harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[y_1, y_1]=y_3, [y_2, y_1]=y_4+\theta y_5, [y_2, y_2]=\frac{\alpha_1\beta_6\gamma_1}{\beta^2_2\gamma_5}y_5, [y_1, y_3]=y_4, [y_2, y_3]=y_5, [y_1, y_4]=y_5.
\begin{array}{l}{{\delta \theta^{\alpha}= \epsilon^{\alpha}, \qquad \delta \bar{\theta}_{\alpha}= \bar{\epsilon}_{\alpha},}}\\{{\delta x^{\bar{\mu}}= - i \bar{\epsilon}_{\alpha}\tilde{\Gamma}^{{\bar{\mu}}\alpha \beta}\bar{\theta}_{\beta}+ i \epsilon^{\alpha}\Gamma_{\alpha \beta}^{\bar{\mu}}\theta^{\beta}, \qquad \delta x^{1 1}= 0 ,}}\\ \end{array}
\mathrm{X}^{l m 0}\partial_{i}\mathrm{D}^{l m}- \partial_{m}( \mathrm{X}^{m l 0}\mathrm{D}_{l i}) - \partial_{l}( \mathrm{X}^{m l 0}\mathrm{D}_{m i}) - q{}\partial_{i}\partial_{l}\partial_{m}\mathrm{X}^{l m 0}= 0 .
||\partial_{r}v_{3}(t)||_{L^{2}(r dr)}\leq C \int_{t}^{\infty}||F_{0,1}(s)||_{L^{2}(r dr)}ds \leq C \int_{t}^{\infty}\frac{\sqrt{\log(\log(s))}}{s^{2}\log^{b+1}(s)}ds &\leq \frac{C \sqrt{\log(\log(t))}}{t \log^{b+1}(t)}
\times{\cal D}h_{\mu\nu}^a\exp\Biggl\{\int d^4x\Biggl[-\frac{1}{24\eta^2}\left(H_{\mu\nu\lambda}^a\right)^2-\frac{3g_m^2}{8}\left(h_{\mu\nu}^a\right)^2+i\pi h_{\mu\nu}^a\bar\Sigma_{\mu\nu}^a\Biggr]\Biggr\}.
\delta_{j0}=\int \theta^*(x)\theta(x-j)dx=\sum_{n=-\infty}^{\infty}|a(n)|^2e^{-i2\pi jn/L}&=|a(0)|^2+\sum_{n=1}^{\infty}(|a(n)|^2+|a(-n)|^2)\cos(2\pi jn/L)-i \sum_{n=1}^{\infty}(|a(n)|^2-|a(-n)|^2)\sin(2\pi jn/L).
f(X(K)) = \{(x_1, \ldots , x_n ) \in (K^\circ)^n \ \big| \ \exists (z_1, \ldots , z_N) \in (K^\circ)^N \ \g_i(z)=0, \i=1\ldots m, f_j(z)=x_j, \j=1 \ldots n \}
\beta{\frac{F}{L}}= e_{0}\beta - \int_{- \infty}^{\infty}{\frac{d p}{2 \pi}}\ln \left( 1 + e^{- \beta \epsilon ( p )}\right) = e_{0}\beta +{\frac{m}{\pi}}\sum_{n = 1}^{\infty}{\frac{( - )^{n}}{n}}K_{1}( n m \beta )
a_{i}^{I}({\bf r}, t ) = \epsilon^{i j}\nabla_{j}\frac{1}{\kappa_{I}}\sum_{p}q_{p}^{I}\int \! d^{2}\!{\bf r}^{\prime}G ({\bf r}-{\bf r}^{\prime}) \rho_{p}({\bf r}^{\prime}, t ) ,
X = \;\{1, 2, 3, 4, 5, 6, 7, 8, 9\}, \mathcal{A}= \;\{\{1, 2, 3\},\{4, 5, 6\},\{7, 8, 9\}, &\{1, 4, 7\},\{2, 5, 8\},\{3, 6, 9\}\},
\dim(\langle{}v_4(A)\rangle{}\cap \langle{}v_4(B)\rangle{})= \ell_A+\ell_B-H_Z(4)-1 = \ell_{A'}+\ell_B-H_{Z'}(4)-1 &=\dim(\langle{}v_4(A')\rangle{}\cap \langle{}v_4(B)\rangle{}),
a = \left[\begin{array}{c}1 2 1 6 2 1 \end{array}\right],B = \left[\begin{array}{cccccc}4 1 4 2 4 4 1 1 4 0 1 7 4 4 4 6 6 7 2 0 6 6 7 9 4 1 6 7 3 0 4 7 7 9 0 3\end{array}\right],d = \left[\begin{array}{c}8 7 6 4 7 6 \end{array}\right].
