harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Ag_{\lambda}\left(\frac{\partial}{\partial x^{i}},\frac{\partial}{\partial y^{i}},\frac{\partial}{\partial z^{j}}\right)=\left(\frac{\partial}{\partial x^{i}}+\lambda \sqrt{2}\frac{\partial}{\partial y^{i}},\frac{\partial}{\partial y^{i}}+\lambda \sqrt{2}\frac{\partial}{\partial x^{i}},\sigma_{Ag}\frac{\partial}{\partial z^{j}}\right)
S= \int d^{4}x\sqrt{|g_{E}|}\left[ \hat{R}(\hat{g}_{E})-{\textstyle\frac{1}{2}}\frac{\partial_{\hat{\mu}}\hat{\lambda}\partial^{\hat{\mu}}\bar{\hat{\lambda}}}{\left(\Im{\rm m}\hat{\lambda}\right)^{2}}+{\textstyle\frac{1}{4}}\hat{F}^{I}\{}^{\star}\tilde{\hat{F}^{I}}\right]\, .
\Delta^+_{\prec}([w_1 | \cdots | w_m]) := [\Delta^+_{\prec}(w_1)\Delta(w_2) \cdots \Delta(w_m)] \Delta^+_{\succ}([w_1 | \cdots | w_m]) := [\Delta^+_{\succ}(w_1)\Delta(w_2) \cdots \Delta(w_m)].
\sigma_{j-1}\left(2-\frac{r}{r_j}\right) + \sigma_{j}\left(2-{r}\right)+\sum_{i=1, i\neq j-1,j}^{2q}2\sigma_i &\qquad=-2d_{j-1}+ \frac{d}{r}+d_{j+1}+3t - \frac{e}{2q}\left(\frac{1}{r_j}+\frac{1}{r_{j+1}}\right)
S^{\prime}={{N^2}\over{16\pi G_{10}}}\int d^{10}x\sqrt{-G^{dual}}(Ne^{\phi})^{-{6\over 7}}\left[{\cal R}^{dual}+{{16}\over{49}}\partial_M\phi\partial^M\phi-{1\over{2\cdot 8!}}{1\over{N^2}}F_{M_1\cdots M_8}F^{M_1\cdots M_8}\right].
\nabla^a\Big(\left(-\frac{2}{r^2}D H_1\cdot H_2 g_{ab}\right)T^b\Big) = -\frac{2}{r^2}D \nabla_T H_1\cdot H_2-\frac{2}{r^2}DH_1\cdot\nabla_T H_2 =_s -\frac{2}{r^2}\nabla_T H_1\cdot D^\dagger H_2-\frac{2}{r^2}DH_1\cdot\nabla_T H_2.
\Psi^{\mu}_d(y) \, \Omega^{\mu}= y(1-y)(\Psi^{\mu}_d)''(y) + a(y)(\Psi^{\mu}_d)'(y) + \Psi^{\mu}_d(y)\, F^{\mu}(y)= \Lambda^{\mu}_d \,\Psi^{\mu}_d(y),
\left[E^+\right]+\left[D_{i_1}^+\right]+\cdots+\left[D_{i_{n/2}}^+\right]=\left[E^-\right]+\left[D_{i_{n/2+1}}^-\right]+\cdots+\left[D_{i_{n}}^-\right] \{i_1,\dots,i_n\}=\{1,\dots,n\}.
v^{(n)}:= \left(\prod_{n^\prime = 0}^{n-1}p(D-n^\prime)\right) \left(\prod_{i \in J_{n-1}}(D-i)\right) u, w^{(n)}:= \left(\prod_{n^\prime = 0}^{n-1}p(D-n^\prime)\right) \left(\prod_{i \in J_n}(D-i)\right) u = \left(\prod_{i \in I_n}(D-i)\right) v^{(n)}, r^{(n)}:= \left(\prod_{n^\prime = 1}^n p(D-n^\prime)\right) \left(\prod_{i \in J_n}(D-i)\right) f,
[\begin{matrix}r\\ \theta\\ \phi\end{matrix}]=[\begin{matrix}\sqrt{x^{2}+y^{2}+z^{2}}\\arccos(z/\sqrt{x^{2}+y^{2}+z^{2}})\\arctan(y/x)\end{matrix}],0\le\theta\le\pi,0\le\phi<2\pi,
\lim_{y\to\infty}\frac{G^{-1}(y)}{1/\log_2 y}=1, G^{-1}\in_\infty(0), \lim_{y\to\infty}\frac{\Gamma(y)}{1/(y\log y(\log_2 y)^2)}=1, \Gamma=g\circ G^{-1}\in_\infty(-1).
...=\sum_{i=1}^{N}(N)_{i}A^{-i}-\sum_{i=1}^{N}(N-1)_{i-1}A^{-i+1}+1=\sum_{i=1}^{N}(N)_{i}A^{-i}-\sum_{i=0}^{N-1}(N-1)_{i}A^{-i}+1
B_{q,1}\leq \frac{\pi}{2 \tanh(\pi d/(2 \alpha))}\int_{-\infty}^{\infty}\frac{\pi}{\pi^{2}+ 4d^{2}\omega^{2}}\, \mathrm{d}\omega=\frac{\pi^{2}}{4d \tanh(\pi d/(2 \alpha))}
B_\alpha : = B\cup \bigcup\left\{A(\beta,\delta) : \beta\in I\ \wedge\ \beta<\alpha\ \wedge\ \delta\in J_\beta\right\}\ J_\alpha : =\left\{\gamma \in J_\alpha' : \left(A(\alpha,\gamma) \setminus A_\alpha\right)\cap B_\alpha =\emptyset\right\}.
