harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{0}^{T}d \sigma^{\prime}\left[ \delta ( \sigma - \sigma^{\prime}) \frac{\partial^{2}}{\partial \sigma^{2}}-{\frac{1}{L}}B ( \sigma^{\prime}, \sigma ) \right] z_{\mu}( \sigma^{\prime}) = [ \delta ( \sigma - \tau_{2}) - \delta ( \sigma - \tau_{1}) ] ( \tau_{1}- \tau_{2}) \frac{u_{\mu}}{L}
\begin{cases}d\tilde Y_t=-{\dot{W}}_{\varepsilon, \eta}(t, B_t) \tilde Y_t dt-Y_t^{\varepsilon, \eta}\nabla_x{\dot{W}}_{\varepsilon, \eta}(t, B_t) I_{[0, t]}(r)dt +\tilde Z_t dB_t,\ \r\leq t\le T \tilde Y_T=D_r^B\xi\,. \end{cases}
u_\beta(t)\beta^\vee(t)u_{-\alpha}(z)=u_{-\alpha}\left( t^{\frac{-2\langle \alpha,\beta\rangle}{\langle \beta,\beta \rangle}}z \right) u_\beta(t)\beta^\vee(t),\;\;\;\;\;\;\;t\in\mathbb{C}^\times, \, z \in{\mathbb{C}}.
X_{s}= 2 i g_{s}\sum_{\alpha \in \Delta_{s}}x_{d}^{( 1 / 2 )}( \alpha \cdot q , \xi ) E_{d}( \alpha ) , \quad Y_{s}= i g_{s}\sum_{\alpha \in \Delta_{s}}y_{d}^{( 1 / 2 )}( \alpha \cdot q , \xi ) E_{d}( \alpha ) , \quad E_{d}( \alpha )_{\Upsilon \Omega}= \delta_{\Upsilon - \Omega , 2 \alpha}.
|g_{f,s}(x)| = |f(\varphi(t,x))| \leq |f(\varphi(t,x))- f(\varphi(t,y_{n_K}))| + |f(\varphi(t,y_{n_K}))| \leq \frac{\varepsilon}{2}+ g_{f,s}(y_{n_K}) & \leq \frac{\varepsilon}{2}+|g_{f,s}(y_{n_K})- f(\varphi(t^*,x))| + |f(\varphi(t^*,x))|\leq \varepsilon + |f(\varphi(t^*,x))|.
\nabla^2\Psi(P_j) = \begin{pmatrix}M_1 0 0 -M_1 0 0 0 M_2 0 0 -M_2 0 0 0 j 0 0 -j -M_1 0 0 M_1 0 0 0 -M_2 0 0 M_2 0 0 0 -j 0 0 j\end{pmatrix},
\frac{1}{\varepsilon^2\ln\frac{1}{\varepsilon}}\int_\Omega\left( v^\tau_\varepsilon-q\ln\frac{1}{\varepsilon}\right)^{p}_+q\ln\frac{1}{\varepsilon}dx\rightrightarrows q^{p+1}(\bar{x})\int_{\hat{B}_\tau(0)}(\hat{U}(y)+\ln\tau)^{p}_+dy.
Q_A \tilde{w}_i = \tilde{c}_i \tilde{u}_i^A, Q_A^T \tilde{u}_i^A = \tilde{c}_i \tilde{w}_i + \alpha_{k+1}v_{k+1}e_{k+1}^T x_i, Q_B \tilde{w}_i = \tilde{s}_i \tilde{u}_i^B, Q_B^T \tilde{u}_i^B = \tilde{s}_i \tilde{w}_i + \check{\beta}_{k}v_{k+1}e_{k}^T \hat{x}_i,
\int_{-T}^{x_n - R_n}q_{-}(t) \, dt = \int_{x_n - T_n}^{x_n - R_n}q_{-}(t) \, dt = \frac{1}{2}\left(-\frac{T_n d_{-}}{2}+ (T_n/2 - R_n)\delta_{-}\right) =c_{-}, \shortintertext{and}\int_{x_n + R_n}^{T}q_{+}(t) \, dt = \int_{x_n + R_n}^{x_n + T_n}q_+(t) \,dt = \frac{1}{2}\left(\delta_+ (T_n/2 - R_n) + \frac{d_+ T_n}{2}\right) = -c_+.
\frac{1}{2}f^{\prime \prime}( 3 - D ) - \frac{1}{2}f \frac{s^{\prime \prime}}{s}+ f^{\prime}\frac{s^{\prime}}{s}\frac{( 3 - D )}{4}+ \frac{1}{4}f \left( \frac{s^{\prime}}{s}\right)^{2}- f \left( \frac{f^{\prime}}{f}\right)^{2}\left( \frac{D^{2}- 9 D + 1 8}{8}\right) = - \alpha^{2}f \frac{( D - 2 )^{2}+ ( D - 2 )}{8}
J_{\varphi}=\frac{\partial (\varphi_{1},\mathellipsis ,\varphi_{m,}\varphi_{m+1},\mathellipsis ,\varphi_{m+n},\thinspace \varphi_{m+n+1},\mathellipsis ,\varphi_{m+n+p})}{\partial (x_{1},\mathellipsis ,x_{m},y_{1},\mathellipsis ,y_{m},z_{1},\mathellipsis ,z_{p})}
\O_u \star \O_v = Q_j \mathcal{O}_{u'}\star \mathcal{O}_v = \sum_{w \in W^P, \beta \in E(X)}N_{u',v}^{w, \beta+l_j}Q^{\beta}\mathcal{O}_w &= \sum_{w \in W^P,\: \beta \in E(X)}N_{u',v}^{w, \beta + l_j}Q^{\beta}
k_3(\langle H\rangle)=\frac{q-1}{768}\left[2(q^2-20q+81)+2 u(-p)^t+3q^2\cdot{_{3}}F_{2}\left(\begin{array}{ccc}\chi_4, \chi_4, \overline{\chi_4}\varepsilon, \varepsilon\end{array}| 1\right)\right].
w_{\rm spinor}= 2{\rm Im}{\cal L}^{(1)}_{\rm spinor}= - \Bigl(\frac{qE}{2 \pi}\Bigr) \Bigl(\frac{qB}{2 \pi}\Bigr) \sum_{j = 0}^{\infty}\sum_{\sigma = \pm 1}\ln \Biggl\{1 - \exp \Bigl(- \pi \frac{m^2 + qB (2 j+1 - \sigma)}{qE}\Bigr) \Biggr\}.