\frac{\big[\tilde{\mathbb{A}}/f\tilde{\mathbb{A}}\big]_{\tilde{\mathbb{A}}}}{\big[\psi(\tilde{\mathbb{A}}/f\tilde{\mathbb{A}})\big]_{\tilde{\mathbb{A}}}}=\frac{f}{f + c(f)p_1t^df(z_1)+ c(f)p_2t^{2d}f(z_1)^2 + \dots +c(f)t^{dr_0}f(z_1)^{r_0}}&=\frac{1}{1 + c(f)p_1t^df(z_1)f^{-1}+ \dots +c(f)t^{r_0d}f^{r_0-1}f(z_1)^{r_0}f^{-r_0}}&=D^{\psi}_f(f^{-1})^{-1}.
\hat h^+_{\mu\nu}= a^2\xi_{\mu\nu}+ 2 a^2 \left(\int \frac{dz}{a^4}\right)\partial_{\mu}\partial_{\nu}\xi - 2 \frac{a^2}{a_+^2}\left(\int \frac{dz}{a^2}\right)\partial_{\mu}\partial_{\nu}\xi-\frac{U_B}{2}\frac{a^2}{a_+^2}\xi \eta_{\mu\nu}
G_9 := E\sup_{t_0 \leq t \leq u}\int_{t_0}^{t}\int_{Z}\{|e_s^n|^{q-2}|\bar \gamma_s^n(z)|^2+|\bar \gamma_s^n(z)|^q \}N(ds,dz) &= E\int_{t_0}^{u}\int_{Z}|e_s^n|^{q-2}|\bar \gamma_s^n(z)|^2 \nu(dz) ds +E\int_{t_0}^{u}\int_{Z}|\bar \gamma_s^n(z)|^q \nu(dz) ds =:H_1+H_2
\forall \gamma^1,x^X\big(\gamma>_\mathbb{R}0\land\tilde{\rho}>_\mathbb{R}-\gamma &\qquad\qquad\qquad\land \exists y^X,z^X(z\in Ay\land x=_Xy+_X \gamma z)\rightarrow \gamma^{-1}(x-_XJ^A_{\gamma}x)\in A(J^A_{\gamma}x)\big),
T ( t ) \approx D_{1}( k ) ( t S ( t ) )^{- 1 / 2}\exp \left[ - i \int^{t}S ( t^{\prime}) d t^{\prime}\right] + D_{2}( k ) ( t S ( t ) )^{- 1 / 2}\exp \left[ i \int^{t}S ( t^{\prime}) d t^{\prime}\right] .
0 \leq \rho\tau\Theta_1 + (1-\rho)\Theta_1 + 2\sigma^2 \rho^2 T + \frac{L^2\gamma^2}{\rho}\sum_{t=1}^T\Delta_t +\frac{\gamma^3\tau^2\eta C_L}{2}\sum_{t=1}^T\Delta_t &~~~~~-\gamma\left(\mu-\frac{\gamma L}{2}-\frac{2}{\eta}\right)\sum_{t=1}^T\Delta_t + (\frac{\gamma\eta}{2}-\rho)\sum_{t=1}^{T+1}\Phi_t
|\phi(t, u - w) - \phi(t, u- w_n)| \leq |\phi(t, u-w) - \phi_{m_n}(t, u-w)| &\phantom{\leq}{}+ |\phi_{m_n}(t, u-w) - \phi_{m_n}(t, u-w_n)| &\phantom{\leq}{}+ |\phi_{m_n}(t, u-w_n) - \phi(t , u-w_n)| &\leq 2 \|\phi - \phi_{m_n}\|_{\infty}+ \frac{T_{n+1}^2}{16}& \leq \frac{2 T_{n+1}^2}{32}+ \frac{T_{n+1}^2}{16}& = \frac{T_{n+1}^2}{8}.
\begin{cases}-\Delta u_k=\left|x\right|^{2\alpha_k}U_{k}e^{bu_k}+\sigma_k\quad\mbox{in}\Omega, \displaystyle{\int_{\Omega}\left|x\right|^{2\alpha_k}U_{k}e^{bu_k}}\leq C, \displaystyle{\max_{\partial \Omega}}u_k - \displaystyle{\min_{\partial \Omega}}u_k \leq C,\end{cases}
f(t,x,v) = -\int_0^t E(s,X_{s,t}(x,v)) \cdot \nabla_v \mu( V_{s,t}(x,v))-\int_0^t e_0 \nabla \Phi(X_{s,t}(x,v)-X(s))\cdot \nabla_v \mu( V_{s,t}(x,v)).
