harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\rho ={\frac{n a_{H}^{n}}{1 6 \pi G_{n + 2}a^{n + 1}}}\left({\frac{a_{H}}{L}}+ k{\frac{L}{a_{H}}}+{\frac{n w_{n + 1}^{2}Q^{2}L}{8 ( n - 1 ) a_{H}^{2 n - 1}}}\right) ,
\prod_{n = 0}^{\infty}\left[ ( 1 - e^{- \alpha \epsilon n + ( \pi - \alpha ) \epsilon + i \epsilon m \beta_{1 2}}) ( 1 - e^{- \alpha \epsilon n + ( \pi - \alpha ) \epsilon - i \epsilon m \beta_{1 2}}) ( 1 - e^{- \alpha \epsilon n - \alpha \epsilon + i \epsilon m \beta_{1 2}}) ( 1 - e^{- \alpha \epsilon n - \alpha \epsilon - i \epsilon m \beta_{1 2}}) \right] .
\begin{matrix}Var(\sum x_{i})=\sum Var(x_{i})=nVar(x_{i})=n\sigma^{2}\\Var(\overline{X})=Var(\frac{\sum x_{i}}{n})=\frac{1}{n^{2}}Var(\sum x_{i})=\frac{n\sigma^{2}}{n^{2}}=\frac{\sigma^{2}}{n}\end{matrix}
V_{0}(\sigma)=\frac{\sigma^{2}}{2\lambda_{0}}- \int_{0}^{\sigma}ds\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{s}{(4\pi)^{\frac{D-1}{2}}}(\omega^{2}+s^{2})^{\frac{D-3}{2}}\Gamma(-\frac{D-3}{2}) \mbox{tr}\mbox{\boldmath$1$}.
0 = G^\prime ( \widetilde{y}, \widetilde{z}) \widetilde{g}( \widetilde{y}, \widetilde{z}) = F^\prime (\widetilde{y}) \widetilde{g}_Y ( \widetilde{y}, \widetilde{z}) - \widetilde{g}_Z ( \widetilde{y}, \widetilde{z}) ,
\big[\mathcal{K}'_{\mathcal{O}}(t)\big]^2:=\int_{\mathbb{R}}|\widehat{\mathcal{O}u}(\xi,t)|^2\cdot \big(|\xi|^{-1}+|\xi|^{2N_{\mathcal{O}}}\big)\mathcal{P}((1+t)^2|\xi|)\,d\xi.
\lim_{z\to 0}\left(\mathcal{D}_z^{2a}\left(\frac{\zeta^d}{1-\zeta^\ell}\right)+\frac{(2a)!}{\ell \left(2\pi i z\right)^{2a+1}}\right)= -\ell^{2a}\frac{B_{2a+1}\left(\frac{d}{\ell}\right)}{2a+1}.
i^{*}{\rm div}\left(\widetilde{J}_{{\Bbb A}_{1}^{+}}^{*}\left(\frac{\chi_{10}}{\theta_{a_{0},b_{0}}}\right)^{8}|_{U}\right)=\sum_{\mu\in\Lambda/\pm1,\,\mu^{2}=-4}\sum_{(a,b)\not=(a_{0},b_{0})}\{c_{a,b}(\mu+\frac{d}{2})+c_{a,b}(\mu-\frac{d}{2})\}\,H_{\mu}.
\left(\begin{matrix}x=0 x=\infty \, \left( \frac12 \right) \begin{matrix}0 0 \theta^0 \end{matrix}\overbrace{\begin{matrix}1 t_2 -t_1/2 \theta^\infty_1/2 -1 t_2 t_1/2 \theta^\infty_1/2 0 0 0 \theta^\infty_2 \end{matrix}}\end{matrix}\right) ,
\langle \chi_{\upsilon(\jmath)}\vartheta_{Z_\jmath}, \eta_{\varsigma(\jmath)}\rangle_{Z_\jmath^F}= \prod_j{|I_j| \choose \nu_j}{|I_j'| \choose \nu_j'}\nu_j! \nu_j'!(\kappa_G \cdot j)^{\nu_j+ \nu_j'}.
\psi(a_1,a_2,e_1,e_2) = q \circ (\psi_1 \times \psi_2)(a_1,e_1,a_2,e_2) &= q \circ (id_{[0,\infty)^{s_1}}\times \overline{\psi}_1 \times id_{[0,\infty)^{s_2}}\times \overline{\psi}_2)(a_1,e_1,a_2,e_2) &= (a_1,a_2,d_1,d_2,w_1,w_2).
E_C^{\mbox{fixed}}(R\to\infty,T)=\frac{RT}{2}\sum_{n=-\infty}^{+\infty}\Omega_n=2\pi R T^2\sum_{n=1}^{\infty}n=2\pi R T^2\zeta (-1)= - \frac{\pi R T^2}{6}\,{,}
C_{CSCC}^{L}(P) = \max_{X_1^L \in \mathcal{T}_{P}^{L}}\frac{I(X_1^L ; Y_1^L)}{L}&= \max_{X_1^L \in \mathcal{T}_{P}^{L}}\left( \frac{H(Y_1^L)}{L}- \frac{H(Y_1^L | X_1^L)}{L}\right) &= \max_{X_1^L \in \mathcal{T}_{P}^{L}}\left( \frac{H(Y_1^L)}{L}- \frac{\sum_{i=1}^{L}H(Y_i | X_i)}{L}\right)
(\Gamma \circ \varphi(g')) \big((g,q) \big) = \Gamma \big( (g'g,q) \big) &= \big(\alpha'(g'g),\alpha(q)\big) &= \big(\alpha'(g')\alpha'(g),\alpha(q)\big) &= \varphi \big(\alpha'(g') \big) \big((\alpha'(g),\alpha(q)) \big) &= (\varphi\big(\alpha'(g')\big) \circ \Gamma)\big((g,q) \big).
