psi(t;\zeta)\ , \partial_t\vec \psi(t;\zeta)=\mathcal{B}(t;\zeta)\vec \psi(t;\zeta)\ ; & \partial_\eta\vec \chi(t;\eta)=\widetilde{\mathcal{Q}}(t;\eta)\vec \chi(t;\eta)\ , \partial_t\vec \chi(t;\eta)=\widetilde{\mathcal{B}}(t;\eta)\vec \chi(t;\eta)\ .
\left[ \begin{array}{ccc}1 0 -\frac{n/2}{n-1}1\end{array}\right]\cdot\left[ \begin{array}{ccc}6 1 0 1\end{array}\right]\cdot\left[ \begin{array}{ccc}1 0 -\frac{3n}{n-1}1\end{array}\right]=\left[ \begin{array}{ccc}0 1 6 0\end{array}\right],
g_{1}([U,V],Z) =\cos^{2}\theta g_{1}(\nabla^{^{M_1}}_{U}V,Z)-g_{2}((\nabla \pi_{\ast})(U,\varphi V)+\nabla_{U}^{\pi}\pi_{\ast}\varphi V,\pi_{\ast}wZ) &-\cos^{2}\theta g_{1}(\nabla^{^{M_1}}_{V}U,Z)+g_{2}((\nabla \pi_{\ast})(V,\varphi U)+\nabla_{V}^{\pi}\pi_{\ast}\varphi U,\pi_{\ast}wZ).
D{\mathcal F}_{X_{k+1}^{p}}(H)=(\mathcal{A}-\mathcal{B}\,{\mathcal{B}}^T \, X_{k+1}^{p})^T \, H\,+\,H \,(\mathcal{A}-\mathcal{B}\,{\mathcal{B}}^T \,X_{k+1}^{p})
\mathbf{Y}_{nm}^u=\begin{cases}\mathbf{I}_Q+\mathcal{E}_{\mathrm{s}}\sum_{v=1}^{U}\mathbf{C}_n^v(\mathbf{C}_n^{v})^{\dag}, n=m\in\mathbb{N}\mathcal{E}_{\mathrm{s}}\sum_{v=1}^{U}\mathbf{C}_n^v(\mathbf{C}_m^{v})^{\dag}, n\neq m\in\mathbb{N}.\end{cases}
\left\langle \int d^2x:\Psi^{\dagger}(x)M\Psi(x):\right\rangle=-\frac{1}{2}\left( \sum_{\rm occupied}\int d^2x\psi_E^{\dagger}(x)M\psi_E(x)-\sum_{\rm unoccupied}\int d^2x \psi_E^{\dagger}(x)M\psi_E(x)\right).
R_5 = -\zeta h \dfrac{\nabla_X b}{h_b^3}\Big(\cosh{(\beta b(X)\sqrt{\mu}|\mathrm{D}|)}\mathrm{sech}{(\sqrt{\mu}|\mathrm{D}|)}-1\Big)\frac{1}{\mu|\mathrm{D}|^2}\Delta_X\psi.
x_1^\prime = \alpha_{11}y_1 + \alpha_{12}y_2 \, , x_2^\prime = \alpha_{21}y_1 + \alpha_{22}y_2 \, ; y_1^\prime = \beta_{11}x_1 + \beta_{12}x_2 \, , y_2^\prime = \beta_{21}x_1 + \beta_{22}x_2 \, .