\dot{W}(x) \leq \sum_{k=1}^{\infty}\frac{2^{-k}}{1+M(k,k)}\dot{V}_k (x) \leq \sum_{k=1}^{\infty}\frac{2^{-k}}{1+M(k,k)}\Big(V_k(x) + \frac{1}{k}\Big) = W(x) + C_2,
\xi_f(j; u, v) + (-1)^j \xi_f(k-2-j; v, -u) = 0, \xi_f(j; u, v) + \sum_{i=0}^{k-2-j}(-1)^{k-2-i}\binom{k-2-j}{i}\xi_f(i; v, -u-v) &+ \sum_{i=k-2-j}^{k-2}(-1)^i \binom{j}{i-k+2+j}\xi_f(i; -u-v, u) = 0, \xi_f(j; u, v) - (-1)^{k-2}\xi_f(j, -u, -v)=0,
H_{b}= \frac{1}{2}\left( p_{x}^{2}+ \omega_{b}^{2}x^{2}\right) = \frac{\omega_{b}}{2}[ a^{+}, a^{-}]_{+}= \omega_{b}\left( N_{b}+ \frac{1}{2}\right) , \quad N_{b}= a^{+}a^{-},
M_x[a_1-1] =M_x[\Phi_B(T_{\delta})^2]+M_x[r_1(T_{\delta})], M_x[(a_1-1)^2]=M_x[\Phi_B(T_{\delta})^2]+2 M_x[\Phi_B(T_{\delta})^3]+\mathtt{Q}_2(T_{\delta}), begin{align*}1.5mm]M_x[(a_1-1)^3] =4 M_x[\Phi_B(T_{\delta})^3]+\mathtt{Q}_3(T_{\delta}),
Q_2 =\frac{1}{k}\left[\begin{array}{cccc}2k_1c\!-\!2\epsilon m_1k mk_1\!-\!m_1k \!+\! \epsilon m_1c k_2 c \epsilon m_2 c * 2\epsilon m m_1 mk_2 \epsilon m m_2 * * -2\epsilon_2 k -m_2k * * * 0\end{array}\right]
\frac{I_n(f)}{I_{n-1}(f)}=\frac{\omega_n}{\omega_{n-1}}\frac{\int_0^\infty f(r)r^{n-1}dr}{\int_0^\infty f(r)r^{n-2}dr}& \leq \frac{\kappa_n}{\kappa_{n-1}}\frac{n}{n-1}\frac{\int_0^\infty f(r)r^{n-2}r_0 dr}{\int_0^\infty f(r)r^{n-2}dr}& \sim r_0\sqrt{\frac{2\pi e}{n}}
V_{n m}^{+}( z , \bar{z}) = \sqrt{2}\cos ( p \phi ( z ){+}\bar{p}\bar{\phi}( \bar{z}) ) ~ ~ , V_{n m}^{-}( z , \bar{z}) = \sqrt{2}\sin ( p \phi ( z ){+}\bar{p}\bar{\phi}( \bar{z}) ) ~ , ~ ~ ~
\|y\|_{L^2(Q)}^2 \leq C (\underset{[0,T]}{}\|y_x(t)\|^2 + \|y_{xx}\|_{L^2(Q)}^2) \leq C (\|f\|_{L^2(\mathcal{O}\times (0,T))}^2 + \|y_0\|^2+ \sum_{i=1}^2 \frac{1}{\mu}\|\phi^i\|_{L^2(\mathcal{O}_i \times (0,T))}^2)
\widehat \mu_{k_n}:=\dfrac{1}{k_n}\sum_{i=1}^{k_n}X_{n,i},\;\;\;\;\widehat \sigma^2_{k_n}:=\dfrac{1}{k_n}\sum_{i=1}^{k_n}(X_{n,i}-\widehat\mu_{k_n})^2.
\mathcal{T}: = \left\{\varphi \in L^\infty(0,T\,; L^p(\Omega)) \cap L^p (0, T\,; W^{1, p}_0 (\Omega)) \,\,\Bigg|\left.\begin{array}{c}\partial_t \varphi \in L^p(\Omega_T), begin{align*}1mm]\varphi (x,0)= \varphi (x,T)=0 \textrm{a.e.}\,\, x \in \Omega \end{array}\right.\right\},
|u(y,t)-Q_0|<2\epsilon \mbox{for}|y|\leq\min I_1(t), &|u(y,t)-Q_k|<2\epsilon \mbox{for}|y|\in[\max I_k(t),\min I_{k+1}(t)],\; k=1,..., n_0-1, &0<u(y,t)<2\epsilon \mbox{for}|y|\geq \max I_{n_0}(t),
J(y_1, y_2, \cdots, y_m) = \frac{1}{p}\|(y_1, y_2, \cdots, y_m )\|_{Y,1}+ \frac{1}{q}\| (\nabla y_1, \nabla y_2, \cdots,\nabla y_m )\|_{Y,3}- \sum_{i = 1}^{n}\frac{1}{1 - \nu}\int_{\Omega}a_i\vert y_i\vert^{1 - \nu}dz - \lambda\int_{\Omega}\vert y_1\vert^{\kappa_1 + 1}\vert y_2\vert^{\kappa_2 + 1}\cdots \vert y_m\vert^{\kappa_m + 1}dz.