(\psi_{\pi/2}''(0)^{-1}\partial, \partial)^{(k_1+1)/2}a_{k_1,k_2,\pi/2}(0) &= (-1)^{k_1}\frac{i^{k_2-n-\frac{k_1+1}{2}}}{2^{\frac{k_1+1}{2}}}\left(i\frac{\pi}{2}\right)^{n+k_1+k_2-1}(k_1+1)! (n+k_1+k_2+m-1).
\left[\begin{array}{cc}\rho_{1}\sigma_{1}\sigma_{1}\tau_{1}\end{array}\right]=\left[\begin{array}{cc}\frac{1}{\alpha_{1}^{2}}-\frac{\beta_{1}}{\alpha_{2}\alpha_{1}^{2}}-\frac{\beta_{1}}{\alpha_{2}\alpha_{1}^{2}}\frac{1}{\alpha_{2}^{2}}+\left(\frac{\beta_{1}}{\alpha_{2}\alpha_{1}}\right)^{2}\end{array}\right]=B_{2}^{-T}B_{2}^{-1},
\int\limits_{\mathbb T^d}h(\xi) v_0(\xi) \d \xi \ = \ \int\limits_{\mathbb R^d}\int\limits_{\mathbb T^d}a (\xi-q) \mu (\xi, q ) (\xi - q ) \dq \, v_0 (\xi) d \xi \ - \b \int\limits_{\mathbb T^d}\, v_0 (\xi) d \xi \ = \ 0.
\Omega_{j k}^{( \alpha \beta )}\, = \, \frac{2 R_{j}^{( \alpha \beta )}R_{k}^{( \alpha \beta )}- ( R^{( \alpha \beta )})^{2}\delta_{j k}}{( R^{( \alpha \beta )})^{4}}, R^{( \alpha \beta )}\equiv R^{( \alpha )}- R^{( \beta )}\; .
(p_2 - p_1 + s) (F E_{1}^{n}F_{2}^{m}\otimes F_2) \cdot w_s = (p_2 - p_1 + s)mn(p_1 + 2p_2 - m) (E_{1}^{n-1}F_{2}^{m-1}\otimes F_2) \cdot w_s & -(n + m + p_1 - p_2 - (s-1))(E_{1}^{n}F_{2}^{m}\otimes F_2) \cdot w_{s+1}.
{\cal L}_{2}= e^{2}\left( - f^{a b c}f^{a d e}A_{\mu}^{b}A^{d \mu}A_{\nu}^{c}A^{e \nu}- \frac{\mu^{2}}{e^{2}v^{2}}\left| \Phi^{\prime}\right|^{4}+ \frac{1}{2}\, A_{\mu}^{a}A^{b \mu}\Phi^{\prime \dagger}\{\lambda^{a}, \lambda^{b}\}\Phi^{\prime}\right) .
\begin{array}{ll}\displaystyle{h^{*}(\theta,x)=h'(\sqrt{2\theta},x)}\displaystyle{b^{*}(\theta,x)=b'(\sqrt{2\theta},x)}\displaystyle{d^{*}(\theta,x)=\frac{d'(\sqrt{2\theta},x)}{\sqrt{2\theta}}}\displaystyle{v^{*}(\theta,x)=\frac{v'(\sqrt{2\theta},x)}{\sqrt{2\theta}}}\end{array}
\vec r={1\over4}\left( \begin{array}{c}-2(\cos{\theta_2}\cos{u}+ \sin{\theta_2}\sin{u}) + \cos{\psi}\sin{2v}2 (\cos{u}\sin{\theta_2}- \cos{\theta_2}\sin{u}) - \cos{2v}\sin{\psi}\sin{2v}\end{array}\right).
C \|f|W^{1}_{1}(\mathbb{R}^{n+1}_{+},\gamma)\|\le \sum\limits_{m \in \mathbb{Z}^{n}}\widehat{\gamma}_{0,m}\|\varphi|L_{1}(Q_{0,m})\|+ \sum\limits_{s=1}^{\infty}\sum\limits_{\alpha \in \widetilde{A}^{s}}\widehat{\gamma}^{s}_{\alpha}\int\limits_{\widetilde{Q}^{s}_{\alpha}}|\varphi^{s}_{\alpha}-\varphi(x)|\,dx .
F_{QR,P}(\Omega)(\zeta_i \otimes \alpha_j)= (2g_C+1)\left(\int_{C-\gamma}\log(f) dz_i \wedge dx_j + 2\pi i \int_{\gamma}dz_i dx_j \right) = (2g_C+1) \int_{\zeta_i}\log(f) dx_j - N \int_{\zeta_i}W(dx_j).
\|\hat g\|_S \leq \lim_{n\to\infty}\|\hat g - \hat g_n\|_S + \|\hat g_n - \hat g_0\|_S+\|\hat g_0\|_S & \leq 2\|\hat g_1 - \hat g_0\|_S + \|\hat g_0\|_S = 2\| F(-W) - F(0)\|_S + \|W\|_S \leq 2\|W\|_S.
\langle B\phi_k,\phi_k\rangle \geq \epsilon_{2}\int_{\ell_{2}}^{\ell_{2}+ \delta_{2}}|\phi_{k}'(x)|^2dx ={2\epsilon_{2}k^2 \over \pi}\int_{\ell_{2}}^{\ell_{2}+ \delta_{2}}\!\!\! \cos^2(kx)\,dx \geq c\, k^2.