(\widetilde{\alpha}\otimes \widetilde{\varphi}) \circ \widetilde{\Delta}(x,m) = (\alpha \otimes \varphi_L) \Delta(x)+( \beta \otimes \varphi_L ) (\rho_{\beta}(m))-( \alpha_M \otimes \varphi_L ) (\tau \circ (\rho_{\beta}(m))).
\begin{aligned}\mathcal{I}_{\mathrm{D}}(\eta_0, \delta ) - \mathcal{I}_{\mathrm{D}}(\eta, 0 ) = \mathcal{I}_{\mathrm{D}}(\eta_0, \delta ) - \mathcal{I}_{\mathrm{D}}(\eta, \delta ) + \mathcal{I}_{\mathrm{D}}(\eta, \delta ) - \mathcal{I}_{\mathrm{D}}(\eta, 0 ) & \leq \Psi (\eta_0-\eta, 0 ) + \Psi (0, \delta ) = \Psi ( |\eta_0-\eta|, 0 ) + \Psi (0, \delta ).\end{aligned}
\left\| \mathbb{G}_{L;0}(x,t)\right\|_{L^2_\xi}\leq C_N\left[\sum_{j=1}^{3}(1+t)^{-1/2}\Big(1+\frac{|x-a_{j}t|^{2}}{1+t}\Big)^{-N}+e^{-ct}\right].
\{U_{m}, U_{n}\}= - 1 6 m n i \int d x \, k_{a_{1}\ldots a_{m - 1}b c}^{( m + 1 )}k_{a_{m}\ldots a_{m + n - 2}d c}^{( n + 1 )}h_{+}^{a_{1}}\ldots h_{+}^{a_{m + n - 2}}\psi_{+}^{' b}\psi_{+}^{' d}\, ,
\mbox{(i)\, $\partial\Omega$ is Uniformly Rectifiable}, &\mbox{(ii)\, $\Omega$ is an NTA domain (and therefore $\Omega$ is a chord-arc domain)}, &\mbox{(iii)\, $\omega\in A_{\infty}$}, &\mbox{(iv)\, $\omega\in\mbox{weak}-A_{\infty}$}.
\int_{\mathcal{M}^\perp}\! e(k,\tau) d\tau = \int_{\mathcal{M}^\perp}\! e(k,\tau_0+\tau-\tau_0) d\tau = e(k,\tau_0) \int_{\mathcal{M}^\perp}\! e(k,\tau-\tau_0) d\tau & = e(k,\tau_0) \int_{-\tau_0+\mathcal{M}^\perp}\! e(k,\tau) d\tau = \underbrace{e(k,\tau_0)}_{\neq 1}\int_{\mathcal{M}^\perp}\! e(k,\tau) d\tau,
u_{0}(z)=\left(\frac{z-a_{1}}{\left|z-a_{1}\right|}\right)^{d_{1}}\left(\frac{z-a_{2}}{\left|z-a_{2}\right|}\right)^{d_{2}}....\left(\frac{z-a_{m}}{\left|z-a_{m}\right|}\right)^{d_{m}}.
\frac{-1}{b(1-b)^2}- \sum_{n=1}^\infty\left(\frac{q^nb}{(1-bq^n)^2}+\frac{q^n/b}{(1-q^n/b)^2}\right) =\frac{(q)^2_\infty}{[1/b]_\infty}\sum_{k=-\infty}^\infty\frac{(-b)^kq^{\frac{k(k+1)}{2}}}{1-bq^k}& -\sum_{\substack{n=-\infty n\neq0}}^\infty\frac{(-1)^nnq^{n(n+1)/2}}{1-q^n}
f(\left<x,y\right>\left<y,x\right>)^2 \ =\ \left<x,y\left<y,x\right>\right>_f^2 \ \leq\ \|x\|_f^2 \, \|y \left<y,x\right>\|_f^2 \ =\ \|x\|_f^2\, f(\left<x,y\right>\left<y,y\right>\left<y,x\right>) \ \leq\ \|x\|_f^2 \, \|\left<y,y\right>\| \, f(\left<x,y\right>\left<y,x\right>),
{\cal M}_{b}^{u u}= \frac{i g^{4}}{2}[ T^{a}T^{b}\otimes T_{a}T_{b}] \int \frac{d^{2}k}{( 2 \pi )^{2}}\left[ \frac{T ( k , p_{1}) T^{*}( k , p_{3})}{w_{k}- w_{p}}\right]
U(k, i_0, \ldots, i_d, \ell_0, \ell_1, \ldots, \ell_k) = \prod_{t=0}^k |E_{i_{\ell_t}}|~ q^{t(\ell_{t+1}-\ell_t-1)}&=\left( \prod_{t=0}^k |E_{i_{\ell_t}}| \right) \left(q^{\sum\limits_{t=0}^k t(\ell_{t+1}-\ell_t-1)}\right),
{\cal{Q}}^{\dagger}{\cal{Q}}\overline{{{\psi}}}= \left\{- \frac{d}{d z}- \frac{3}{2}\dot{A}\right\}\left\{\frac{d}{d z}- \frac{3}{2}\dot{A}\right\}\overline{{{\psi}}}= 2 m^{2}\overline{{{\psi}}}
\begin{cases}f_1(t, B_t) = -f(t, B_t) \frac{c^2 + t +B_t^2}{2(c^2+t)^2}, f_2(t,B_t) = f(t, B_t) \frac{B_t}{c^2 + t}, f_{22}(t,B_t)= f(t, B_t) \frac{c^2 + t +B_t^2}{(c^2+t)^2}. \end{cases}
G_{i - 1}( x ) \ = \ \frac{\Gamma \left( \frac{d}{2}- i \right)}{( 4 \pi )^{\frac{d}{2}}}\left( \frac{x^{2}}{4}\right)^{i - \frac{d}{2}}\hspace{2 . 7 c m}\mathrm{f o r ~}i \geq 1
u_b(x)=\exp\left\{\frac{b (2-n) r^2}{(n-1) n}F\left[\begin{array}{c}1,1,2-\frac{n}{2}2,1+\frac{n}{2}\end{array}; r^2\right]\right\}\left(1-r^2\right)^{\frac{b}{(n-1)}}.