dS^2 ={H_6}^{-1/2}[ du dv - \mu^2\sum_{i=1}^4{z^i}^2 du^2 + \sum_{i=1}^4 dz^i dz^i + d\tilde x^2 ] +{H_6}^{1/2}(dr^2 + r^2 d{\Omega_2}^2),
{\cal L}= - \frac{1}{4}\exp ( \frac{\phi}{f_{\phi}}) F_{\mu \nu}{}^{a}F^{\mu \nu}{}_{a}- \frac{1}{2}\partial^{\mu}\phi \cdot \partial_{\mu}\phi + \sum_{f = 1}^{N_{f}}\overline{{{q}}}_{f}( i \gamma^{\mu}\partial_{\mu}+ g \gamma^{\mu}A_{\mu}{}^{a}X_{a}- m_{f}) q_{f},
\sum_{j=0}^{n}\frac{(-1)^{j}q^{(j-1)j/2}}{(q;q)_{j}{}^{2}}\left(\sum_{k=0}^{n-j}(q^{k+1};q)_{j}(q^{n-j-k+1};q)_{j}\, a^{k}\right)\! m_{j}(a,q)=0\ \ \ \n\geq1.
g^{( + )}= \left( \begin{array}{c c c}{{\frac{1}{2}\alpha^{( + ) 2}}}&{{}}&{{}}\\{{- \frac{i}{\sqrt{2}}\alpha^{+}}}&{{- \frac{1}{2}\alpha^{( + ) 2}}}&{{\tau_{m}}}\\{{- s^{( + )}\tau^{* n}}}&{{}}&{{}}\\ \end{array}\right)
\lambda(g_{-s}E_{T,c}\cap c^{-1}g_{-t}E_{T,c}) = \lambda\left(g_{\tau}\left(g_{-s}E_{T,c}\cap c^{-1}g_{-t}E_{T,c}\right)\right) =\lambda(g_{\tau-s}E_{T,c}\cap c^{-1}g_{\tau-t}E_{T,c})
\mathrm{R}_t{}^\delta_{\alpha\beta\gamma}=\frac{\mathbf{g}_{\alpha\gamma}\delta^\delta_\beta -\mathbf{g}_{\beta\gamma}\delta^\delta_\alpha}{2c},~~~~~ \mathrm{Ric}_t{}_{\beta\gamma}=\mathrm{R}_t{}^\alpha_{\alpha\beta\gamma}=-\frac{\mathbf{g}_{\beta\gamma}}{2c},~~~~~ \mathfrak{R}_t=\mathrm{Ric}_t{}_{\beta\beta}=-\frac{1}{c}.
\P (\varrho_{R, i}Y_{R, i}>S_{R,i}+ x - \gamma(KL)\,|\,{\cal F}^c) =\beta^*_{K', L'}(S_{R,i}+ x) e^{-\sqrt{2d}(S_{R,i}+ x)}\to 0 \mbox{as}KL\to \infty\,.
\int_{0}^{\infty}d \nu \int_{m}^{\infty}d k \ [ k^{2}- m ]^{1 / 2 - s}\frac{\partial}{\partial k}\frac{t^{j}}{\nu^{n}}= \frac{( m R )^{2 - n}}2 \frac{\Gamma ( \frac 3 2 - s ) \Gamma ( 1 + \frac{j - n}2 ) \Gamma ( s + \frac{n - 3}2 )}{\Gamma ( \frac j 2 )}.
\footnotesize\aligned{\bf g}^{\sf red}:=\left( \begin{array}{cccc}a_1^{p}\overline a_1^{q}0 \cdots 0 \vdots \ddots 0 0 0 \cdots \overline a_1 0 0 \cdots 0 a_1 \end{array}\right).\endaligned
\|(\lambda I-BA) u \| \|u\| =\|B(\lambda B^{-1}-A) u \| \|u\| &\geq \frac{1}{\|B^{-1}\|_{\mathcal{L(H)}}}\|(\lambda B^{-1}I-A)u \|\|u\| &\geq \frac{1}{\|B^{-1}\|_{\mathcal{L(H)}}}|(\lambda B^{-1}u-Au,u)| &=\frac{|(B^{-1}u,u)|}{\|B^{-1}\|_{\mathcal{L(H)}}}|\lambda-\frac{(Au,u)}{(B^{-1}u,u)}|.
\mathfrak{M}(L,j,\delta)= \bigg\{\gamma\in \Gamma(i,\alpha_1^j\alpha_2^j)\colon u_{\nu}(\gamma_{\nu}.P_{\nu},P_{\nu})\leq \delta_{\nu}\nu \alpha_1,\alpha_2\in \mathcal{P}(L) \bigg\}
f = (f_0 + f_1 + f_2 + f_3)^4 + (f_1 + f_3)^4 (g+1) + (f_2 + f_3)^4 (g+1)^2 + f_3^4 (g+1)^3 &= f_0^4 + (f_3 g)^4 g^{-1}+ (f_2 g)^4 g^{-2}+ (f_1 g)^4 g^{-3}&= f_0^4 + (t^{-1}f_1)^4 t^4 g + (t^{-2}f_2)^4 (t^4 g)^2 + (t^{-3}f_3)^4 (t^4 g)^3.