M_{i}(q_{1},\ldots ,q_{N}):=L_{iN}\left( \begin{smallmatrix}1 0 0 q_{N}\end{smallmatrix}\right)_{i}\cdots L_{i2}\left( \begin{smallmatrix}1 0 0 q_{2}\end{smallmatrix}\right)_{i}L_{i1}\left( \begin{smallmatrix}1 0 0 q_{1}\end{smallmatrix}\right)_{i}\, ,
{\bf C}_n=\begin{pmatrix}0 0 \Phi_{02}0 \dots 0 0 0 0 \Phi_{13}\dots \Phi_{1(n-1)}\Phi_{20}0 0 0 \dots 0 0 \Phi_{31}0 0 \dots \Phi_{3(n-1)}0 0 \Phi_{42}0 \dots 0 \vdots \vdots \vdots \vdots \ddots \vdots 0 \Phi_{(n-1)1}0 \Phi_{(n-1)3}\dots 0\end{pmatrix}.
\begin{aligned}X^{N,i}_t(k) =X^{N,i}_0(k)+\int_0^t\mathcal{A}^NX^{N,i}_s(k)\,ds &\quad+\int_0^t\int_0^{I^N_{s-}(k)}\int_{E}J_i( y,X^N_{s-}(k))\big(\mathcal{N}-\mathcal{N}^{\prime}\big)\big(\{k\},dy,dr,ds\big),\end{aligned}
|\Re C_n| = \bigg|\frac{\alpha_n^{1-2\gamma}\sqrt{\rho_2}c}{\delta^2}+\Re t_n\bigg|\geq \frac{\alpha_n^{1-2\gamma}\sqrt{\rho_2}c}{\delta^2}-| \Re t_n|\sim \frac{\alpha_n^{1-2\gamma}\sqrt{\rho_2}c}{\delta^2}\to \infty.
L = \frac{1}{2}\frac{m}{1+\lambda r^2}\left(\sum_i v_i^2 + \lambda \sum_{i<j}(x_iv_j - x_jv_i)^2\right) -{\cal V}(r),{\cal V}(r) =\frac{1}{2}m \alpha^2 \frac{r^2}{1+\lambda r^2}.
\frac{\partial}{\partial y_l}\bar{H}(y) = \frac{1}{M}y_l + \frac{1}{N}\sum_{j \in B(l)}s_j + \frac{1}{N}\mathbb{E}_{\mu_{N,m}(dx|y)}\left[ \sum_{i=1}^N \sum_{j \in B(l)}M_{ij}X_i + \sum_{i \in B(l)}\delta \psi ' (X_i)\right].
\mathcal{U}_{1}(\tilde{g}) - \min_{g_q\in \mathbb{Q}}\mathcal{U}_{1}(\tilde{g}g_q) = (1-\cos\theta^*) \max_{u_q\in \mathbb{U}}\Delta_Q(u_q,v), \forall v\in \mathcal{E}(Q) & \geq (1-\cos\theta^*) \min_{v\in \mathcal{E}(Q)}\max_{u_q\in \mathbb{U}}\Delta_Q(u_q,v) & \textstyle = (1-\cos\theta^*) \Delta^*_Q > \delta,
|\widehat{\widetilde{H}}(r_1,r_2,\xi)|, |\widehat{H}(r_1,r_2,\xi) | \lesssim\begin{cases}(\lambda r_1 r_2)^{-\frac 12}|\xi| \lesssim ( \lambda r_1 r_2)^{\frac{1}{2}}\frac{1}{|\xi|}|\xi| \gg ( \lambda r_1 r_2 )^{\frac{1}{2}}\end{cases}.
A_{j}A_{k}\mathcal{L}^{n-2}=[A_{k}]_{A_{j}}\mathcal{L}_{A_{j}}^{n-2}=\sum_{k_{1}+...+k_{r}=n-2}\frac{(n-2)!}{k_{1}!...k_{r}!}a_{1j}^{k_{1}}...a_{jj}^{k_{j}}...a_{kj}^{k_{k}+1}...a_{rj}^{k_{r}}l_{j}
\begin{array}{c}{{d \tau = \frac{d^{4}k_{1}}{( 2 \pi )^{4}}\frac{d^{4}k_{2}}{( 2 \pi )^{4}}( 2 \pi )^{4}\delta^{4}( p_{1}+ p_{2}- k_{1}- k_{2}) \pi \delta ( k_{1}^{2}- M^{2}) \theta ( k_{1}^{0})}}\\{{\times \pi \delta ( k_{2}^{2}- M^{2}) \theta ( k_{2}^{0})}}\\ \end{array}
\MoveEqLeft \int e^{-3i \Phi(\xi,\eta)}\frac{1}{3\partial_{\eta}\Phi(\xi,\eta)}\partial_\eta K(S,S)(\eta) \bar S(\eta-\xi) (\varphi_1(\eta/\xi) - \phi(\xi^{\gamma}\eta)) d\eta = \int_{\xi^{-\gamma}/2}^{2\xi}O \left(\frac{1}{\xi^2|\eta|^{3/2}}\right)= O(|\xi|^{-2+\gamma/2}).
dZ_{n}(t) \le e^{- \int_0^t V_{n}(X(s)) ds}[ V_{n}(X_{n}(t)) - V_{n}(X_{n}(t))] \, |\eta_{n}(t)|^2 dt + 2 \sum_{i=1}^d \langle \eta_{n}(t), (\sigma_{n})_i'(X_{n}(t)) \eta_{n}(t) \rangle dW^i(t)\Big]
\simeq \frac{-\lambda_1(a^+ \Gamma_1 \Lambda_3) + i\lambda_2 (\Lambda_3 \Gamma_2 \Gamma_3 a^+)}{\sqrt{2}}u_R (\lambda_1, \lambda_2) \mid 0 \, \rangle \mid \psi (R, \Lambda_3) \, \rangle.