\mathbb{E}(X_iX_{i+1}X_jX_{j+1})= \sum_{\substack{0\leq l\leq i \gcd(l,i)\{\rm{is}}\k-{\rm{free}}}}C_{\alpha}(i,l)\sum_{\substack{0\leq m_1\leq 1 \gcd(l+m_1,i+1)\{\rm{is}}\k-{\rm{free}}}}C_{\alpha}(1,m_1) &\Big(f_k(j)f_k(j+1)+O_{\alpha}\Big(\frac{\tau_3(j)\tau_3(j+1)}{\sqrt{j-i-1}}\Big)\Big).
\begin{aligned}\int_{\Xi_{0}}e^{W / \hbar}\Omega = 1 + (y_0-\tfrac18 y_2^2) \hbar^{-1}+ O(\hbar^{-2}) \int_{\Xi_{1}}e^{W / \hbar}\Omega = y_1 \hbar^{-1}+ O(\hbar^{-2}) \int_{\Xi_{2}}e^{W / \hbar}\Omega = y_2 \hbar^{-1}+ O(\hbar^{-2})\end{aligned}
(A-1I)x_{2}=(\begin{matrix}0&0&0&0&0\\ 3&0&0&0&0\\ 6&3&1&0&0\\ 10&6&3&1&0\\ 15&10&6&3&1\end{matrix})(\begin{matrix}1\\ -15\\ 30\\ -1\\ -45\end{matrix})=(\begin{matrix}0\\ 3\\ -9\\ 9\\ -3\end{matrix})=x_{1},
\underbrace{\begin{bmatrix}H+\rho I +\Theta^\dagger -A^T A \delta I \end{bmatrix}}_{\mathcal{N}_{\rho,\delta,\Theta}}\begin{bmatrix}\Delta \mathbf{x}\Delta \mathbf{y}\end{bmatrix}=\begin{bmatrix}\xi_d^1 \xi^2_d+X^{-1}_{\mathcal{C}}\xi_{\mu,\sigma}\xi_p \end{bmatrix},
\left. \frac{1}{M_{6}^{4}}f_{0}( \rho ) \right|_{\rho = \rho^{\#}}\leq \left. \frac{3}{8 M_{6}^{4}}f_{\theta}( \rho ) \right|_{\rho = \rho^{\#}}\! - \frac{C}{\epsilon^{2}}\ .
y ( x ) = A \ \mathrm{e}^{- \frac{x^{2}}{4}}\, \mathrm{M}\left( \frac{a}{2}+ \frac{1}{4}, \frac{1}{2}, \frac{x^{2}}{2}\right) + B \x \ \mathrm{e}^{- \frac{x^{2}}{4}}\, \mathrm{M}\left( \frac{a}{2}+ \frac{3}{4}, \frac{3}{2}, \frac{x^{2}}{2}\right) \, ,
\mathrm{Fix}(S_{n})=\mathrm{Fix}\left(\mathcal{T}_{\left\{C_{m,r}\left(n\right)\left(j\right)\right\}_{j=1}^{r}}\right)=\mathrm{Fix}\left(\mathcal{V}_{\left\{C_{m,r}\left(n\right)\left(j\right)\right\}_{j=1}^{r}}\right)\supset\cap_{i=1}^{m}C_{i}.
\begin{array}{rclrcl}A_{\mu}^{\prime}= A_{\mu}\, , (k^2)^{\prime}= k^2\, , & B_{\mu}^{\prime}= B_{\mu}- \beta V_{\mu}+{\textstyle \frac{1}{2}}\beta^{2}A_{\mu}\, ,\hspace{1.2cm}\ell^{\prime}= \ell+\beta\, , & V_{\mu}^{\prime}= V_{\mu}- \beta A_{\mu}\, . \end{array}
\begin{aligned}\left\|w^j\right\|_{C^{2,\alpha}\left(B_{3r/5}^{m}(0)\right)}\leq C(r)\left\|\Delta_{h}z^j\right\|_{C^{0,\alpha}\left(B_{3r/5}^{m}(0)\right)}\leq C(r)\Psi(\tau\,|\, n,\alpha, Q).\end{aligned}
n\: C \;\alpha_1^{1-p}\:\beta_1^{1-q-\gamma}\left(\prod_{j=2}^p \;\:\alpha_1^{a_j - \alpha_j}\:\alpha_j^{-a_j}\right)\beta_{i_0}^\gamma \left( \prod_{k=2}^q \beta_1^{b_k-\beta_k}\:\beta_k^{-b_k}\right),
M_\infty = \begin{pmatrix}\frac{-1}{m}\frac{-\lambda^2}{m-2}\lambda \lambda^m \frac{-1}{m}\frac{-(1+\lambda)^2}{m-2}1+\lambda (1+\lambda)^m 0 \frac{-\lambda}{n-1}1 m\lambda^{m-1}0 \frac{-(1+\lambda)}{n-1}1 m(1+\lambda)^{m-1}\end{pmatrix}
G_{M N}d x^{M}d x^{N}= H_{p}^{-{\frac{4 ( n - 1 )}{( p + n ) \Delta_{p}}}}\left[ \eta_{\mu \nu}+ h_{\mu \nu}\right] d x^{\mu}d x^{\nu}+ H_{p}^{\frac{4 ( p + 1 )}{( p + n ) \Delta_{p}}}d{\bf y}\cdot d{\bf y}
\mathcal{N}_{( r , s ) , ( r^{\prime}, s^{\prime})}^{( r^{\prime \prime}, s^{\prime \prime})}\medskip = \sum_{( k , l ) \in E ( p , p^{\prime})}\frac{S_{( r , s ) , ( k , l )}\S_{( r^{\prime}, s^{\prime}) , ( k , l )}\S_{( k , l ) , ( r^{\prime \prime}, s^{\prime \prime})}}{S_{( 1 , 1 ) , ( k , l )}}
\int_{B^+_1}| D^\alpha v(t,x) -g_\alpha(t)|^2 \,dx=\int_{B^+_1}|\int_{B^+_1}( D^\alpha v(t,x)- D^\alpha v(t,y))\xi(y)\,dy|^2dx &\le N\int_{B^+_1}\int_{B^+_1}|D^\alpha v(t,x)-D^\alpha v(t,y)|^2\,dy\,dx \le N\int_{B^+_1}|D^{|\alpha|+1}v(t,y)|^2\,dy.