:{\rho}_{\pm}( 0 ) : | m_{1}; m_{2}; . . . ; m_{n}; A ; \pm \rangle = \pm \sum_{k = 1}^{n}\mathrm{s i g n}({\varepsilon}_{m_{k}, \pm}) | m_{1}; m_{2}; . . . ; m_{n}; A ; \pm \rangle .
{x_{T}}={x_{T-1}}+{A}_{\cdot , T}\epsilon_{T-1}={x_{T-2}}+{A}_{\cdot,T-1}\epsilon_{T-2}+{A}_{\cdot,T}\epsilon_{T-1}& \cdots \cdots={A}_{\cdot, 1}\epsilon_{0}+....+{A}_{\cdot,T}\epsilon_{T-1}=A{z}
(J\circ r_1\circ J^{-1})(x,y,z) =J(r_1(x,u,\lambda_u^{-1}(v))) =J(y,\rho_u(x),\lambda_u^{-1}(v)) =(y,\lambda_y(\rho_u(x)),\lambda_y(\lambda_{\rho_u(x)}(\lambda_u^{-1}(v)))) =(y,\sigma_y(x),z) =s_1(x,y,z).
\eta_{j,N}\le \frac 1N \sum_{h=1}^{\infty}\frac{d^{2h}}{h! (h-1)!}(1+x_j)^h \left(1-x_j - \frac 1N \right)^{h-1}&\le \frac 1N \sum_{h=1}^{\infty}\frac{d^{2h}}{h! (h-1)!}(1- x_j^2)^{h-1}\left(1+x_j \right)\,.
T r [ \, Q_{2}\lambda_{S U ( 2 )}^{2}\, ] = - \frac{2}{3}\sum_{k = 1}^{3}T r [ \, \gamma_{k , 7_{4}, ( 0 )}\lambda_{U ( 1 )_{2 , ( 0 )}}\, ] \, T r [ \, \gamma_{k , 7_{4}, ( 0 )}^{- 1}\lambda_{S U ( 2 )}^{2}\, ]
\lambda(P\oplus_{[m]}P_{\mathcal{G}}) =\max\left\{\lambda_{\mathcal{E}}(x)+\lambda(\mathcal{G}) \left(\sum_{i=1}^mx_i^r\right) : x=(x_1, \dots, x_{m})\in \Delta_{m-1}\right\}&=1-(1-\lambda(\mathcal{G}))\max_{x\in \Delta_{m-1}}\sum_{i=1}^mx_i^r &=1-\frac{1-\lambda(\mathcal{G})}{m^{r-1}}.
\int_{\mathbb{R}^n\setminus \overline{B_\rho}}|(\nabla\Delta)_0\varphi|^2dx= (n-6)^2(n-4)^2 \int_{\mathbb{R}^n\setminus \overline{B_\rho}}\frac{u_\epsilon^2r^2}{(\epsilon^2+r^2)^6}\Big[(n+2)\epsilon^2+4r^2\Big]^2dx \leq C\int_{\rho/\epsilon}^\infty\sigma^{5-n}d\sigma=O\big(\epsilon^{n-6}\big).
\tilde \Psi_{y}= \tilde \Psi_{xx}+2 (\alpha-1)(\tilde \Psi\tilde \Psi^*) \Psi_x+ 2\alpha \tilde \Psi^2\tilde \Psi^*_x -\alpha(1+\alpha)(\tilde \Psi\tilde \Psi^*)^2 \tilde \Psi.
e^{2 A}b \left( - \dot{b}\dot{\phi}- b \ddot{\phi}- 2 \dot{A}b \dot{\phi}\right) + 4 A^{\prime}\phi^{\prime}- \frac{b^{\prime}\phi^{\prime}}{b}+ \phi^{\prime \prime}= b^{2}\frac{\partial U}{\partial \phi}\; .
{\frac{P_{i j}}{P_{j i}}}={\frac{F ( \Delta E )}{F ( - \Delta E )}}= e^{- 2 \pi \Delta E}\left|{\frac{\cosh \alpha + \sinh \alpha e^{i \beta}e^{\pi \Delta E}}{\cosh \alpha + \sinh \alpha e^{i \beta}e^{- \pi \Delta E}}}\right|^{2}.