\Omega ( p ) = \frac{4 \Lambda^{2}c^{2}}{\pi}\left[ 2 \lambda_{0}+ 8 \pi \frac{\epsilon}{\Lambda^{2}}\partial_{1}^{2}I_{1}( 0 ) \right] e^{- 8 \pi I_{1}( 0 )}+ \frac{4 c^{2}\epsilon}{\pi}e^{- 8 \pi I_{1}( 0 )}p_{1}^{2}
\frac{1}{4}\left[ k^{2}- G^{- 2}( k ) \right] + \frac{\pi^{2}}{g^{2}}\left[ 2 z k^{- 2}G^{- 3}( k ) - 4 \pi^{2}\frac{\delta z}{\delta G ( k )}\int \frac{d^{2}p}{( 2 \pi )^{2}}p^{- 2}G^{- 2}( p ) \right] = 0
\begin{pmatrix}1 -1 2 0 0 0 0 0 \cdots 0 0 1 -1 2 0 0 0 \cdots 0 0 0 0 1 -1 2 0 \cdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \vdots \ddots\end{pmatrix}\begin{pmatrix}x_0 y_0 x_1 y_1 \vdots\end{pmatrix}=\vec 0.
\sum_{m~\chi\notin \mathfrak{M}_m}|\hat{1_S}(\chi)|^3 = \binom{n}{m}\sum_{\substack{\chi_1,\dots,\chi_m\neq 0 (\chi_1,\dots,\chi_m,0^{n-m})\notin \mathfrak{M}_m}}|\hat{1_S}(\chi_1,\dots,\chi_m,0,\dots,0)|^3.
\textrm{rank}([{\bf V}_{2,3}^p~{\bf V}_{2,3}^c~{\bf V}_{2,3}^r])=d_{2,3}, &\textrm{rank}([{\bf V}_{3,1}^p~{\bf V}_{3,1}^c~{\bf V}_{3,1}^r])=d_{3,1}.
\int_M |\nabla_gu(x)|^2{\rm d}v_g -\mu\int_M\frac{u^2(x)}{d^2_g(x_0,x)}{\rm d}v_g+\int_Mu^2(x){\rm d}v_g=\lambda\int_M \alpha(x)\xi_x u(x){\rm d}v_g,
\begin{bmatrix}0 p 0 0 p 0 0 0 0 0 0 p 0 0 p 0\end{bmatrix},\begin{bmatrix}0 p^2 0 0 1 0 0 0 0 0 0 p 0 0 p 0\end{bmatrix},\begin{bmatrix}0 p^2 0 0 1 0 0 0 0 0 0 p^2 0 0 1 0\end{bmatrix}.
\Delta_{\tilde{g}}\Phi =-u^{-1}\Delta_gu - u^{-1}\underbrace{(\tilde{g}^{ij}-g^{ij})\nabla_j\nabla_iu}_{O_{3}(r^{-2\delta-2})}+ O_3(r^{-2\delta - 2})=-u^{-1}\Delta_gu + O_3(r^{-2\delta-2}), &=-c_nR_g + O_3(r^{-2\delta-2}),
\lim_{\epsilon \to 0}\int_{0}^{\epsilon}d \sigma \int d^{3}p ~ \frac{e^{- i k \times p \sigma}}{p^{2}( k - p )^{2}}\left\{( k \times p ) ( k_{\mu}l_{\nu}- k_{\nu}l_{\mu}) - l^{2}( k_{\mu}p_{\nu}- k_{\nu}p_{\mu}) \right\}= 0 .
|u|_L:= \|\boldsymbol{D}^s u\|_{L^2(\mathbb{R})}~~~~\widetilde{W}^{s}_L(\mathbb{R}) ~~~~ |u|_R:= \|\boldsymbol{D}^{s*}u\|_{L^2(\mathbb{R})}~~~~\widetilde{W}^{s}_R(\mathbb{R}),
M_1 = \ \sum_k [\bar R_k Y_{<k,\alpha}- R_{<k,\alpha}\bar Y_k ]+ [R_k \bar Y_{<k,\alpha}- \bar R_{<k,\alpha}Y_k], M_2 = \ \sum_k\partial_\alpha(\bar P [ \bar R_k Y_k] + P [R_k \bar Y_k])- (\bar R_{k,\alpha}Y_k + R_{k,\alpha}\bar Y_k).
\bar h_{\bf i}= \sum_{{\bf j}\in \mathcal{P}(n, k)}h_{\bf j}+ \sum_{1 \leq \ell \leq k}(-1)^{\ell}\sum_{\substack{{\bf j'}\in \mathcal{P}(n, \ell) :{\bf j'}\subset{\bf i}}}h_{\bf j'},
\mathcal O_{X^{(s)}}(k(K_{X^{(s)}/Y}+\Delta_{X^{(s)}})) \simeq p_s^*\mathcal O_X(k(K_{X/Y}+\Delta))\otimes q_s^*\mathcal O_{X^{(s-1)}}(k(K_{X^{(s-1)}/Y}+\Delta_{X^{(s-1)}})),
h(s) = \begin{dcases}1 (p<p_{subc}(n,\alpha)), \log(t_0+s) (p = p_{subc}(n,\alpha)), (t_0+s)^{\frac{1}{2-\alpha}\left( n - \alpha \frac{p+1}{p-1}\right)}(p> p_{subc}(n,\alpha)). \end{dcases}
\lfloor\frac{-2mp^i}{q-1}\rfloor+\lfloor\frac{mdp^i}{q-1}\rfloor+\lfloor\frac{-m(d-1)p^i}{q-1}\rfloor-\lfloor\frac{-mp^i}{q-1}\rfloor+1 &=\sum_{h=1}^{d-2}\lfloor\langle\frac{hp^i}{d-1}\rangle-\frac{mp^i}{q-1}\rfloor+\lfloor\langle\frac{p^i}{2}\rangle-\frac{mp^i}{q-1}\rfloor+\sum_{h=1}^{d-1}\lfloor\langle \frac{-hp^i}{d}\rangle+\frac{mp^i}{q-1}\rfloor.