e_{r,r+1}(w) :=- \frac{\prod_{j=1}^{r+1}(w_{r1}-w_{r+1,j})}{\prod_{j\neq 1}^{r}(w_{r1}-w_{rj})}, e_{r+1,r}(w) := \frac{\prod_{j=1}^{r-1}(w_{r1}-w_{k-1,j})}{\prod_{j\neq 1}^{k}(w_{r1}-w_{rj})}, e_{rr}(w) :=r-1+\sum\limits_{i=1}^{r}w_{ri}-\sum\limits_{i=1}^{r-1}w_{r-1,i}
|\{m \in M_k\mid m>i_k\}| begin{align*}3pt] \quad=|M_k|-|\{m \in M_k\mid m\leq i_k\}| begin{align*}3pt] \quad=(n-k+1-c_{k-1})-(n-k+1-c_k) begin{align*}3pt] \quad=c_k-c_{k-1}.
\begin{array}{lll}\overline{\mathrm{Ric}}_{00}=2(\beta-1), \ \ \ \overline{\mathrm{Ric}}_{11}=\overline{\mathrm{Ric}}_{22}=\beta(\beta-1)-\alpha^2, \ \ \ \overline{\mathrm{Ric}}_{12}=\alpha. \end{array}
\left| \sum_{\alpha \beta}\langle \Phi_{\alpha}\Phi_{\beta}\rangle_{c}\left. \right/ \sum_{\alpha \beta}\langle \Phi_{\alpha}\Phi_{\beta}\rangle \right| = \left| \langle \Phi^{2}( x ) \rangle_{c}\left. \right/ \langle \Phi^{2}( x ) \rangle \right| ( n = 2 ) ,
\begin{array}[c]{ll}-\left[ \alpha_{3}(t)X(t)+\beta_{3}(t)Y(t)+\gamma_{3}(t)Z(t)+L_{3}(t)\right] =p(t)\left[ \alpha_{1}(t)X(t)+\beta_{1}(t)Y(t)+\gamma_{1}(t)Z(t)+L_{1}(t)\right] -A(t)X(t) & \ \ +q(t)\left[ \alpha_{2}(t)X(t)+\beta_{2}(t)Y(t)+\gamma_{2}(t)Z(t)+L_{2}(t)\right] -C(t).\end{array}
{A}_t\setminus{P}_t=({A}_t\setminus{P}_t)\cap \bigcup_{s=1}^{t-1}\left[({B}_s\setminus \cup_{j=s+1}^{t-1}D_s^j)\setminus\bigsqcup_{k=1}^{s-1}D_k^t \right]=\bigsqcup_{s=1}^{t-1}D_s^t.
a^i(\theta_t)=\frac{(\theta_t^i)^{\frac{1}{\gamma}}}{\sum_j(\theta_t^j)^{\frac{1}{\gamma}}}b^i(\theta_t)=\frac{\gamma\hat{\phi}}{\eta}\left[\frac{(\theta_t^i)^{\frac{1}{\gamma}}}{\sum_j(\theta_t^j)^{\frac{1}{\gamma}}}-\frac{\phi^i}{\hat{\phi}}\right],
\left\|A_QA_R^{-1}\right\|^p =\sup_{\vec z\in\mathbb{C}^m,\,|\vec z|=1}\left|A_QA_R^{-1}\vec z\right|^p &\sim\sup_{\vec z\in\mathbb{C}^m,\,|\vec z|=1}\left[|c_Q|+\ell(Q)\right]^{-d}\left|A_R^{-1}\vec z\right|^p &\sim\left[\frac{|c_R|+\ell(R)}{|c_Q|+\ell(Q)}\right]^d.
A=\sinh(ka/2)\cosh(ka/2)\left[-\frac{a}{2}+\cosh^{2}(ka/2)\left(\frac{2a}{3}-\frac{1}{2(a/2)k^{2}}+\frac{\sinh(ka/2)\cosh(ka/2)}{2(a/2)^{2}k^{3}}\right) \right]^{-1/2}
\frac{1}{2 \pi i}\frac{\partial}{\partial \tau}g_j(\tau) = g_{j-2}(\tau) + \rho_R g_{j-1}(\tau), \left(\frac{1}{2 \pi i}\right)^2 \frac{\partial^2}{\partial \tau^2}g_j(\tau) = g_{j-4}+ 2 \rho_R g_{j-3}+ \rho_R^2 g_{j-2}.
\sum_{\ell=0}^{k}\sum_{m=\ell}^n q(n-m,k-\ell)q(m,\ell) = \frac{2}{an+(b-1)k+2}\binom{an+(b-1)k+2}{k}\frac{k!}{n!}B_{n,k}(x) = \frac{2}{an+(b-2)k+2}\binom{an+(b-1)k+1}{k}\frac{k!}{n!}B_{n,k}(x) = \frac{2(an+(b-1)k+1)}{an+(b-2)k+2}\,q(n,k).