\left\|\frac{k}{\pi\sigma(p\tau-q)}\right\|_{L^1}=\frac{1}{\pi}\int_D\left|e^{-2(p\eta_3-q\eta_1)z}\cdot \frac{\sigma(z+p\tau-q)}{\sigma(p\tau-q)\sigma(z)}\right|\,\frac{\sqrt{-1}}{2}dz\wedge \overline{z}
p^\star_{_k}=\frac{1}{\hat{\lambda}^{2}_{_k}}\left(\sqrt{\left(\beta_k+\frac{\alpha_k}{2}\right)^2+\left(\frac{\alpha_k\hat{\lambda}^{2}_{_k}}{\mu}-\beta^2_k-\alpha_k\beta_k\right)}-\left(\beta_k+\frac{\alpha_k}{2}\right)^2 \right),\quad\forall k
\hbox{d}^\mathbf{A}p_I + \frac{1}{2}\hat{e}^{(N-2)}_{IJK}\wedge\Omega^{JK}- \mathbf{F}{^J}_{Ib}\p{_J}^{jb}\ \hat{e}^{(N)}_j= \Lambda_0\hat{e}^{(N)}_I
(d_A\omega)(X_1,\dots, X_{k+1})= \sum_{i=1}^{k+1}(-1)^{i+1}\rho(X_i)[\omega(X_1,\dots, \widehat{X}_i,\dots,X_{k+1})] &+\sum_{i<j}(-1)^{i+j}\omega([X_i,X_j],X_1,\dots, \widehat{X}_i,\dots,\widehat{X}_j,\dots, X_{k+1})
R_{pqr}=\frac{f_{qr}f_{rp}}{f_{pq}}\,,\left(f_{pq}\right)^{rN}=\prod_{k=1}^{rN-1}\left(\frac{\mu_{q}(x_{q}-\omega^ky_{p})}{\mu_{p}(x_{p}-\omega^ky_{q})}\frac{(1-\omega^k)(x_{p}y_{p}-\omega^kx_{q}y_{q})}{(x_{p}-\omega^kx_{q})(y_{p}-\omega^ky_{q})}\right)^{k}\,.
x_n = \kappa_0 \prod_{i=0}^{n-2}\varphi_t(s_0+i\alpha), x'_n = \kappa'_0 \prod_{i=0}^{n-2}\varphi_{t'}(s'_0+i\alpha)= \kappa_0' \prod_{i=0}^{n-2}\varphi_{t'}(s_0+u+i\alpha),
\{\frac{1-a_0}{4}-3T|k|(C^{\Omega}_{H^1,L^4})^2 \overline{m},\frac{b(1-\delta)}{2}-3|k|(C^{\Omega}_{H^1,L^4})^2 \overline{m}, \frac{c^2}{4}\}\,|||u|||^2 \leq \ |k|(C^{\Omega}_{H^1,L^4})^2 \overline{m}(T+1)\max\{1,T\}\,|||v|||^2,
\left<\phi^2(x)\right>_{\mu^2}=\frac{a^{2-N}}{2^{N-1}\pi^{N/2}\Gamma\left(\frac N2\right)}\sum_{n=0}^{\frac N2-1}c^N_{2n+1}\left[{1\over 2}(-a^2b)^n\left( d_n-\ln(b/\mu^2)\right)-2H_n(1;a\sqrt b)\right].
\left\Vert I_{2}(g)\right\Vert_{L^{p}\left( L^{\widetilde{q}}\right)}\lesssim_{\, q}\left( \int\left\Vert g(z,~\cdot~)\right\Vert_{q}^{p}~dz\right)^{1/p}& =\left\Vert g\right\Vert_{L^{p}\left( L^{q}\right)}.
\mathcal{J}_{\varepsilon_{j}}(v_{j},P_{\rho})={}\mathcal{J}_{\varepsilon_{j}}(u_{j},A_{j})+\mathcal{J}_{\varepsilon_{j}}(v_{j},S_{j})+\mathcal{J}_{\varepsilon_{j}}(\widetilde{u}_{j},B_{j}) & +2\mathcal{J}_{\varepsilon_{j}}(v_{j},S_{j},A_{j}\cup B_{j})+2\mathcal{J}_{\varepsilon_{j}}(v_{j},A_{j},B_{j})~.
\int_{y<f_q^\epsilon(x)-\epsilon}(f_q^\epsilon(x)-\epsilon-y)^{q-1}d\rho_x(y)\geq\int_{y<f_q(x)}(f_q^\epsilon(x)-\epsilon-y)^{q-1}d\rho_x(y) &\geq\int_{y<f_q(x)}(f_q(x)-y)^{q-1}d\rho_x(y)=\int_{y>f_q(x)}(y-f_q(x))^{q-1}d\rho_x(y)
\rho_{n}( x ) = \frac{1}{( x^{2})^{\frac{n - 2}{2}}}+ \sum . . \sum_{q \neq 0}\{\frac{1}{| x - q |^{n - 2}}- \frac{1}{| q |^{n - 2}}[ 1 + \frac{( n - 2 )}{q^{2}}( q \cdot x + \frac{1}{2 q^{2}}( n ( q \cdot x )^{2}- q^{2}x^{2}) ) ] \}
s_1 =\frac{1-t_1^2+t_1^2t_2^2}{-1+4t_1-t_1^2+t_1^2t_2^2}, s_2 =\frac{2(-1+2t_1-2t_1^2+t_1^3+2t_1^2t_2-t_1^3t_2-t_1^3t_2^2+t_1^3t_2^3)}{-1+5t_1-5t_1^2+t_1^3-t_1t_2+4t_1^2t_2-t_1^3t_2+t_1^2t_2^2-t_1^3t_2^2+t_1^3t_2^3}, s_3 =\frac{2(t_1t_3-t_1^3t_3+t_1^3t_2^2t_3)}{(1-t_1+t_1t_2)(-1+4t_1-t_1^2+t_1^2t_2^2)^2}.