\Psi_{1}= b_{\alpha}^{\dot{\alpha}}\left( \bar{D}_{\dot{\alpha}}\sigma^{\alpha}+ k \beta_{\dot{\alpha}}^{\alpha}\right) + \left( \bar{\sigma}^{\dot{\alpha}}D_{\alpha}+ \bar{k}\bar{\beta}_{\alpha}^{\dot{\alpha}}\right) \bar{b}_{\dot{\alpha}}^{\alpha}.
\langle \bar W^4| W^4|W^4|\bar W^4 \rangle = \frac{1}{x^8_{13}x^8_{42}}\left[a_2 + \frac{b_4}{s}+ \frac{c_1}{s^2}+ \frac{b_1}{s^3}+ \frac{a_1}{s^4}\right] \;,
\lim_{\ell\to\infty}\frac{\langle H\rangle^{\ell}_{min}}{E^{\ell}_{0}}=\lim_{\ell\to\infty}\frac{(\ell +1)^2}{(\ell +\frac{3}{2})}\Big(\frac{\Gamma(\ell+1)}{\Gamma(\ell+\frac{3}{2})}\Big)^2=1.
\left(\begin{array}{l}f_n(r) g_n(r)\end{array}\right)=\left(\begin{array}{c}C^{(1)}_n(E)J_{n-\Phi^{(0)}+\Upsilon}(kr)+C^{(2)}_n(E)Y_{n-\Phi^{(0)}+\Upsilon}(kr){ik\over E+m}\bigl[C^{(1)}_n(E)J_{n+1-\Phi^{(0)}+\Upsilon}(kr)+C^{(2)}_n(E)Y_{n+1-\Phi^{(0)}+\Upsilon}(kr)\bigr]\end{array}\right),
C_{\operatorname*{DtN},k}:=\left\Vert T_{k}\right\Vert_{\mathbf{H}_{\operatorname*{div}}^{-1/2}\left( \Gamma\right) \leftarrow\mathbf{H}_{\operatorname*{curl}}^{-1/2}\left( \Gamma\right)}<\infty.
\begin{array}{l}\qquad H_+=\frac{8}{\pi}\sum\limits^\infty_{-\infty}L_ne^{i2n(\tau-\sigma)}, \qquad L_n\equiv\frac12\sum\limits^\infty_{-\infty}\alpha^\mu_{n-k}\alpha^\mu_k=0, \qquad H_-=\frac{8}{\pi}\sum\limits^\infty_{-\infty}\bar L_ne^{i2n(\tau+\sigma)}, \qquad \bar L_n\equiv\frac12\sum\limits^\infty_{-\infty}\bar\alpha^\mu_{n-k}\bar\alpha^\mu_k=0,\end{array}
\prod_{\substack{\gcd(j_1,j_2,j_3,j_4,j_5,k)=1 j_1,j_2,j_3,j_4,j_5<k j_1,j_2,j_3,j_4,j_5\geq 1; k \geq 2}}\left(\frac{1}{1-y^{j_1+j_2+j_3+j_4+j_5}z^k}\right)^{\frac{1}{k}},
\int_0^t \frac{aq^-}{p}{\Vert u_{k}\Vert}^2d\tau \le \ \big({\Vert x \Vert}^2 + 2\int_0^k\tau b \Vert f\Vert \cdot \Vert u_{k}\Vert \, d\tau \big) + \int_0^t \frac{aq^+}{p}{\Vert u_{k}\Vert}^2d\tau \le \ \big({\Vert x \Vert}^2 + 2DE \big) + E^2a_+(\infty)\Big(\int_0^{\infty}\frac{q^+}{p}\, d\tau \Big) :=F<\infty.
\delta^{s}S \simeq \int_{t_{1}}^{t_{2}}d t \; \epsilon \; \left[ U_{_{( \nu )}}^{s}\Omega_{_{( \nu )}}- y^{s}\Omega_{_{( 1 )}}- z_{_{( \nu )}}^{s}\left({\frac{\partial \Omega_{_{( \nu )}}}{\partial \pi}}\pi - \Omega_{_{( \nu )}}\right) \right] \; .
d_{S(G,t+1)}(x^{t+1},y^{t+1})\le d_{S(G,t+1)}(x^{t+1},x(u_1)^t)+\sum_{i=0}^{s-2}d_{S(G,t+1)}(u_{i+1}(u_i)^t,u_{i+1}(u_{i+2})^t)+ &+d_{S(G,t+1)}(y(u_{s-1})^t,y^{t+1})+s = 2^t-1+\sum_{i=0}^{s-2}2\left(2^t-1\right)+2^t-1+s = 2^t-1+2\left(2^t-1\right)(d_G(x,y)-1)+2^t-1+d_G(x,y) = (2^{t+1}-1)d_G(x,y).
\frac{\partial U}{\partial z_{1}}(\sqrt{\tilde{A}(0)}z) =U^{\prime}(\sqrt{\tilde{A}(0)}z)\frac{z_{1}}{|z|}, \frac{\partial}{\partial z_{1}}\left( \frac{U^{\prime}(\sqrt{\tilde{A}(0)}z)}{|z|}\right) =\left( \frac{\sqrt{\tilde{A}(0)}U^{\prime\prime}(z)}{|z|^{2}}-\frac{U^{\prime}(z)}{|z|^{3}}\right) z_{1},
\frac{E_{s}(\alpha_{N})-I_{s}(\sigma) N^{2}}{N^{1+s}}=\sum_{k=1}^{\tau(N)}\left(\frac{2^{n_{k}}}{N}\right)^{1+s}\left[2^{s+1}\,\mathcal{R}_{s}(2^{n_{k}+1})\,\sum_{j=k+1}^{\tau(N)}2^{n_{j}-n_{k}}+\mathcal{R}_{s}(2^{n_{k}})\left(1-\sum_{j=k+1}^{\tau(N)}2^{n_{j}-n_{k}+1}\right)\right].
|\{u\leq M\}\cap G(y,3r/2)| =|T\bigl(\{\tilde{u}\leq M\}\cap G((y_1/|y_1|,0),3r/(2|y_1|))\bigr)| &=|y_1|^3|\{\tilde{u}\leq M\}\cap G((y_1/|y_1|,0),3r/(2|y_1|))| &\geq \frac{\nu}{2}|y_1|^3|G((y_1/|y_1|,0),3r/(2|y_1|))| &\geq \frac{\nu}{2}|G(y,3r/2)|.