\eta^{\star}= \frac{C_0^{U^{\star}}C_1^{S^{\star}}+ C_1^{U^{\star}}- C_0^{S^{\star}}C_1^{U^{\star}}}{C_0^{U^{\star}}C_0^{L^{\star}}+ (1-C_0^{S^{\star}})\sigma_v^2/A^2},
\frac{J^{w_1}(X_{i:n})}{J^{w_1}(X_{i+1:n})}=\frac{c_{i,n}}{c_{i+1,n}}\left(\frac{2i-1}{2i+1}\right)\frac{E\left(M(B_{2i:2n})\right)}{E\left(M(B_{2i+2:2n})\right)}&=\frac{i(2n-2i-1)}{(2i+1)(n-i)}\frac{E\left(M(B_{2i:2n})\right)}{E\left(M(B_{2i+2:2n})\right)}&\le \frac{E\left(M(B_{2i:2n})\right)}{E\left(M(B_{2i+2:2n})\right)}.
f_{\mu}^{a}(\psi_{,\mu}^{b})-f_{\mu}^{a}(\psi^{a},\lambda_{a}^{\mu})-\lambda_{b}^{\nu}\frac{\partial f_{\nu}^{b}}{\partial \lambda_{a}^{\mu}}+\frac{\partial L}{\partial\lambda_{a}^{\mu}}=0.
\left<\begin{pmatrix}u v w\end{pmatrix},\begin{pmatrix}I T_i^* T_j^* T_i T_i^* T_i T_j^* T_i T_j T_i^* T_j T_j^* T_j\end{pmatrix}\begin{pmatrix}u v w\end{pmatrix}\right>_{\mathcal{C}_d \times \mathcal{C}_d \times \mathcal{C}_d}= \hdots
=\rho\left[ \frac{\partial u_{1}}{\partial t}+u_{1}\frac{\partial u_{1}}{\partial x}+u_{2}\frac{\partial u_{1}}{\partial y}\right] +\frac{\partial}{\partial x}\frac{Ke^{f(\frac{r}{a(t)})}}{a(t)^{2}}+\rho\frac{\partial}{\partial x}\Phi
\hat p_{d,t+1}=\frac{D_t}{D_{t+1}}\ \frac{\hat p_{d,t}^{{\eta_{t+1}}/{\eta_t}}e^{-\eta_{t+1}\ell(f_{d,t},y_t)}}{\sum_{k=1}^{D_{t}}\hat p_{k,t}^{{\eta_{t+1}}/{\eta_t}}e^{-\eta_{t+1}\ell(f_{k,t},y_t)}}\,.
h_{\mu}(\alpha\vee \beta,G)=h_{(\sigma_{i})_{i},\mu}(\alpha\vee \beta:\gamma) \leq h_{(\sigma_{i})_{i},\mu}(\beta:\gamma)+h_{(\sigma_{i})_{i},\mu}(\alpha|\beta:\gamma) &= h_{\mu}(\beta,G)+h_{(\sigma_{i})_{i},\mu}(\alpha|\beta:\gamma).
\begin{cases}\beta_1\gamma_1=\alpha_1\gamma_3 \beta_1\gamma_2+\beta_2\gamma_5=\alpha_1\gamma_4+\alpha_2\gamma_6 \beta_4\gamma_1=\alpha_4\gamma_3 \beta_4\gamma_2+\beta_5\gamma_5=\alpha_4\gamma_4+\alpha_5\gamma_6 \gamma_3\gamma_5=\gamma_1\gamma_6\end{cases}
\theta_1 = \frac{1}{(1-2r^2)^2}\begin{pmatrix}\phi_1 \phi_2\end{pmatrix}\begin{pmatrix}2-4w_1^2 4w_1w_2 4w_1w_2 2-4w_2^2\end{pmatrix}\begin{pmatrix}\phi_1 \phi_2\end{pmatrix}.
\limsup_{n \rightarrow \infty}\sup_{q \in (0,1)}\sup_{k \in \{k_0^*,\ldots,k_1^*\}}\sup_{f \in \mathcal{F}_{d,\theta}}\mathbb{P}_f\bigl(I_{n,q}\ni H(f)\bigr) - (1-q) \leq \inf_{L \geq 1}\frac{2}{L(2\pi)^{1/2}}= 0.
\begin{aligned}q^{-33}L_3(q^{32}) = \sum_{\substack{n \geq 1 -n \leq j \leq n-1}}q^{(16n-1)^2 - 2(8j+1)^2}+ q^{(16n+1)^2 - 2(8j+1)^2}& + \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{(16n+7)^2 - 2(8j+3)^2}+ q^{(16n+9)^2 - 2(8j+3)^2}.\end{aligned}
v(t,x)=\frac{e^x}{2}\int_{-\infty}^x e^y n(t,y)dy +\frac{e^x}{2}\int_{x}^{\infty}e^{-y}n(t,y)dy, v_x(t,x)=-\frac{e^{-x}}{2}\int_{-\infty}^x e^y n(t,y)dy +\frac{e^x}{2}\int_{x}^{\infty}e^{-y}, n(t,y)dy.
S_{\beta}( x , x^{\prime}) = i \left( - i \partial_{\tau}- \alpha \sigma^{( r )}\partial_{r}+ \frac 1 r \sigma^{( r )}\vec{\sigma}\cdot \vec{L}+ \frac{1 - \alpha}r \sigma^{r}\right) G_{\beta}( x , x^{\prime}) \ . \
P(J_{i_{1}\cdot\cdot\cdot i_{r}})=\sqrt{\frac{N^{r-1}}{J^{2}\pi r!}}exp\{-\frac{N^{r-1}}{J^{2}r!}(J_{i_{1}\cdot\cdot\cdot i_{r}}-\frac{J_{0}r!}{2N^{r-1}})\}
\begin{tabular}{c c c}$m_{14}+n_{12}=\lambda_1$ $n_{14}+l_{12}=\mu_1$ $l_{14}+m_{12}=\nu_1$ \cr$m_{24}+n_{13}=\lambda_2$ $n_{24}+l_{13}=\mu_2$ $l_{24}+m_{13}=\nu_2$ \cr$m_{34}+n_{14}=\lambda_3$ $n_{34}+l_{14}=\mu_3$ $l_{34}+m_{14}=\nu_3$ \end{tabular}
D ( 0 , 0 , \alpha ) = \left( \begin{matrix}{{\cos \alpha}}&{{- \sin \alpha}}&{{0}}&{{0}}\\{{\sin \alpha}}&{{\cos \alpha}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{matrix}\right) .