\sum_{l=-r}^{r}\dfrac{1}{\omega_{l}}\, \biggl(u_{(2r+1)+l}+\sum_{p\ge 2}\dfrac{\overline{\lambda}^{p-1}}{\omega_{(p-1)(2r+1)+l}\dots\omega_{(2r+1)+l}}u_{p(2r+1)+l}&+\sum_{p\ge 1}\dfrac{\omega_{l-p(2r+1)}\dots\omega_{l-(2r+1)}}{\overline{\lambda}^{p}}u_{-p(2r+1)+l}\biggr){(\lambda -A)}^{*}{e_{l}}=0
f(z) = 2\pi i \cdot \frac{e^{2\pi i z}}{e^{2\pi i z}-1}= z^{-1}+ \pi i - \frac{\pi^2}{3}z - \frac{\pi^4}{45}z^3 \cdots, g(z) = (2\pi i)^2 \cdot \frac{e^{2\pi i z}}{(e^{2\pi i z}-1)^2}= z^{-2}+ \frac{\pi^2}{3}+ \frac{\pi^4}{15}z^2 + \cdots,
\textrm{diam}(\varphi_t(\omega,B(0,r)) = \sup_{x,y \in B(0,r)}\left| \varphi_t(\omega,x) - \varphi_t(\omega,y) \right| &\leq \sup_{x,y \in B(0,r)}\left| \varphi_t(\omega,x) - a(\theta_t \omega) \right| + \left| a(\theta_t \omega) - \varphi_t(\omega,y) \right| & \rightarrow 0 \textrm{as}t \rightarrow \infty
\left(\delta_t e_{\boldsymbol\sigma}^n,\bar{\boldsymbol \sigma}_h\right)+ \langle B_h e_{\boldsymbol\sigma}^n, \bar{\boldsymbol \sigma}_h\rangle = 2(e_u^n\nabla u(t_n)+u^n_h \nabla e_u^n ,\bar{\boldsymbol \sigma}_h) + (\xi_2^n,\bar{\boldsymbol \sigma}_h),\ \ \forall \bar{\boldsymbol\sigma}_h\in{\boldsymbol \Sigma}_h,
\frac{\left( e_{j}^{( \alpha )}\cdot e_{l}^{( \alpha )}\right)_{C}}{| e_{j}^{( \alpha )}|_{C}| e_{l}^{( \alpha )}|_{C}}= \frac{C_{j l}}{\sqrt{C_{j j}C_{l l}}}= c_{j l}\equiv \cos \tau_{j l}.
\limsup_{n\to\infty}\frac{1}{b_c^{(n)}}\log P(S_n(\kappa_n(x))\leq\lfloor\kappa_n(x)\rfloor-a_n) &\,\,\,\,\,\,\leq\limsup_{n\to\infty}\frac{1}{b_c^{(n)}}\log P\left(\frac{n-a_n-S_n(\kappa_n(x))}{f_2(n)}\geq x(1-\delta)\right) &\,\,\,\,\,\,\leq-\inf_{[x(1-\delta),+\infty)}H(\ell_2 y)=-H(\ell_2 x(1-\delta))
U_n A_n V_n = S(A_n) \Rightarrow \det(U_n) \det(A_n) \det(V_n) = \det(S(A_n)) \Rightarrow \det(A_n) \det(V_n) = \pm|\det(A_n)| \det(U_n) = \pm 1) \Rightarrow \det(V_n) = \pm 1 \det(A_n) \neq 0).
=\left(\int_{2at}^{2bt}A(2u)P(2u,k)du \right) \mu_+((b-k)\sqrt t)-\int_{2at}^{2bt}\left(\int_{2at}^x A(2u)P(2u,k)du\right)\frac{\mu'_+((x-2k t)/(2\sqrt t))}{2\sqrt t}dx\,.
\partial_u H= [(u+u^*+e_2)(u-u^*)]/[(u+e_2)u],\partial_v H=\varrho(v-v^*)/v, begin{align*}0.2cm] \partial_{uu}H= \left[u^*(u^*+e_2)(2u+e_2)\right]/[(u+e_2)u]^2,\ \ \partial_{vv}H=\varrho v^*/v^2.
\inf_{\xi\in M}\inf_{v\in C(\xi)}\|D_\xi T_t^{-n}v\|> \tilde c\tilde \nu^{-n}\|v\| & \inf_{\xi\in M}\inf_{v\not\in C(\xi)}\|D_\xi T_t^n v\|> \tilde c \tilde\lambda^{n}\|v\|
\mathcal{H}^{(d^\prime)}(A) \; = \; \lim_{R \searrow 0}\mathcal{H}_{R}^{(d^\prime)}(A) &\mathrm{with}\mathcal{H}_{R}^{(d^\prime)}(A) \; := \; \inf_{\mathrm{covering}}\Bigl\{\sum_i R_i^{d^\prime};\; A \subset \bigcup_i P_{R_i}(z_i), \, 0 < R_i < R \Bigr\}.