y[m] = \frac{1}{{\sqrt{{\bf{w}}_t^H{{\bf{w}}_t}}}}\sum\limits_{l =1}^N{\frac{{{{\bar \lambda}^{(0)}}}}{{\lambda_l^{(0)}}}{\lambda_l}s[m-\Delta_l]}+{\bf{w}}_r^T{\bf{n}}.
\lim_{n\to\infty}\mathbb{P}\left\lbrace \sup_{t\in[0,T]}\left\lvert X^{n,r}_t-X^{r}_t\right\rvert\ge \varepsilon\right\rbrace=0,\lim_{n\to\infty}\tilde{\mathbb{P}}\left\lbrace \sup_{t\in[0,T]}\left\lvert Y^{n,r}_t-Y^{r}_t\right\rvert\ge \varepsilon\right\rbrace=0.
\frac{1}{\zeta(1+2\delta)}\frac{1}{2\pi i}\int\limits_{(2A+1)/\log q -iq}^{(2A+1)/\log q +iq}H_1(0,w_2,\delta,t) \Psi_{\Upsilon,M}(w_2) dw_2 \hskip 120pt = \frac{1}{\zeta(1+2\delta)}\bigg( 1 + \mathcal{O}_{\varepsilon}\left(M^{-(1-\Upsilon)/2+\varepsilon}\right)\bigg),
{\rm lim}_{L\rightarrow +\infty}\frac{k_{\gamma}^{L}}{\sqrt{L}}=\frac{|\frac{d}{dt}(\omega(\dot{\gamma}(t)))|}{\frac{1}{2}\left(-e^{-\gamma_3}\dot{\gamma_1}+e^{\gamma_3}\dot{\gamma_2}\right)^2+\dot{\gamma}_3^2},~~if ~~\omega(\dot{\gamma}(t))= 0~~and~~\frac{d}{dt}(\omega(\dot{\gamma}(t)))\neq 0.
\sum_{r \in \mathbb{Z}}c^F_{-r}\big(r_l(\pi_F)^{(l)}(w_n)W,\sigma_F^{(l)}(w_{n-1})W'\big)q_F^{-\frac{r}{2}}X^{-lr}= \gamma\big(X^l, r_l(\pi_F)^{(l)}, \sigma_F^{(l)},\overline{\psi}_F^l\big)\sum_{r \in \mathbb{Z}}c^F_r(W,W')q_F^{\frac{r}{2}}X^{lr}.
B_k^{m+1}=\left(1-(2\lambda+2m+1)r^2\right)R_k^m(r)+(1+r^2)\,rR_k^{m\,\prime}(r), C_k^{m+1}=2\,(\lambda+m)\,rR_{k-1}^m(r)-2r^2R_{k-1}^{m\,\prime}(r).
u(x)= \begin{cases}o\left( \left(\frac{1}{|x|}\right)^{\frac{n-2}{n}[(n-2)\lambda+\alpha]}\right), \frac{b}{1-a}<n-2 o\left( \left(\frac{1}{|x|}\right)^{\frac{n-2}{n}(\frac{b\lambda}{1-a}+\alpha+\varepsilon)}\right), \frac{b}{1-a}\geq n-2 \end{cases}
\frac{\|(J_\varepsilon u)(t+\tau)\|_{H^k,0}^2-\|(J_\varepsilon u)(t)\|_{H^k,0}^2}{\tau}=\frac{1}{\tau}\int_0^\tau\frac{d}{ds}\|J_\varepsilon u(s)\|_{H^k,0}^2ds\leq \frac{C_1}{\tau}\cdot\int_0^\tau1+\|u(s)\|_{H^k,0}^2ds.
\rho_gJAJ\delta_h =\rho_gJA\delta_{h^{-1}}=\rho_gJ\left(\sum_{u\in G}\zeta_u (A\delta_{h^{-1}}) \delta_u\right)=\sum_{u\in G}\zeta_u (A\delta_{h^{-1}}) \rho_gJ\delta_u &=\sum_{u\in G}\zeta_u (A\delta_{h^{-1}}) \rho_g\delta_{u^{-1}}=\sum_{u\in G}\zeta_u (A\delta_{h^{-1}}) \delta_{u^{-1}g^{-1}}.