\|\mathcal{T}h-\mathcal{T}\tilde{h}\|_{\mathcal{B}_{\alpha}^{2}}\leq \left(\frac{1}{1-\alpha}\sum_{j=2}^{\infty}\frac{(r_{0}+r_{1})^{j-1}}{(j-1)!}\right)\|h-\tilde{h}\|_{\mathcal{B}_{\alpha}^{2}}.
\tilde{c}_{21}(f): =\bigg( 2|D|^{-1/2}\kappa_1-(\log|D|)\kappa_0\bigg) \tilde{c}_1(\tilde{d}_2(f)) +|D|^{1/2}\kappa_0 \sum_{v}(\log q_v)\bigg((e(\psi_v)+1)\tilde{c}_{2,v}(I(f_v))-\tilde{a}_{2,v}(f_v)\bigg) \prod_{v'\neq v}\tilde{c}_{2,v'}(f_{v'}),
\frac{\partial^{2}{Z_0}_l(r)}{\partial r_*^{2}}-V(r){Z_0}_l(r)=-\frac{4 \pi q_s m_0 f(r)Y_{l}(0)}{r^{(d-2)/2}}(\frac{dt}{d\tau})_{r_0}^{-1}\delta(r-r_0) \,,
1 = [\nabla^{k,g_P}\eta_i]_{C^{\alpha}(B^{g_{P}}(p_i,\delta_iR_i))}>\; \epsilon [\nabla^{k,g_P}\eta_i]_{C^{\alpha}(B^{g_{P}}(p_i,R_i))}+ i[\nabla^{k-1,g_P}L^{g_P}\eta_i]_{C^{\alpha}(B^{g_{P}}(p_i,R_i))}&{+}\sum_{j=1}^k i R_i^{-k+j-\alpha}\|\nabla^{j,g_P}\eta_i\|_{L^\infty(B^{g_P}(p_i, \delta_iR_i))}.
\mathrm{tr}(R_0(h^1b_{E/F'}(\Sigma^{S'}_{\phi}(X)))) +o_{E,F',n,S_0}(X) &=\sum_{\pi'}\mathrm{tr}(\pi')(h^1)\mathrm{Res}_{s=1}\left( \widetilde{\phi}(s)X^sL(s,(\pi'_E \times \pi'^{\tau \vee}_E)^{S_0})\right)
\langle \theta_{2}, \theta_{1}| T_{\mu}^{\mu}| \theta_{1}, \theta_{2}\rangle_{\mathrm{s y m}}= 8 \pi m^{2}\left( \frac{2 \sin \pi \alpha \cosh \theta_{1 2}}{\sinh^{2}\theta_{1 2}+ \sin^{2}\pi \alpha}\right) \cosh^{2}\frac{\theta_{1 2}}{2}.
\left( \begin{matrix}{T_{1 1}}&{}&{T_{1 2}}\\{T_{2 1}}&{}&{T_{2 2}}\\ \end{matrix}\right) = \left( \begin{matrix}{\exp{i \alpha}\cos{\eta}}&{}&{\exp{i \beta}\sin{\eta}}\\{- \exp{i \gamma}\sin{\eta}}&{}&{\exp{i \delta}\cos{\eta}}\\ \end{matrix}\right) ,
[x_0^2:x_0x_1:x_1^2:x_2] = \varphi(|x_0:x_1:x_2|) = \varphi\left(\lambda^{\frac12}\cdot|x_0:x_1:x_2|\right) = \varphi\left(|\lambda^{\frac12}x_0:\lambda^{\frac12}x_1:\lambda x_2|\right) = [\lambda x_0^2:\lambda x_0x_1:\lambda x_1^2:\lambda x_2].
{\bf M}={\bf \hat{r}}{\frac{i m}{r}}- \mathrm{\boldmath{\hat{\theta}}}{\frac{\partial}{\partial r}}, \quad{\bf N}={\bf \hat{r}}i k{\frac{\partial}{\partial r}}- \mathrm{\boldmath{\hat{\theta}}}{\frac{m k}{r}}-{\bf \hat{z}}d_{m},
\tilde{N}= N + \vec{n}\cdot \left( \frac{1}{L}\, \vec{s}- \frac{1}{L^{2}}\, \vec{n}\right) = N + \frac{1}{L}\, \vec{n}\cdot \vec{s}- \frac{1}{L^{2}}\,{\vec{n}}^{\, 2}
\sup_{y\in B^0\big(0,(1-O(r))r\big)}\frac{d(f(y),f(0))}{r}\geq \big(1-O(r)\big)\frac{\|Df(0)\|d(y_0,0)}{r}-\sup_{y\in B(0,r)}\frac{d(f(y),Df(0)y)}{r}&=\big(1-O(r)\big)^2(1-\varepsilon)\|Df(0)\|-\sup_{y\in B(0,r)}\frac{o\big(d(0,y)\big)}{r}.
T_{f}= \left( \frac{1}{h^{2}}\right)^{x / ( 1 - q y )}\left( \frac{\sqrt{m_{D p}}m^{d_{\perp}- q - 1}}{a B V_{p}}\right)^{y / ( 1 - q y )}\sim \left( \frac{1}{V_{p}}\right)^{\frac{x + y / 2}{1 - q y}}
\frac{|k(\theta)|}{d(f,g)}\le \frac{|k(\theta_0)|}{d(f,g)}+\left|\frac{k(\theta)}{d(f,g)}-\frac{k(\theta_0)}{d(f,g)}\right|<\frac \kappa 2+|l(\theta)-l(\theta_0)|<\frac \kappa 2+\frac \kappa 2=\kappa.