\hat{\hat{K}}_{\hat{\hat{a}}\hat{\hat{b}}\hat{\hat{c}}}\;=\;{\textstyle\frac{1}{2}}\left(\hat{\hat{T}}_{\hat{\hat{a}}\hat{\hat{c}}\hat{\hat{b}}}\; +\; \hat{\hat{T}}_{\hat{\hat{b}}\hat{\hat{c}}\hat{\hat{a}}}\; -\; \hat{\hat{T}}_{\hat{\hat{a}}\hat{\hat{b}}\hat{\hat{c}}}\right) \; ,
Im\{\zeta(\alpha,t)-z_j(t)\}\geq \inf_{\alpha\in\mathbb{R}}Im\{\zeta(\alpha,0)-z_j(0)\}+(\frac{|\lambda|}{10\pi x(0)}-6\epsilon-(C\epsilon^2+K_s^{-1}\epsilon))t \geq 1+\frac{|\lambda|}{18\pi x(0)}t.
{\cal F}_{N}\star_{N}{\cal G}_{N}({\cal W}) ={\cal F}_{N}\,{\cal G}_{N}({\cal W}) + \sum_{m = 1}^{N}\frac{( N - m ) !}{N ! m !}\,{\cal F}_{N}({\cal W}) \, \, \vdots \underbrace{( \partial^{^{\! \! \! \! \! \leftarrow}}\, \Omega \, \vec{\bar{\partial}}) \cdots ( \partial^{^{\! \! \! \! \! \leftarrow}}\, \Omega \, \vec{\bar{\partial}})}_{m f a c t o r s}\, \vdots \,{\cal G}_{N}({\cal W}) \, ,
\begin{aligned}2\Delta t((D(\mathbf{u}_h^{n})^{-1}(\zeta^{n}_\sigma),\zeta^{n}_\sigma)_{\mathcal{T}_h}\geq C\Delta t\|\zeta^{n}_\sigma\|_{\mathcal{T}_h}^2, &2(\phi(\zeta^{n}_c-\zeta^{n-1}_c),\zeta^{n}_c)_{\mathcal{T}_h}\geq(\phi\zeta^{n}_c,\zeta^{n}_c)_{\mathcal{T}_h}-(\phi\zeta^{n-1}_c,\zeta^{n-1}_c)_{\mathcal{T}_h}.\end{aligned}
(\Pi \times ev)_{|\mathcal{W}_g}^{-1}(m,x) = \left( M_{m}\times_{X^r}\{x\}\right) \times_{X^r}(g\prod_i X(u_i)) &= M_m \times_{X^r}\left( \{x\}\times_{X^r}(g\prod_i X(u_i)) \right) &= M_m \times_{X^r}\{x\}&= (\Pi \times ev)^{-1}(m,x),
\sum\limits_{\mu=0}^{k-1}f(c(\mu+1))\sum\limits_{v=0}^{k-1}f^{\ast}(bv)B_{2}\left( \frac{v}{k}\right) =2B_{0}(A)P_{2}\left( 0,B_{b}\right) , \sum\limits_{v=0}^{k-1}f^{\ast}(bv)\sum\limits_{\mu=0}^{k-1}f(c(\mu+1))B_{2}\left( \frac{\mu+1}{k}\right) =2B_{0}\left( B\right)P_{2}\left( 0,A_{c}\right)
e^{ix^{0}P_{0}}M_{j0}e^{-ix^{0}P_{0}}=M_{j0}+ix^{0}[P_{0},M_{j0}]+\frac{i^2}{2!}(x^{0})^2\underbrace{[P_{0},[P_{0},M_{j0}]]}_{=0}+\cdots &= M_{j0}+x^{0}P_{j},
\mathcal{R}^+_{}=\left\{Q \in \mathcal{L}^+: \begin{matrix}q_{13}\!=\!q_{14},q_{23}\!=\!q_{24},q_{31}\!=\!q_{32},q_{41}\!=\!q_{42}q_{12}q_{23}\!=\!q_{21}q_{13},q_{34}q_{13}\!=\!q_{12}q_{31},q_{43}q_{13}\!=\!q_{12}q_{41},\ldots\end{matrix}\right\},