\begin{multlined}[t][12.5cm]r_{x}=\frac{\pi^{d_{T}-1}\mathcal{D}(m)(\|\Phi_{T}\|+\|\eta_{Mh}\|^2)}{d_{\nu_{G}}\left(\sqrt{2}\right)^{-d_{G}-d_{T}+1-d_{N_{m}}}}.\end{multlined}
\begin{aligned}\bar P_b\approx \frac{1}{12}\int_0^\infty \exp\left(-{\frac{\rho\zeta^2x}{4(1+\rho(1-\zeta^2)\sigma_e^2L)}}\right)f(x)dx & +\frac{1}{4}\int_0^\infty\exp\left(-{\frac{\rho\zeta^2x}{3(1+\rho(1-\zeta^2)\sigma_e^2L)}}\right)f(x)dx.\end{aligned}
,q^{h}q^{h'}=q^{h+h'},q^0=1\,, &q^{h}E_i(z)q^{-h}=q^{(h,\alpha_i)}E_i(z)\,,q^{h}F_i(z)q^{-h}=q^{-(h,\alpha_i)}F_i(z)\,,q^hK^\pm_i(z)=K^\pm_i(z)q^h\,.
(a)= \int_{\Omega}w^{1+\alpha+n}_\eta G \, U^j_r[D_\eta G]^j_r dx + \int_{\Omega}(w^{1+\alpha+n}),_sA^s_r A^k_i G^i,_{k}U_{r}^jG^j dx + \int_{\Omega}w^{1+\alpha+n}(A^s_r A^k_i U_{r}^j),_s G^i,_{k}G^j dx
< \frac{1}{2}, m_{s}|{\psi}_{+}^{( \frac{1}{2})}( g ) > \equiv D_{m_{s}+ \frac{1}{2}}^{( \frac{1}{2})}( g ) = < g , l | T_{m_{s}+}^{\frac{1}{2}}| g , l + \frac{1}{2}> ,
\sum_{v\in V_j}\sum_{u\in N(v)\cap V_i}\Big(z_{uv}e(G)/\binom r 2k_{[r]}-1\Big)=\sum_{v\in V_j}\Big(z_ve(G)/\binom r 2k_{[r]}-d(v,V_i)\Big) &=k_{[r]}e(G)/\binom{r}{2}k_{[r]}-\sum_{v\in V_j}d(v,V_i)=e(G)/\binom{r}{2}-d(V_j,V_i),
T_l\left(\sum_{\mu \in \mathcal{P}, U \in I(\mathcal{C})}c_\mu^{(U)}p_\mu^{(U)}[U]\right) = \sum_{\mu \in \mathcal{P}, U \in I(\mathcal{C})}c_\mu^{(U)}p_\mu^{(U)}\otimes T_l(U)
G(\cos \theta_{\rm c}\cdot f)= \frac{N_0}{\alpha_{\rm c}}\left( \! \left( \frac{\mu}{2}\! \int\limits_{\frac{\cos \theta_{\rm c}\cdot f}{f_{\rm max}}}^{\frac{\cos \theta_{\rm c}\cdot f}{f_{\rm min}}}\frac{S_0(\frac{\cos \theta_{\rm c}\cdot f}{u})}{u}{\rm d}u \right)^{\!\!-1}\!\!\! - \frac{1}{S_0(f)}\right)_{\!+}\!\!.
G(z) \approx - \frac{\pi}{N}\sum_{k=0}^{N}{}^{''}\frac{b-a}{2}\sin\left ( \frac{2\pi k}{N}\right ) \frac{f\left ( \frac{b-a}{2}\cos\left ( \frac{2\pi k}{N}\right ) + \frac{b+a}{2}\right )}{z - \left ( \frac{b-a}{2}\cos\left ( \frac{2\pi k}{N}\right ) + \frac{b+a}{2}\right )},
\mathbf{A}=\begin{bmatrix}1 0 0 0 \dots 0 0 0 -1 1 0 0 \dots 0 0 0 0 -1 1 0 \dots 0 0 0 0 0 -1 1 \dots 0 0 0 \vdots \vdots \vdots \vdots \ddots \vdots \vdots \vdots 0 0 0 0 \dots -1 1 0 0 0 0 0 \dots 0 -1 1 \end{bmatrix}.
\rho_{1}^{v_{2,n}}\psi_{1}^{v_{2,n}}(i)=\biggl[\sum_{j\in S}\psi_{1}^{v_{2,n}}(j)\pi_{ij}(v_{1,n}^{*}(i),v_{2,n}(i))+c_1(i,v_{1,n}^{*}(i),v_{2,n}(i))\psi_{1}^{v_{2,n}}(i)\biggr].
\begin{aligned}S_+ \equiv (1+\Delta\alpha)S_a\in[S_a, 2S_a], r \equiv \frac{1-\Delta\alpha}{1+\Delta\alpha}\in [0,1], L_{ac}\equiv L_a\cos(2\pi\phi)\in[-L_a,L_a], L_{as}\equiv L_a\sin(2\pi\phi),\{\rm and \where}\L_{ac}^2+L_{as}^2=L_a^2.\end{aligned}
1\;=\;{\int d A \exp\left\{-N\beta\,{\rm Tr}\,\left[ \left( A - \sum_{i=1}^d U_i\right)\left( A^\dagger - \sum_{i=1}^d U_i^\dagger\right)\right]\right\}\over\int d A \exp \left\{-N\beta\,{\rm Tr}\,\left[ A A^\dagger \right]\right\}}\;.
\nabla\cdot ({f \rho + g \bar\rho}) =\nabla\cdot (f \rho) + \sum_{i=1}^{m}\nabla \cdot (g_i\rho u_i) = -\mathcal{P}_{f}\rho - \sum_{i=1}^m \mathcal{P}_{g_i}\bar\rho_i \approx \Psi^T (\frac{I - P_0^T}{\Delta t}v) + \Psi^T(\frac{I - (P_i-P_0)^T}{\Delta t}w_i).