\phi^*(j_1) = 4J_2(3J_{1}^2J_2-2)^5 \ , \phi^*(j_2) = \frac{1}{2}J_2(3J_{1}^2J_2-2)^3 \ , \phi^*(j_3) = 2^{-3}J_2(2J_{1}^2J_2-2)^2(4J_{1}^2J_2+2^5\cdot3^2J_1-3) \ .
\int_{\Delta_{\theta,t}^m}\prod_{j=1}^m f_j(s_j) ds_m \dots ds_1 \int_{\Delta_{\theta,t}^n}\prod_{j=m+1}^{m+n}f_j(s_j) ds_{m+n}\dots ds_{m+1}&= \sum_{\sigma\in S(m,n)}\int_{\Delta_{\theta,t}^{m+n}}\prod_{j=1}^{m+n}f_{\sigma(j)}(w_j) dw_{m+n}\cdots dw_1.
\nabla_{E_i}^{nc}\begin{pmatrix}\rho_j E_j 0 \end{pmatrix}= \begin{pmatrix}* \nabla^g_{E_i}E_j +\rho_j E_i 0 \end{pmatrix}\in \Gamma \left(\mathcal{H}^{\bot}\cap \mathcal{I}^{\bot}\right).
f \star g = \frac{1}{(2\pi )^2}\sum_{m, n, P', P''}\int_{-\pi}^{+\pi}d\theta 'd \theta '' \delta_{P', P_{\theta}+m}\delta_{P'', P_{\theta}-n}e^{[-2iA(\tilde{\bf r}, \tilde{\bf r}', \tilde{\bf r}'')]}f(\tilde{\bf r}') g (\tilde{\bf r''})
{\cal R}_n(g) = \sum_{x \in \xi}g\Bigl(\phi_{l^x_n}\bigl({\cal M}({\cal S}^\prime(\widehat{\pi}^{(r^x_n,0)}, \widehat{\pi}^{(l^x_n,0)}))\bigr),\phi^\prime_{l^x_n}\bigl({\cal M}(\widehat{\cal S}(\widehat{\pi}^{(r^x_n,0)}, \widehat{\pi}^{(l^x_n,0)}))\bigr)\Bigr).
\mathbb{E}\left[e^{uZ_{t}^{i}}\right]=\exp\left\lbrace \int_{0}^{t}\int_{0}^{\infty}\left(e^{z\psi(s,u)}-1\right)\nu_{i}(\mathrm{d}z)\mathrm{d}s\right\rbrace ,i=1,2,\ (t,u)\in\mathbb{R}_{\geqslant0}\times\mathcal{U}.
\|U_k\|_{H^1(B_{R_0})}\le C\sum_{j=0}^{N_k}|\gamma_j^{(k)}|(j+1) &\le C\Big(\sum_{j=0}^{N_k}|\gamma_j^{(k)}|^2(j+1)^{2s}\Big)^{1/2}\Big(\sum_{j=0}^{N_k}(j+1)^{2(1-s)}\Big)^{1/2}<\infty.
L_{0}= ( 2 n + 1 ){\frac{| h |}{2}}+ | h | d_{0}^{\dagger}d_{0}-{\frac{| h |}{2}}- | h | \sum_{k = 1}^{\infty}\left( d_{k}^{\dagger}d_{k}- \tilde{d}_{k}^{\dagger}\tilde{d}_{k}+ a_{k}^{\dagger}a_{k}- b_{k}^{\dagger}b_{k}\right) + L_{0}^{f r e e}.
\xi_p \left [\begin{matrix}a_{11}1 a_{21}a_{22}\end{matrix}\right ]=\left [\begin{matrix}a_{11}+ p 1 a_{21}a_{22}+p\end{matrix}\right ]+ v^{-a_{11}- a_{22}-3}\ \overline{[p]}\left [\begin{matrix}a_{11}+p+1 0 a_{21}-1 a_{22}+p +1\end{matrix}\right ].
ds^{2}=\left[\left(1-\frac{q^{2}}{r_{1}^{2}}\right)\left(1-\frac{p^{2}}{r_{2}^{2}}\right)\right]^{-\frac{1}{2}}dt^{2}-\left[\left(1-\frac{q^{2}}{r_{1}^{2}}\right)\left(1-\frac{p^{2}}{r_{2}^{2}}\right)\right]^{\frac{1}{2}}d\vec{x}^{2}\, .
\displaystyle\check{\mu}(E) =\begin{bmatrix}\int_E\,1\,d\mu(\xi) \int_E\,e^{-2\pi i (x_1-x_2)\cdot \xi}\,d\mu(\xi) \, \, \int_E\,e^{-2\pi i (x_2-x_1)\cdot \xi}\,d\mu(\xi) \int_E\,1\,d\mu(\xi) \end{bmatrix}
\nabla_X(Y\wedge Z) =(\nabla_X Y)\wedge Z+(-1)^{(k-1)l}Y\wedge\nabla_X Z &=(\widetilde{\nabla}_X Y)\wedge Z+F^{k,l}(X,Y)\wedge Z &+(-1)^{(k-1)l}Y\wedge \widetilde{\nabla}_XZ+(-1)^{(k-1)l}Y\wedge F^{k,m}(X,Z).