B(b) = \left| Z_{b^{-1}}\phi(0,0)\right|^2 = \frac{\pi}{2}\left|b \, \vartheta'_1(0,i b) \frac{\vartheta_3(0, i b)}{\vartheta_4(0,i b) \vartheta_4(0, i b^{-1})}\right|^2.
{\cal K}_{{\bf C}^{n}}( z , \bar{z}) = N \mathrm{e x p}\left( \mathrm{i}\sum_{j = 1}^{n}\bar{z}^{j}z^{j}\right) \mathrm{d}z^{1}\wedge \ldots \wedge \mathrm{d}z^{n}\wedge \mathrm{d}\bar{z}^{1}\wedge \ldots \wedge \mathrm{d}\bar{z}^{n},
a^{1234}=a^{1243}=a^{1324}=a^{1342}=a^{1423}=a^{1432}= a^{2134}=a^{2143}=a^{2314}=a^{2341}=a^{2413}=a^{2431}= a^{3124}=a^{3142}=a^{3214}=a^{3241}=a^{3412}=a^{3421}= a^{4123}=a^{4132}=a^{4213}=a^{4231}=a^{4312}=a^{4321}
\hat{\hat{T}}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}{}^{\hat{\hat{\rho}}}= - \left( i_{\hat{\hat{k}}_{( n )}}\hat{\hat{C}}\right)_{\hat{\hat{\mu}}\hat{\hat{\nu}}}Q^{n m}\hat{\hat{k}}_{( m )}{}^{\hat{\hat{\rho}}}.
S_{ii}= \frac{1}{n}\sum_{j=1}^p A_{ij}^2 \sum_{t=1}^n Z_{jt}^2 + \frac{1}{n}\sum_{t=1}^n \sum_{j_1\neq j_2=1}^p A_{ij_1}A_{ij_2}Z_{j_1 t}Z_{j_2 t}=: S_i(1)+S_i(2)\,.
\mathcal{R}(U)\subseteq \mathcal{R}(A) \Longleftrightarrow E_{A}U=0,\mathcal{R}(V)\subseteq \mathcal{R}(A^{\ast})\Longleftrightarrow V^{\ast}F_{A}=0, \mathcal{R}(U^{\ast})\subseteq \mathcal{R}(S_{A}) \Longleftrightarrow UE_{S_{A}}=0, \mathcal{R}(V^{\ast})\subseteq \mathcal{R}(S_{A}^{\ast})\Longleftrightarrow F_{S_{A}}V^{\ast}=0.
Q^{(1)}_{\alpha}=-\frac{i}{2}\,[A_{\mu},A_{\nu}]^{ij}\Gamma^{\mu \nu}_{\alpha \beta}\, \frac{\partial}{\partial \bar{\psi}_{\beta}^{ij}}+i\Gamma_{\mu \alpha \beta}\psi_{\beta}^{ij}\,\frac{\partial}{\partial A_{\mu}^{ij}}
\Phi({\bf x},\theta)=\chi+\bar\theta\lambda+\frac12\bar\theta\theta F,~~D_\alpha=\frac{\partial}{\partial\bar\theta_\alpha}-(\hat\partial\theta)_\alpha,~~\bar D_\alpha=\frac{\partial}{\partial\theta_\alpha}-(\bar\theta\hat\partial)_\alpha.
n_{\ref{main}}=\max\left(\frac{n_{\ref{conn}}}{\gamma^2/4-14\gamma^3},\; n_{\ref{abs59}},\; \frac{n_{\ref{res}}}{1-\gamma},\; \frac{n_{\ref{clic}}}{1-\gamma-\gamma^2/2},\; 10^{12},\;\frac2{\gamma^3}\right).
\sup_{a\in\mathcal{M}_2}S_{\mathcal{R}_X}\Bigl(\frac{a}{X}\Bigr) =\Delta X\sup_{\substack{b\le q q\le (\log{X})^C c\le (\log{X})^C}}\sum_{m<X^{1-\eta/3}}\frac{\Lambda_{\mathcal{C}}(m)}{m\phi(q)}\sum_{\substack{1\le r<q (r,q)=1}}e\Bigl(\frac{b r m}{q}\Bigr)\sum_{1\le j< \Delta^{-1}}e(j\Delta c) &\qquad+O_{C,\eta}\Bigl(\frac{X}{(\log{X})^{4C}}\Bigr).
I(X^n;W,Z^n,F) &\; \overset{(a)}{\leq}n\big(I(X;Z|A,U)+I(X;A,V,Y)-I(X;Y|A,U) \big) &+n\big(I(V;Z|A,X,U)\!-\!I(V;Y|A,X,U)\big) &+n(I(U;Z|A) -I(U;Y|A)+\epsilon^{\prime\prime}) &\;\overset{(b)}{=}n\big(I(V,X;Z|A)+I(X;A,V,Y)-I(V,X;Y|A) \big) &+n\epsilon^{\prime\prime}
\ell \leq \frac{2^{x + 1}- z}{3}\leq \frac{2^{x + 1}+ 1}{3}= \frac{2(2^x) + 1}{3}\leq \frac{2(3^{k + 1}- 1) + 1}{3}&= \frac{2(3^{k + 1}- 1)}{3}< \frac{2(3^{k + 1})}{3}= 2(3^k).