\sum_{k=0}^{2^{n-2}}=\sum_{k=0}^{k_{0}-2}+\sum_{k=k_{0}-1}^{k_{0}+2}+\sum_{k=k_{0}+3}^{2^{n-2}}\stackrel{\rm def}=\Sigma_1+\Sigma_2+\Sigma_3.
\hbox{$\alpha(x) = \alpha_c (x)$ \,and \,$\gamma(x) \subset \gamma_c(x)$ \,on $X_\Gamma$, \,$\beta_{cc}= \beta$ on $Y_\Gamma$, \,and \,$\Gamma \subset \Gamma_{(\alpha_c, \gamma_c, \beta_{cc})}.$}
\mathcal{\tilde{R}}_{1,\epsilon}(t)=\int^{t}_{0}\big(b(\varphi(s))-b(X^{\epsilon}(s)))\big)ds, &\mathcal{\tilde{R}}_{2,\epsilon}(t)=\lambda(\epsilon)\int^{t}_{0}\big(\sigma(\varphi_{m}(s))-\sigma(X^{\epsilon}(s))\big)\theta_{\epsilon}(s)ds.
R_{M N}-{\frac{1}{2}}G_{M N}R +{\frac{1}{2}}{\frac{\sqrt{-{\widehat G}}}{\sqrt{- G}}}~{\delta_{M}}^{\alpha}{\delta_{N}}^{\beta}{\widehat G}_{\alpha \beta}~{\widetilde f}_{D - 2}~ \delta^{( 2 )}( x^{i}) = 0 ~ ,
\left|\frac{\partial(u,v,w)}{\partial(t_1,t_2,t_3)}\right|(\mathbf{t})=\left|\widetilde{P}_1'(t_2)\widetilde{P}_2'(t_3)-\widetilde{P}_1'(t_3)\widetilde{P}_2'(t_2)\right|\asymp 2^{-\mathfrak{c}|l|}|t_2-t_3|,
{[}\hat{D}_{\lambda}^{( + )}( \vec{k}) , \hat{Q}^{a}] = - k_{0}\varepsilon_{\lambda \lambda^{\prime}}\hat{C}_{\lambda^{\prime}}^{a ( + )}( \vec{k}) , \; \; \varepsilon_{\lambda \lambda^{\prime}}= - \varepsilon_{\lambda^{\prime}\lambda}, \; \; \varepsilon_{1 2}= - 1 ,
|\partial^{\alpha}_x \partial_{\xi}^{\beta}e_{\pm}(x,\xi)| \le C_{\alpha\beta}\langle\xi\rangle^{-3+2\alpha-\beta}\,, |\partial^{\alpha}_x \partial_{\xi}^{\beta}\tilde{e}_{\pm}(x,\xi)| \le C_{\alpha\beta}\langle\xi\rangle^{3+2\alpha-\beta}\,.
H(z_2,\dots,z_m) := \prod_{j= b_1+b_2+1}^{2b_1}(1-\frac{z_2}{p^j}) \prod_{j= b_1+b_2+b_3+1}^{b_1+2b_2}(1-\frac{z_3}{p^j}) \cdots\prod_{j= b_1\dots+b_m+1}^{b_1+\dots+b_{m-2}+2b_{m-1}}(1-\frac{z_m}{p^j})=\sum_{d_2,\dots,d_m \geq 0}e_{d_1,\dots,d_m}z_2^{d_2}\cdots z_m^{d_m}.
F_j^{\mu_1,\mu_2}(\phi) = \xi^+_j \Big[ P_{\frac{j}{2}}^{(\mu_2-1/2,\mu_1-1/2)}(\cos 2\phi) - \cos(\phi)\sin(\phi)P_{\frac{j}{2}-1}^{(\mu_2+1/2,\mu_1+1/2)}(\cos 2\phi)\Big].
r(t)-\widetilde{r}(t)=P(t)-\widetilde{P}(t)+\int\limits_{0}^{t}\left[Q(t,\tau )-\widetilde{Q}(t,\tau )\right] r(\tau )d\tau +\int\limits_{0}^{t}\widetilde{Q}(t,\tau )\left[ r(\tau )-\widetilde{r}(\tau )\right] d\tau .
\sin(X) :=\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}X^{2n+1}, \cos(X) :=\sum_{n=0}^\infty\frac{(-1)^n}{(2n)!}X^{2n}, \tan(X) :=\frac{\sin(X)}{\cos(X)}, \sinh(X) :=\sum_{k=0}^\infty\frac{X^{2k+1}}{(2k+1)!}, \arcsin(X) :=\sum_{n=0}^\infty\frac{(2n)!}{(2^nn!)^2}\frac{X^{2n+1}}{2n+1}, \arctan(X) :=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}X^{2k+1}.
\int_{B_{H_0}}E_0\,dy = \int_0^{\frac 12}t\int_{\{E_0=t\}}\frac{d\mathcal H^{n-1}}{|\nabla E_0|}\, dt = \int_0^{\frac 12}t^{\frac n2}\int_{\{E_0=1\}}\frac{d\mathcal H^{n-1}}{|\nabla E_0|}\, dt = \frac 1{(n+2)2^{\frac n2}}\int_{\{E_0=1\}}\frac{d\mathcal H^{n-1}}{|\nabla E_0|}\, dt,
\sum_{n=0}^\infty \frac{(-1)^nq^{n^2+n}}{(q^2;q^2)_n}=\frac{1}{(q;q)_{\infty}}\sum_{n=0}^{\infty}\sum_{|m|\leq n}(-1)^{m}(1-q^{2n+1})q^{2n^2+n-m^2}&=\frac{1}{(q;q)_{\infty}}(f_{1,3,1}(q^2,q^2;q^2) + q^3f_{1,3,1}(q^6,q^6;q^2)) &=\frac{J_1J_2}{(q;q)_{\infty}}&=J_2,
L_{01}= \{(x_0^- + u_{00}x_0^- + u_{01}x_1^+, x_1^+ + u_{11}x_1^+ + u_{10}x_0^-) \mid x_0^- \in V_0^-, x_1^+ \in V_1^+\}, L_{12}= \{(x_1^- +v_{11}x_1^- + v_{12}x_2^+, x_2^- + v_{22}x_2^+ + v_{21}x_1^-) \mid x_1^- \in V_1^-, x_2^+ \in V_2^+ \}.