\langle{\cal T}(a_1){\cal T}(a_2)\rangle =(-1)^2{\delta^2\Gamma_{SC}\over\delta\hat\mu(a_1)\delta\hat\mu(a_2)}\bigg\vert_{\hat\mu= 0}= \kappa\partial^2_1D_1\left({\theta_1-\theta_2\over z_1-z_2-\theta_1\theta_2}\right)
R~=~{\sqrt{\pi \over 2}}\omega^{3/2}\left[{\alpha \over 2}\frac{({\rho \over 2})^{- \epsilon}}{\Gamma(2-\epsilon)}~+~{\beta \over 2}\left(\left({\rho \over 2}\right)^{-\epsilon}\cot \pi (1 - \epsilon)-\frac{\epsilon}{{\mbox{sin}}\pi(1-\epsilon)}\left({\rho \over 2}\right)^{-2+\epsilon}\right) \right]~.
\begin{array}{ll}(|\mathcal{D}^{\alpha}\phi|^2)_t = -2\sum\limits_{|\alpha_1|>0}\mathcal{D}^{\alpha}\phi\mathcal{D}^{\alpha_1}v\cdot\nabla(\mathcal{D}^{\alpha_2}\phi) -v\cdot\nabla|\mathcal{D}^{\alpha}\phi|^2. begin{align*}6pt]\end{array}
E_0(x)> \frac{11}{480}\frac{1}{4x^4}-\frac{29}{336}\frac{1}{5 (x-\frac 12)^5}= \frac{11}{1920}\frac{1}{(x+3)^4}+\frac{\Psi_1(7;x)(x-6)+2164192911}{13440 x^4 (3 + x)^4 (-1 + 2 x)^5}> \frac{11}{1920}\frac{1}{(x+3)^4}.
\begin{array}{l}\mu_+ x^+=q^2x^+\mu_+~,~~~~~\mu_+\partial_+=q^{-2}\partial_+\mu_+~, \Lambda x^k=q^2 x^k \Lambda~,~~~~~\Lambda\partial_k=q^{-2}\partial_k\Lambda~,~~~{\rm for}~k=0,\pm~.\end{array}
\dfrac{M_{2}(\mathcal{C}(G))}{|e(\mathcal{C}(G))|}- \dfrac{M_{1}(\mathcal{C}(G))}{|v(\mathcal{C}(G))|}=\dfrac{2^{6k}(3\cdot2^{k}-11)+2^{k}(8\cdot2^{4k}-6\cdot2^{2k}-1)+2^{2k}(8\cdot2^{2k}-1)}{2^{5k}(2^{2k}-2\cdot2^{k}-3)+(4\cdot2^{4k}-2^{2k}-2)+2^{k}(6\cdot2^{2k}-5)}:= \dfrac{f(k)}{g(k)}
\partial_{s}^{\ell}\kappa - \nabla_{s}^{\ell}\kappa =\left[\sum_{\substack{[\![a,b]\!] \leq [\![\ell-1,2]\!] c\leq \ell-1}}P^{a,c}_{b}(\kappa) \right] T +\sum_{\substack{[\![a,b]\!] \leq [\![\ell-2,3]\!] c\leq \ell-2}}P^{a,c}_{b}(\kappa)
\begin{aligned}g_{_{X,f}}(d) \geq E f(x_\epsilon+dX) &\geq (\sup_x f(x)-2\epsilon)\mu_X(\{y:||dy||<\delta_\epsilon\}) &\ \ \ - |\inf_x f(x)|\mu_X(\{y:||dy||\geq\delta_\epsilon\}) \end{aligned}
(\alpha f_1+\beta f_2)'(x;l) = \alpha f'_1(x;l)+\beta f'_2(x;l) \,, (f_1\cdot f_2)'(x;l) = f_1(x)f'_2(x;l)+f_2(x)f'_1(x;l) \,, \left(\frac{f_1}{f_2}\right)'(x;l) = -\frac{f_1(x)f'_2(x;l)-f_2(x)f'_1(x;l)}{[f_2(x)]^2}
K(a_1-1,\ldots,a_d) = (a_1-1)K(a_2,\ldots,a_d)+K(a_3,\ldots,a_d) = a_1K(a_2,\ldots,a_d)+K(a_3,\ldots,a_d)-K(a_2,\ldots,a_d) = K(a_1,\ldots,a_d)-K(a_2,\ldots,a_d) =m-n.
\{(x^{(0)}_k,x^{(1)}_k,\dots,x^{(\ell)}_k;\,t^{}_{1,k},\dots,t_{\ell, k};\,\xi^{(0)}_k,\xi^{(1)}_k,\dots,\xi^{(\ell)}_k;\,\tau_{1,k},\dots,\tau_{\ell, k})\}_{k=1}^\infty
\frac{1}{2\pi i}\int\limits_{(2A+1)/\log q -iq}^{(2A+1)/\log q +iq}\frac{\zeta(1-w_2+2it)\zeta(1+w_2+2\delta-2it)H_2(-2\delta-w_2,w_2,\delta,t)}{\zeta(1-2\delta-w_2)\zeta(1-w_2-2\delta+2it)\zeta(1+w_2)\zeta(1+w_2-2it)}\hskip 240pt \cdot \Psi_{\Upsilon,M}(-2\delta-w_2) \Psi_{\Upsilon,M}(w_2) dw_2.
0= \int_{\Delta'}(\dot{u}^*_{t,i,j}u_t^{*,i,j}P_{DH}'+\dot{u}^*_{t,i}u_{t,j}^{*,i,j}P_{DH}'-\dot{u}^*_{t}u_{t,j}^{*,i,j}P_{DH,i}'-\dot{u}^*_{t}u_t^{*,i,j}P_{DH,i,j}')dp
v^i(x_1^\ast,y_1^\ast) = \sum\limits_{x_2,\dots,x_{k-1},x_k\in X}p(x_k|x_{k-1},\mu_{k-1}^\ast) p(x_{k-1}|x_{k-2},\mu_{k-2}^\ast)\dots & p(x_2|x_1^\ast,\mu_1^\ast) v^i(x_k,y_k^\ast).