\begin{gathered}e^{-r^{2}}H_{m,n}\left(re^{i\theta},re^{-i\theta}\right)=\frac{ie^{i\left(m-n\right)\theta}}{2^{m+n+1}\sqrt{\pi}}\times\int_{0}^{2\pi}H_{n+m+1}\left(r\cos\phi\right)\exp\left(-r^{2}\cos^{2}\phi+i\left(n-m\right)\phi\right)d\phi,\end{gathered}
{S_{\mathrm{\scriptsize ~ e f f}}}={\frac{1}{4 \pi \alpha^{\prime}}}\int{d^{2}}\xi{\partial_{a}}{X_{0}^{\mu}}{\partial^{a}}{X_{0 \mu}}+{\frac{g}{2 \sqrt{2 \alpha^{\prime}}}}{\int_{- \infty}^{+ \infty}}d \tau \int{d^{2 6}}k \exp \left[ i k \cdot{X_{0}}( \tau ) \right]{\hat{\mathcal{L}}_{\mathrm{\scriptsize ~ B}}}( k , \tau ) + \cdots ,
\left(\frac{F(x)^p - (1+\lambda x) G(f)}{\lambda^p}\cdot f - (1+\lambda x) \right)\cdot (1/y)^p + \frac{pf}{\lambda^{p-1}}F(x)^{p-1}\cdot (1/y)^{p-1}\qquad+ \cdots + \frac{pf}{\lambda}F(x)\cdot (1/y) + f = 0
\prod_{j=1}^m \gamma_{j,i}= \sigma_r^\frac{1}{n'}\cdot \prod_{j=1}^m a_j^{-1}\cdot \mathrm{exp}\left(2 \pi \sqrt{-1}(b' - b \, \frac{k'}{n'}) \cdot \iota(P-O_r)\right).
\tilde{V}(K)=\frac{1}{a}\int_{-a/2}^{a/2}dxV(x)e^{-iKx}=\frac{1}{a}\int_{-a/2}^{a/2}dx\sum_{n=-\infty}^{\infty}A\cdot\delta(x-na)e^{-iKx}=\frac{A}{a}.
\int_\Omega f \varphi\, dx =\sum_{i=1}^N\int_{\Omega_i}\varepsilon \nabla u \cdot \nabla \varphi\, dx =-\sum_{i=1}^N\int_{\Omega_i}\nabla \cdot \Big( \varepsilon \nabla u \Big) \varphi\, dx+\sum_{e \in \Gamma}\int_e [\varepsilon \nabla u \cdot \nu] \varphi\, ds.
w_n(t,x)=\sum_{j=1}^\ell \frac{1}{(\lambda_n^{(j)})^{\frac{d-2}{2}}}U^{(j)}\left(\frac{t-t_n^{(j)}}{\lambda_n^{(j)}}, \frac{x-x_n^{(j)}}{\lambda_n^{(j)}}\right)+r_n^{(\ell)}(t,x),
\begin{cases}B_x(x,t)\geq 0 \ \ \mbox{for}\x\in(-X(x_0, t),X(x_0, t)); B_x(x,t)\leq 0 \ \ \mbox{for}\x\in(-\infty, -X(x_0, t))\cap(X(x_0, t), \infty).\end{cases}
\tilde \Omega_jf=\lim_{\mbox{\scriptsize supp}\psi_a\downarrow\mathcal{T}_a}\lim_{t\to\infty}U_a(t)^*\varphi(H^a)(\psi_{aj}(H^a)-P^a_j)P_a^{{\varepsilon}_a}(t)e^{-itH}f.
\mathbf{N}_E/\langle \mathbf{e}_F\rangle=\mathbf{N}_E/\langle \mathbf{e}_{E \setminus F}\rangle \longrightarrow \mathbf{N}_F \oplus \mathbf{N}_{E \setminus F}, \mathbf{e}_I \longmapsto \mathbf{e}_{I \cap F}\oplus \mathbf{e}_{I \setminus F}.