R_j((u\cdot\nabla)u)(x) = -\int_{\mathfrak D^c}\underbrace{\sum^3_{k,\ell=1}\partial y_{\ell}\mathfrak Z_{jk}(x,y) u_k(y)u_{\ell}(y) dy}_{_1}&+ \int_{\partial \Omega}\sum^3_{k,\ell = 1}\underbrace{\mathfrak Z_{jk}(x,y) u_k (y) n_{\ell}(y) u_{\ell}(y) d \partial y}_{_2}
\sum_{i\in I}\mathbb{E}(\Phi_*)_{\left[ - \lambda, \lambda \right]}(f_i) (x) =\ \frac{1}{2 \lambda}\int_{-\lambda}^{\lambda}\left( \sum_{i\in I}f_i (\Phi_{-t}(x) ) \right) \d t =\ \frac{1}{2 \lambda}\int_{-\lambda}^{\lambda}1 \: d t \ = 1 .
\limsup_{t-s\to\infty}\frac{1}{t-s}\ln\frac{\|\xi(t)\|}{\|\xi(s)\|}\leq-\alpha+(M+\alpha)\limsup_{t-s\to\infty}\frac{\log_2(t/s)}{t-s}&\leq-\alpha+(M+\alpha)\limsup_{t-s\to\infty}\frac{\log_2(1+(t-s)/4)}{t-s}= -\alpha\,,
T_t=\left\{\vphantom{\sqrt{x_2^2+\ldots+x_d^2}\leq r(t)+h(t)}x=(x_1,\ldots, x_d)\in \mathbb{R}^d:|x_1|\leq kt+h(t),\right. \left. \ \ \ \sqrt{x_2^2+\ldots+x_d^2}\leq r(t)+h(t)\right\},
\mathcal{M}_{g,d}^r(a) := \{[C,x] \in \mathcal{M}_{g,1}: \exists \, \ell\in G^r_d(C) \,\,\mbox{with}\,\, h^0(C,\ell\otimes \mathcal{O}_C(-n\cdot x))\geq r+1-t \}
\begin{aligned}\Lambda_n(H_\theta) \ge \Lambda_n(Q^{1+\varepsilon}_\theta\oplus R)=\Lambda_n(Q^{1+\varepsilon}_\theta\oplus Q^{1+\frac{1}{\varepsilon}}_{\pi-\theta}) &\ge \min\Big\{\Lambda_n(Q^{1+\varepsilon}_\theta), -\Big(1+\frac{1}{\varepsilon}\Big)^2\Big\}.\end{aligned}
A=\sum_{e_1,e_2\in Q}\lim_{\delta\rightarrow 0}VT{|\beta_1 +\beta_2|\over (2\pi)^2}\int_\delta^\infty dt t^{-1}\int{d^{24}p\over (2\pi)^{24}}e^{-\pi t p^2/2}T^{\perp}T^{\parallel}.
\left\Vert D^1 f - \tilde{f}^{(1)}\right\Vert_\infty = \max_{t\in [0,T]}\left| D^1 f(t) - \tilde{f}^{(1)}(t) \right| \leq \frac{E'' \Vert f \Vert_{H^s}}{N^{s-3/2}}.
\left(b^*(g)\Psi\right)^{(p)}(x_1,\ldots,x_p) := \frac{1}{\sqrt{p}}\sum_{i=1}^p g(x_i)\Psi^{(p-1)}(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_p)\,, \left(b(g)\Psi\right)^{(p)}(x_1,\ldots,x_p) := \sqrt{p+1}\int dx \, \overline{g(x)}\Psi^{(p+1)}(x,x_1,\ldots,x_p)\,.
{\displaystyle \mathbf{E}^{0,h_0}=-\nabla \phi^{0,h_0}\stackrel{\mathrm{w}}{\rightharpoonup}-\nabla\widetilde{\phi}^0=\widetilde{\mathbf{E}}^0 \hbox{weakly\,\,\,in}(L^2(\Omega))^3 \hbox{as}h_0\rightarrow 0.}
-{\frac{2 l \alpha ( l^{2}+ r^{2})}{( l^{2}+ \alpha^{2})}}d \beta^{2}+ \beta^{2}{\frac{2 ( l^{2}+ \alpha^{2})}{l \alpha ( l^{2}+ r^{2})}}\left[ \alpha d \chi +{\frac{( r^{2}- \alpha^{2})}{( 1 + \alpha^{2}l^{- 2})}}d \phi \right]^{2}.
\mathcal{C}=\mathbb{E}_{\mathbf{W}_i}\left[\log_2 \det\left(\mathbf{I}_{m_i}+\frac{a_i}{p_i}\mathbf{W}_i \right)\right]=m_i\int_0^\infty \log_2(1+\lambda)P_\lambda(\lambda)\d\lambda.
D^2_{\rm G}[U,U'] = \sum\limits_{x,i}\sum\limits_{\alpha =0}^3 \left[ ( (u_\alpha)_{x,i}- (u'_\alpha)_{x,i})^2 +( (\dot u_\alpha)_{x,i}- (\dot u'_\alpha)_{x,i})^2 \right] \, ,
\mathrm{I m}\, Z ( \varphi ) \, = \, - \mathrm{I m}\, \log \frac{\vartheta ( \alpha , \tau )}{\vartheta ( 0 , \tau )}\, - \, \tau_{1}\pi \varphi_{1}^{2}\, + \, \pi \varphi_{1}\varphi_{2}\, = \, \mathrm{I m}\, \Gamma ( \varphi ) \; .