\Psi ( t , \theta , \phi ) = e^{- i E t}\exp{i ( n + \frac{1}{2}- \frac{1}{2}\sigma_{_{3}}) \phi}\left( \begin{array}{c}{{A J_{_{| \frac{1}{\beta}( n + \frac{1 - \beta}{2}) |}}( \alpha c r )}}\\{{B J_{_{| \frac{1}{\beta}( n + \frac{1 - \beta}{2}) + 1 |}}( \alpha c r )}}\\ \end{array}\right) ,
\sum_{k=0}^{n-1}\frac{(q,q^{-1};q^2)_kq^{2k}}{(q^2;q^2)_k^2}\equiv 0~n\geq 3~~n\equiv1\pmod 2, \sum_{k=0}^{n-1}\frac{(q^4,q,q^{-2},;q^3)_kq^{3k}}{(q^3;q^3)_k^3}\equiv 0\quad~n\geq 5~~n\equiv 2\pmod 3.
X =\frac{1}{2}\left[ -q_{N}(1 - q_{N}) + \sum_{j=1}^4 \Theta^{(N)}_{\gamma_j}(1 - \Theta^{(N)}_{\gamma_j}) - \sum_{\Gamma_{cut}}\Theta^{(N)}_{\gamma_+}(1 - \Theta^{(N)}_{\gamma_+}) \right].
\| \Psi' \|_{H^{-1}}=\sup_{0\neq v\in H_0^1(\Omega)}\frac{|\langle \Psi', v\rangle|}{\|v\|_{H^1}}\leq \tilde{C}(1+\|\Psi_0\|_{L^2}^2+\|F\|_Y^2) \bigl( \| F \|_{L^2}+ \| \Psi \|_{H^1}\bigr) .
\left \|\sum\limits_{j=1}^{\infty}\lambda_{j}a_{j}\right \|_{h_{\omega}^{p}}\leq C\left \|\sum\limits_{j=1}^{\infty}\frac{\lambda_{j}\chi_{Q_{j}}}{\omega(Q_{j})^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}
\widehat{M}^{n+1}f\left( x\right) =\widehat{M}\widehat{M}^{n}f\left(x\right) & \quad=\sum_{k=0}^{n}\binom{n}{k}x\phi_{n-k}\left( x\right) \left(xD\right)^{k}f\left( x\right) +\sum_{k=0}^{n}\binom{n}{k}\left(xD\right) \left[ \phi_{n-k}\left( x\right) \left( xD\right)^{k}f\left(x\right) \right] .
\begin{gathered}t^{r+r_0+2\ell}\Lambda^{\prime}\subset\nabla(t^{r+r_0+\ell}\Lambda) + I t^{r+\ell}\Lambda^{\prime}\subset\nabla(t^{r+r_0+\ell}\Lambda) + I (\nabla(t^r\Lambda)+I t^{r-r_0}\Lambda^{\prime}) = \nabla(t^{r+r_0+\ell}\Lambda) + I \nabla(t^r\Lambda) \subset\nabla(t^r\Lambda)\end{gathered}
g_1\cdot p = \lambda_1(-\tfrac{1}{2}e_{126}-\tfrac{1}{2}e_{349}+2e_{358}-e_{457}+e_{789}) + \lambda_2(-2e_{137}-\tfrac{1}{4}e_{249}-e_{258}-\tfrac{1}{2}e_{456}-\tfrac{1}{2}e_{689})
- \sum_{A , B}\sum_{\vert J \vert = 1}\sum_{\vert K \vert = 1}\int_{\Omega}D_{J}\biggl ( E_{A}^{J}( h ) I_{A B}^{( K )}E_{B}^{K}( f ) \biggr ) - \sum_{A , B}\sum_{\vert K \vert = 2}\int_{\Omega}E_{A}^{0}( h ) I_{A B}^{( K )}E_{B}^{K}( f ) .
\Lambda(\beta)=[\begin{matrix}\frac{1}{\sqrt{1-\beta^{2}}}&-\frac{\beta}{\sqrt{1-\beta^{2}}}&0&0\\ -\frac{\beta}{\sqrt{1-\beta^{2}}}&\frac{1}{\sqrt{1-\beta^{2}}}&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix}]
H_4(f, g)\ = \ \sum_{\mu_d = 1}^f \sum_{\mu_1 = 1}^{f- \mu_d}\frac{1}{\mu_1 ! \mu_d !}\binom{f - \mu_1 - \mu_d}{g - \mu_1}\sum_{d=1}^{f- \mu_1 - \mu_d}\sum_{\substack{\mu_2 + \cdots + \mu_{d-1}= f - \mu_1 - \mu_d \mu_i \ge 1}}\frac{(-1)^{d-2}}{\mu_2! \cdots \mu_{d-1}!}.
\sigma_{\mathrm{t o t}\, i}^{( 1 )}= \frac{( 2 \pi )^{4}\, \hbar^{2}}{v_{i}}\frac{1}{\pi}\mathrm{I m}R_{i i}^{( 1 )}\exp \left[ - \frac{1}{\hbar}\int_{- \infty}^{+ \infty}d t \, \Gamma^{( 1 )}( t ) \right] \, .
df^*(E_i)=D_{E_i}f^*=(D_{E_i}\bar{p},D_{E_i}(q\bar{p}))=(\overline{D_{E_i}p}, (D_{E_i}q)\bar{p}+q(\overline{D_{E_i}p}))\overset{\eqref{df2}}{=}(-\alpha_i\bar p, q(\beta_i-\alpha_i)\bar p)
{\not \! \nabla}^{2}= \nabla^{\mu}\nabla_{\mu}- e_{a}^{\mu}e_{b}^{\nu}e_{c}^{\lambda}J^{a b}T_{\mu \nu}^{c}\nabla_{\lambda}+ \frac{1}{2}e_{a}^{\mu}e_{b}^{\nu}J^{a b}J^{c d}R_{c d \mu \nu},