\left\langle D_{y}^{m}w_{p}\right\rangle_{\omega\gamma,y}^{(\gamma)}=1,\quad\sup_{y,h\geq\varepsilon y}(y+h)^{\omega\gamma}\frac{|D_{y}^{m}w_{p}(y+h)-D_{y}^{m}w_{p}(y)|}{h^{\gamma}}\geq\frac{1}{2},
B_1(a)B_1(b)=B_1(a)B_1(\sigma(b))=B_1(B_1(a_1)\sigma(b)S(B_2(a_2))) = B_1(B_1(a)\sigma(b)+\sigma(b)S(B_2(a)))=B_1(B_1(a)\sigma(b)-\sigma(b)B_2(a)).
(f\circ \gamma)''(t) = f_{; ij}|_{\gamma(t)}\dot \gamma^i(t) \dot \gamma^j(t) \left(\frac{D^2 X(\gamma(t))}{dt^2}\right)^k = (\nabla_{\dot \gamma}\nabla_{\dot\gamma}X)^k = X^k_{\; ; ij}|_{\gamma(t)}\dot\gamma^i(t) \dot \gamma^j(t).
\frac{X_i(t^*) e^{-s(k^* - i)(t - t^*)}}{X_{k^*}(t^*)}\leq 2 \lambda_i \bigg( \frac{s}{\mu}\bigg)^{k^* - i}(k^*)^{k^* - i}e^{st^*(i - k^*)}e^{-s(k^* - i)(t - t^*)}&= 2 \lambda_i \bigg( \frac{s}{\mu}\bigg)^{k^* - i}(k^*)^{k^* - i}e^{-s(k^* - i) t}.
\liminf_{n\to \infty}E_n := \{x \in X : \lim_{n\to \infty}{\rm dist}\, (x, E_n) = 0 \}, & \limsup_{n\to \infty}E_n := \{x \in X : \liminf_{n\to \infty}{\rm dist}\, (x, E_n) = 0 \},
\frac{1}{2}|C_k|e^{\frac{r_i^2}{4}}r_i^{-(m+2\lambda)}\le |C_k||1+g_k(\frac{r_i^2}{4})|e^{\frac{r_i^2}{4}}r_i^{-(m+2\lambda)}&=|\int_{S^{m-1}}u(r_i,\theta)\varphi_k| &\le (\int_{S^{m-1}}( u(r_i,\theta))^2)^{\frac{1}{2}}(\int_{S^{m-1}}\varphi_k^2)^{\frac{1}{2}}&\le C\sqrt{\omega_n}e^{\frac{r_i^2}{4}}r_i^{-(m+2\lambda+\epsilon)},
\int|f|^4\lesssim \sum_{\substack{R^{\frac{1}{2}}\le W\le R W\,}}\sum_{\ell(\tau)=\frac{W}{R}}\sum_{U\|U_{\tau,R}}|U|^{-1}\Big(\int\sum_{\theta\subset\tau}|f_\theta|^2W_U\Big)^2.
\mu[0,x]^c = -\log \P[X\le x] &= \int_{E}\max_{1\leq i\leq p}\frac{f_i(u)}{x_i}\nu(du) &= \int_{E}\int_0^\infty 1_{[0,x]^c}(r \vec{f}(u))r^{-2}dr \nu(du)
\lim\limits_{t\rightarrow\infty}\|\chi_{S_{vt}^c}e^{-itH}\psi\|\leq \lim\limits_{t\rightarrow\infty}\sup_{\tau\geq 0}\|\chi_{S_{vt}^c}e^{-i\tau H}\psi\|=\lim\limits_{R\rightarrow\infty}\sup_{\tau\geq 0}\|\chi_{S_{R}^c}e^{-i\tau H}\psi\|
\sum_{j=1}^{n}e^{2 \pi i k \varepsilon_j}e^{-2 \pi i k \frac{j}{n}}= 2\pi i k \sum_{j=1}^{n}\varepsilon_j e^{-2 \pi i k \frac{j}{n}}+ \mathcal{O}\left(k^2 \sum_{j=1}^{n}\varepsilon_j^2 \right),
\Delta_{C L C Q}( x^{+}, x^{-}) = -{\frac{i}{2 \pi}}\int_{0}^{\infty}{\frac{d p^{+}}{p^{+}}}\sin [{\frac{1}{4}}{\frac{m^{2}x^{+}}{p^{+}}}+ p^{+}x^{-}] \hat{f}^{2}( p^{+}) .
X_{C}\left(\{x^i,x^j\}_{TM}\right) -\{X_{C}(x^i),x^j\}_{TM}-\{x^i,X_{C}(x^j)\}_{TM}= &=-\{v^i({\bf x}),x^j\}_{TM}-\{x^i,v^j({\bf x})\}_{TM}=0,
\partial_\xi F^f(x, z, \xi) = F^f(x, z, \xi)\cdot g^0(x)G(x,z,\xi)^{-1}\xi, \partial_z F^f(x, z, \xi) = F^f(x, z, \xi)\cdot \left(\frac{n\partial_zf}{2(1+f)}\Big|_{(x,z)}+ tr(G^{-1}\partial_z G)|_{(x,z,\xi)}\right)\,,
P^{z,i,\downarrow}(t\geq 0\xi_t \geq a) =E^{z,i,\downarrow}[\mathbf 1_{(\tau_{[a,0]}<\infty)}] &=E^{z,i,\dag}\left[\frac{U^+_{J_{\tau_{[a,0]}}}(\xi_{\tau_{[a,0]}})}{U^+_i(z)}\mathbf{1}_{(\tau_{[0,a]}< \tau^+_0)}\mathbf 1_{(\tau_{[0,a]}<\infty)}\right].
||w^{-\frac{1}{p}}\mathcal{A}\big(x,\nabla u_{n_k}\big)||^\frac{p}{p-1}_{L^\frac{p}{p-1}(\Omega)}=\int_{\Omega}w^{-\frac{1}{p-1}}(x)\big|\mathcal{A}\big(x,\nabla u_{n_k}(x)\big)\big|^\frac{p}{p-1}dx &\leq||u_{n_k}||_{X}^p\leq\,c^p.