\pi^i(t) ={n \choose n-i}\sum_{k=0}^i(-1)^{i-k}{i\choose i-k}\exp(Q_{(n-k)\lambda}t)x(0) & ={n \choose i}\sum_{k=0}^i(-1)^{i-k}{i\choose k}\exp(Q_{(n-k)\lambda}t)x(0).
{}\sum_{j=1}^{q}\int_{0}^{T}z_{j}(t)\left [a_{j}(t)-\sum_{i=1}^{p}B_{ij}(t)w_{i}(t)+\sum_{i=1}^{p}\int_{t}^{T}K_{ij}(s,t)w_{i}(s)ds\right ]dt\leq 0
\int_{\mathbb{R}^d}\,\epsilon^d\, \bigg|\sum_{i=1}^{N}\,\hat{F_i}(\epsilon\lambda)\,e^{-2\pi i x_i\cdot\lambda}\bigg|^2\,d\mu(\lambda)\geq \left(G(\delta,\epsilon)-I(\delta,\epsilon)\right)^2 &\ge\left(\sqrt{\mathcal{D}^{-}_N(\check{\mu})-\rho'}-\sqrt{\rho'}\right)^2\,\|f\|_2^2\geq (\mathcal{D}^{-}_N(\check{\mu})-\rho)\,\|f\|_2^2.
A_{\delta}^{(p)}=A_\delta-p^{k-1}A_{p\delta}=-\frac{\tau(\phi)}{2N^k}\phi(|c|)(1-p^{k-1})L(\phi^{-1}, 1-k)=-\frac{\tau(\phi)}{2N^k}\phi(|c|) L_p(\phi^{-1}\omega_p, 1-k).
\alpha[u\omega] + \sum_{i,l\geq0}\alpha(\lambda_{i,1,l}[u^{i+1}\omega d^{l+1}] - \alpha\mu_{i,1,l}[u^{i+2}\omega d^l]) = =\sum_{i\geq1,l\geq0}\lambda_{i,1,l}\frac{\alpha^{i}-1}{\alpha-1}[u^i\omega d^l] - \sum_{i\geq0,l\geq1}\alpha\mu_{i,1,l}\frac{\alpha^{l}-1}{\alpha-1}[u^{i+1}\omega d^{l-1}].
\MoveEqLeft -\int_{t_0^S}^{t_0^S(a)}a^{-1}\Big( f (\Theta_a^{Q_x}(t;aS),x+atu) + f (\Theta_u (t;S),x)\Big) dt &=-\int_{t_0^S}^{t_0^S+a\psi_0^S(x)}a^{-1}\Big( f (\Theta_a^{Q_x}(t;aS),x+atu) + f (\Theta_u (t;S),x)\Big) dt+o(1)
A_j(x'')=\frac{1+\Lambda_{1 j}^2(R^{-1}(x''))}{S_j(x'')(1-\Lambda_{1 j}^2(R^{-1}(x''))},B_j(x'')=\frac{\Lambda_{1 j}(R^{-1}(x''))}{S_j(x'')(1-\Lambda_{1 j}^2(R^{-1}(x''))}.
\ln\left\lbrace\prod_{l\neq j}\frac{|\sin\pi(\theta-\theta_l)|}{|\sin\pi(\theta_j-\theta_l)|}\right\rbrace=\sum_{l\neq j}\ln|\sin\pi(\theta-\theta_l)|-\sum_{l\neq j}\ln|\sin\pi(\theta_j-\theta_l)|< (2sq_{n-n_0}-1)\epsilon.
S_{\mathrm{b o s o n}}^{\mathrm{r o o t}}= \int d^{4}x \left\{\frac 1{4 g^{2}}F_{\mu \nu}^{2}+ \bar{\nabla}_{\mu}\bar{q}_{A f}\nabla_{\mu}q^{f A}+ \frac{g^{2}}8 [ \mathrm{~ T r ~}( \bar{q}\tau^{m}q ) ]^{2}\right\},
\widetilde{\Pi}_{x}\left[f(\tau_{1},\widetilde{\xi}_{\tau_{1}-},\widetilde{\xi}_{\tau_{1}})1_{\{\tau_{1}\le t\}}\right] =\widetilde{\Pi}_{x}\left[\int_{0}^{t\wedge \tau_{1}}q(\widetilde{\xi}_{s}){\rm d}s\int_{E}f(s,\widetilde{\xi}_{s},y)\pi^{h}(\widetilde{\xi}_{s},{\rm d}y)\right].
\| \hat{x}- \hat{x}_{k}\|^2_{\hat{A}^T\hat{A}}= (\hat{x}- \hat{x}_{k})^T \hat{A}^T\hat{A}(\hat{x}- \hat{x}_{k}) = (x - x_k)^T L L^{-1}A^TA L^{-T}L^T (x - x_k) = \|{x}-{x}_{k}\|^2_{{A}^T{A}}.
\begin{aligned}\|v\|_*\leq~ 2\rho_1\int_0^T\left\|\left(\frac{e^{u_1}}{\int_{M}e^{u_1}}-\frac{e^{u_2}}{\int_{M}e^{u_2}}\right)\right\|_{L^2}\mathrm{d}s &+2\rho_2\int_0^T\left\|\left(\frac{e^{-u_1}}{\int_{M}e^{-u_1}}-\frac{e^{-u_2}}{\int_{M}e^{-u_2}}\right)\right\|_{L^2}\mathrm{d}s.\end{aligned}
\frac{\partial \Phi_{j+1}}{\partial s_j}= \frac{\zeta'(s_j)[\xi(s_{j+1})-\xi(s_{j})]-\xi'(s_j)[\zeta(s_{j+1})-\zeta(s_{j})]}{[\xi(s_{j+1})\zeta(s_{j})-\xi(s_{j})\zeta(s_{j+1})]^2}=:\frac{\zeta'(s_j)N_3-\xi'(s_j)N_4}{D_2^2}
\frac{d}{dt}|\varphi_t (\omega^0,x) - \varphi_t ( \omega^0,y ) |^2 = 2 \left\langle b( \varphi_t (\omega^0,x) ) - b(\varphi_t (\omega^0,y)), \varphi_t (\omega^0,x) - \varphi_t (\omega^0,y) \right\rangle &\leq -4 \left| \varphi_t (\omega^0,x) - \varphi_t (\omega^0,y) \right|^2.