\lesssim \left\|\mu(D(z_j,r))\right\|_{\ell^{\frac{p}{p-1}}}\Bigg(\sum_{j=1}^\infty \int_{D(z_j,2r)}|f(z)|^p e^{-\frac{\alpha p}{2}|z|^2}dV(z)\Bigg)^{\frac{1}{p}}
\begin{array}{l}{{| h = \frac{k D}{2 ( k + \check{g})}, q = \frac{k D}{( k + \check{g})}\rangle = | a_{1}, . . . . , a_{D}\rangle}}\\{{\}}\\{{| h = \frac{k D}{2 ( k + \check{g})}, q = - \frac{k D}{( k + \check{g})}\rangle = | \bar{a}_{1}, . . . , \bar{a}_{D}\rangle}}\\ \end{array}
S_{GF-ghost}=\int d^{d}x\left\{-\frac{1}{2\xi}\partial_{\mu}A^{\mu}(x)\star\partial_{\nu}A^{\nu}(x) +\partial^{\mu}\overline{c}(x)\star\left(\partial_{\mu}c(x)-i[A_{\mu}(x),c(x)]_{M}\right)\right\}
|B_{P_{R,k-1}}| \sigma(P_{R,k-1}) \lesssim \Big| \int_{P_{R,k-1}}B_{P_{R,k-1}}b\,d\sigma\Big| &= \Big|\int_{P_{R,k-1}}\Delta_{P_{R,k}}f\,d\sigma\Big| \le \sigma(P_{R,k-1})^{1/2}\| \Delta_{P_{R,k}}f\|_{L^2(\sigma)}.
\left(\int_{\mathbb{R}^N}\Psi_\varepsilon(U)^{p^*}\,dx\right)^\frac{p}{p^*}\le \frac{1}{\alpha}\,\|U\|^{p^*+\alpha-1}_{\infty}\,|\{U>K_0\}| &+\left(\int_{\{U\le K_0\}}U^{p^*}\,dx\right)^\frac{p^*-p}{p^*}\left(\int_{\{U\le K_0\}}\Big(\psi_\varepsilon(U)\,U^{p-1}\Big)^\frac{p^*}{p}\,dx\right)^\frac{p}{p^*},
S^{\prime}={\frac{N^{2}}{1 6 \pi G_{1 0}}}\int d^{1 0}x \sqrt{- G^{d u a l}}( N e^{\phi})^{-{\frac{6}{7}}}\left[{\cal R}^{d u a l}+{\frac{1 6}{4 9}}\partial_{M}\phi \partial^{M}\phi -{\frac{1}{2 \cdot 8 !}}{\frac{1}{N^{2}}}F_{M_{1}\cdots M_{8}}F^{M_{1}\cdots M_{8}}\right] .
\int \limsup_{N \to \infty}\left| \frac{1}{N}\sum_{n=1}^N f(U^ny) \right|^2 \, d\nu \leq \frac{C}{H}+ C\left(\frac{1}{H}\sum_{h=1}^H \left| \int f \circ f \cdot U^h \, d\nu \right|^2 \right)^{1/2}.
\Pi(B_n(Kf_0,\epsilon_n)) \geq \Pi\bigl( \tilde{J}_n \leq J \leq 2\tilde{J}_n, \|a - a_0 \|_n \leq c'\tilde{J}_n^{1/2}\epsilon_n \bigr) &\geq \exp\bigl(- \tilde{J}_n ( c_d(\log \tilde{J}_n)^t + c_2 \log(\tilde{J}_n^{-1/2}\epsilon_n^{-1}))\bigr).
\Phi_{\alpha , \bar{\alpha}s}- \Phi_{\bar{\alpha}, \alpha s}+ \Phi_{\beta , \bar{\beta}s}- \Phi_{\bar{\beta}, \beta s}+ [ ( q^{- x}\cdot \Phi_{\bar{\alpha}, s}\Phi_{\alpha , r s}+ q^{x}\cdot \Phi_{\alpha , r}\Phi_{\bar{\alpha}, s s}) - ( q^{- x}\cdot \Phi_{\bar{\alpha}, r}\Phi_{\alpha , s s}