\frac{1}{n^2}\sum\limits_{i=1}^{\lfloor n\lambda\rfloor}\sum\limits_{j=\lfloor n\lambda\rfloor+1}^n1_{\left\{G(\xi_i)\leq G(\xi_j)\right\}}\overset{P}{\longrightarrow}\frac{\lambda(1-\lambda)}{2}
d\boldsymbol{r}^{\prime}\boldsymbol{I}_{12}^{+}\boldsymbol{P}_{1}^{\pm}=2d\boldsymbol{x}^{1}\boldsymbol{I}_{12}^{+}\boldsymbol{P}_{1}^{\pm}=\pm 2\boldsymbol{I}_{12}^{+}\boldsymbol{P}_{1}^{\pm}.
\i< \eta, \pi_K\circ \Omega_{\zeta_i}(a) \eta > = \sum_{k,l=1}^m \i< \xi_k \otimes \eta, ( V_i^* \pi_Y^\infty (a_k a a_l ) V_i \xi_l ) \otimes \eta > &\rightarrow \sum_{k,l=1}^m \i< \xi_k \otimes \eta, (\varphi (a_k a a_l ) \xi_l ) \otimes \eta > &=\pi_K \circ \Omega_\xi (a).
G_{\varphi}(x) = F^{-}(x) \left(F^{-}(x)\right)^{-1}G_{\varphi}(x) \left(F^{+}(x)\right)^{-1}F^{+}(x) \equiv F^{-}(x) G_{1, \varphi}(x) F^{-}(x)
{\cal H}_{n}^{( 2 )}={}- \frac 1 2 \frac{d^{2}}{d r^{2}}+ \frac{c^{2}}{3 2}r^{6}+ \frac{c_{0}c}8 r^{4}+ \frac 1 8 \left( c_{0}^{2}- 2 c ( 2 n - m ) \right) r^{2}+ \frac{m^{2}- \frac 1 4}{2 r^{2}}- \frac 1 2 ( n - m ) c_{0}.
P_{\tau ,Q}= \sum_{j=0}^{2}P_{\tau ,Q}^{j}, \textup{where}P_{\tau , Q}^{0}(n) = P_{\tau ,Q}(n) \mathbf 1_{\mathbb S_{\tau}}(n) , \textup{and}P_{\tau , Q}^{1}(n) = P_{\tau ,Q}(n) \mathbf 1_{0< \textup{dist}(n, \mathbb S_{\tau}) < \tau Q^{1+ \epsilon}/N},
\dim(\langle B_{i_0}, A_j \rangle \cap A_{i_0}) \leq \dim(\langle B_{i_0}, A_j \rangle) + \dim(A_{i_0}) - \dim(\langle A_j, A_{i_0}\rangle) \leq (\dim(\langle B_{i_0}\rangle) + \dim(A_j) - 1) + \dim(A_j) - (2\dim(A_j) - 1) = \dim(\langle B_{i_0}\rangle) \leq m.
\prod_{k=n}^{m-1}\frac{\|\xi(2^{k+1})\|}{\|\xi(2^k)\|}\leq\prod_{k=n}^{m-1}e^{M+\alpha}e^{-\alpha(2^{2k+1}-2^{2k})}&=e^{(m-n)(M+\alpha)}e^{-\alpha (2^m-2^n)}\,.
\sum_{N\leq k<t}\frac{(\alpha t)^k}{k!}k^{\beta+\varepsilon}\sim\sum_{k=N}^\infty\frac{(\alpha t)^k}{k!}k^{\beta+\varepsilon}\gtrsim(\alpha t)^{\beta+\varepsilon}e^{\alpha t}\quad\mbox{as $t\to\infty$, by Theorem \ref{Thm:Kummer}}.
\phi_T(X) =\begin{cases}\phi^{-}_T(X) = \phi^{+}_T(X)+c_0L(X) \;\; X\in T_p^-, \phi^{+}_T(X) = \sum_{i\in\overline{\mathcal{I}^-}}c_i\psi_{i,T}(X)+\sum_{i\in\mathcal{I}^+}v_i\psi_{i,T}(X) \;\; X\in T_p^+,\end{cases}
\bar{R}^\mathrm{J}=\log{|\mathbf{I}+{{\mathbf{\Omega}_\mathrm{z}^{+}}}^{-1}{\mathbf{\tilde{B}}}{\mathbf{\Omega}_\mathrm{z}^{+}}^{-\mathrm{H}}(\mathbf{Q}^{\prime}_\mathrm{z}+\sigma^2{\mathbf{\Omega}_\mathrm{z}^{+}}^{-1}{\mathbf{\Omega}_\mathrm{z}^{+}}^{-\mathrm{H}})^{-1}|}.
\begin{array}{ccl}n_J^{\omega}x^J = f_J^*\pi_!(c_{\omega}(ET^k\times_{T^k}TM)) = (\pi_J)_!\tilde{f_J}^*c_{\omega}(ET^k\times_{T^k}TM) = (\pi_J)_!(c_{\omega}((\prod_{i=1}^{k}S^{2j_i+1})\times_{T^k}TM)).\end{array}
\int_{0}^{\pi}u^{\varepsilon}(t,x) \phi(x) \, \textrm{d}x = \int_{0}^{t}\int_{0}^{\pi}u^{\varepsilon}(s,x) \phi''(x) \, \textrm{d}s \, \textrm{d}x + \int_{0}^{t}\int_{0}^{\pi}\frac{f(u^{\varepsilon}(s,x))}{\sigma(\varepsilon)}\phi(x) \,L^{\varepsilon}(\textrm{d}s, \textrm{d}x)