\|\hat{A}^T\hat{A}- A^TA\|_{\mathrm{op}}= \|(\hat{A}-A)^T\hat{A}+ A^T(\hat{A}-A)\|_{\mathrm{op}}\leq (\|\hat{A}\|_{\mathrm{op}}+ \|A\|_{\mathrm{op}})\|\hat{A}-A\|_{\mathrm{op}}&\leq (2\sigma_1 + \|\hat{A}-A\|_{\mathrm{op}})\|\hat{A}-A\|_{\mathrm{op}}.
\psi_{l}( k , r ) = 2^{- 1 / 2}( - 1 )^{m}i^{| l |}e^{i \delta_{l}}( J_{\scriptscriptstyle{W}}( k r ) \cos \Delta_{\scriptscriptstyle{l}}- Y_{\scriptscriptstyle{W}}( k r ) \sin \Delta_{\scriptscriptstyle{l}}) ,
E_{U_{\mathcal{C}}}(e^{S_{\mathcal{C},h}\mu^0_{\mathcal{C}}(\tilde{f})}) = \frac{\cosh\left(h\mu^0_{\mathcal{C}}(D)+\mu^0_{\mathcal{C}}(\tilde{f})\right)}{\cosh\left(h\mu^0_{\mathcal{C}}(D)\right)},
\frac{1}{r^{2}}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho \lambda''(s)}{\sqrt{(s-t)^{2}-\rho^{2}}}\left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(1+\rho^{2}-r^{2})^{2}+4r^{2}}}+F_{3}(r,\rho,\lambda(s))\right) = \frac{-v_{3}(t,r)}{r}
ds^2_{deS}= -{\ell^2\over 4}\left({d\rho\over \rho}\right)^2 +{1\over \rho}\left[\left(1+{\ell^2\rho\over 4}\right)^2 d\vec n\cdot d\vec n\right] = -{\ell^2\over 4}{d\rho^2\over \rho^2}+{1\over \rho}\,g_{ab}(x,\rho)\,dx^a dx^b\,,
\Gamma_{j\bar{\jmath}}\sim c_{j_0,j}c_{j_0,\bar{\jmath}}+ d_{j_0,j}d_{j_0,\bar{\jmath}}+ \frac{1}{2}c_{j_0,j}d_{j_0,\bar{\jmath}}+\frac{1}{2}d_{j_0,j}c_{j_0,\bar{\jmath}},
Q_{m,n}(1,1) = \prod_{k=1}^{n-2}{\rm Cat}_{k}\prod_{1\le i < j \le n-1}\frac{2(m+1)+i+j-1}{i+j-1}, \sum_{k \ge 0}\iota ({\cal{P}}_{n,m};k) \beta^k = \frac{Q_{m,n}(1,\beta-1)}{(1-\beta)^{{n+1 \choose 2}+1}},
{\eta}_{2}( x_{-}, x_{+}) ={\chi}_{2}( x_{+}) \exp \{\frac{i}{2}\int_{a \mathrm{L}}^{x_{-}}d z V ( z , x_{+}) - \frac{i}{2}( E + \frac{1}{2}( e_{1}- e_{2}) b ) ( x_{-}- a \mathrm{L}) \},
{}^{M\!L\!}\textrm{F}_{p}(a,b;c;z)= \frac{1}{B(b,c-b)}\int_{0}^{\infty}u^{b-1}( 1+u)^{a-c}(1+u(1-z))^{-a}E_{\alpha, \beta}^{\gamma}\left(-p\left(2+u+\frac{1}{u}\right)\right)du.
\mathbf{1}_X \tau^\vee_i = \mathbf{1}_X \Big(T_i + \frac{t^{-\frac12}-t^{\frac12}}{1-Y^{-1}_i Y_{i+1}}\Big)= \mathbf{1}_X \Big(t^{\frac12}+ \frac{t^{-\frac12}-t^{\frac12}}{1-Y^{-1}_i Y_{i+1}}\Big) = \mathbf{1}_X \Big(\frac{t^{-\frac12}-t^{\frac12}Y^{-1}_iY_{i+1}}{1-Y^{-1}_i Y_{i+1}}\Big) = \mathbf{1}_X c^{Y^{-1}}_{i,i+1}.
\left( C \Gamma^{n}\Gamma_{l}S \Gamma_{n m_{1}\ldots m_{2 q - 1}}\right)_{\left( s y m \right)}= - \left( D - 2 q - 1 \right) \cdot \left( C S \Gamma_{l m_{1}\ldots m_{2 q - 1}}\right) \quad ,
\sum_{i=0}^{h}\sum_{j=0}^{k}c_{ij}^{\tilde R}D_s^{h}B_{i,n_1-1}^{{\tilde R}}(a_{\tilde R})D_t^{k}B_{j,n_2-1}^{{\tilde R}}(c_{\tilde R})=\sum_{i=n_1-1-h}^{n_1-1}\sum_{j=n_2-1-k}^{n_2-1}c_{ij}^R D_s^{h}B_{i,n_1-1}^{R}(b_R)D_t^{k}B_{j,n_2-1}^{R}(d_R),