\big[\psi(\tilde{\mathbb{A}}/ f \tilde{\mathbb{A}})\big]_{\tilde{\mathbb{A}}}=\det_{\mathbb{F}_q(z_1,\dots,z_n,t)[X]}(X-\psi_{\theta}\mid \tilde{\mathbb{A}}/f\tilde{\mathbb{A}}\otimes_{\mathbb{F}_q(z_1,\dots,z_n,t)}\mathbb{F}_q(z_1,\dots,z_n,t)[X])_{|X=\theta}.
\left( e^{- 2 i \omega_{j}\pi}- 1 \right) \left( e^{- 2 i \omega_{j^{\prime}}\pi}- 1 \right) \frac{e^{- i ( [ \omega_{j}] - [ \omega_{j^{\prime}}] ) \theta}}{\operatorname{s i n}^{2}\theta / 2}.
'd_2^{1,0}(\overline{z^1}) =\overline{d^{(0,1),1}(d^{\underline{0},2}(z^1_1))+d^{(0,1),2}(d^{\underline{0},2}(z^1_1))+d^{(1,0),1}(d^{\underline{0},1}(z^1_2))-d^{(1,0),2}(d^{\underline{0},1}(z^1_2))}& = \overline{d^{(1,0),2}d^{\underline{0},1}(z^1_1-z^1_2)}.
(1-\nu)I+\nu X \geqslant K(h^{\frac{1}{2^{n}}},2)^{r_{n}}X^{\nu}+\sum_{k=0}^{n-1}r_{k}[X^{\frac{m_k}{2^k}}-2X^{\frac{2m_k+1}{2^{k+1}}}+ X^{\frac{m_k+1}{2^k}}].
\left[\begin{array}{ll}H_{j,e}G_{j,e}H_{j,o}G_{j,o}\end{array}\right]=\left[\begin{array}{ll}H_{j,e}G_{j,e}^{[0]}H_{j,o}G_{j,o}^{[0]}\end{array}\right]\left[\begin{array}{ll}I_{n_j}-U_j 0 I_{n_j'}\end{array}\right]
2\Phi^i : = -k_0 y^i + \frac{1}{2}h_{00}\bar{k}^i - \dfrac{2m\pi}{m+1}\frac{h_{00}^{\frac{m+1}{2}}}{W_0^m}s_0^i &- \frac{2ms}{s^2 - ms^2 + mb^2}\Bigl(r_{00}+ \dfrac{2m \pi}{m+1}\frac{h_{00}^{\frac{m+1}{2}}}{W_0^m}s_0\Bigr),
\begin{cases}\mathbb{L}'_e(\hat{{\mathbf U}},\hat{\Psi})\dot{{\mathbf U}}^{\natural}=\mathbf{F}=\mathbf{f}-\mathbb{L}'_e(\hat{{\mathbf U}},\hat{\Psi})\tilde{{\mathbf U}}\Omega_T, \mathbb{B}'_e(\hat{{\mathbf U}},\hat{\Psi})(\dot{{\mathbf U}}^{\natural},\varphi)=0 \Gamma_T.\end{cases}
\put ( 0 , 0 ){\oval ( 1 0 0 , 4 0 ) [ t ]}\put ( 0 , 0 ){\oval ( 1 0 0 , 4 0 ) [ b l ]}\put ( 0 , - 2 0 ){\line ( 1 , 0 ){1 0 0}}\put ( 5 0 , 0 ){\vector ( 0 , - 1 ){2 0}}
\phi_{1,{1}}(\sigma)(a+b\alpha) =\sum_{i=0}^{n-1}(a+b\alpha)^{q^{i}}&=\sum_{i=0}^{n-1}a+b\sum_{i=0}^{n-1}\alpha^{q^{i}}&=na+b~\phi_{1,{1}}(\sigma)(\alpha) =b~\phi_{1,{1}}(\sigma)(\alpha).
A_n=\underset{k_j\in\mathbb{N}_0}{\sum_{\sum_{j=1}^{n}jk_j=n}}N_1^{(\sum_{j=1}^{n}k_j)}\left(A_{2_0}\right)\prod_{j=1}^{n}\frac{A_{2_j}^{k_j}}{k_j!},\ \forall\n\in\mathbb{N}.
\sum_{j=1}^{n}e^{2 \pi i k \varepsilon_j}= n + \sum_{j=1}^{n}\left( e^{2 \pi i k \varepsilon_j}- 1 \right)= n + \sum_{j=1}^{n}\left( e^{2 \pi i k \varepsilon_j}- 1 - 2 \pi i k \varepsilon_j \right).
\psi(t) = \sum_{k=1}^\infty (k-2) (p_k-\zeta_k(t)) - 2\int_{[0,t] \times [0,1]}{{1}}_{[0,r_0(\boldsymbol{\zeta}(s)))}(y) \, \varphi_0^\varepsilon(s,y) ds\,dy,{\tilde{\psi}}(t) = \sum_{k=1}^\infty (k-2) (p_k-{\tilde{\zeta}}_k(t)) - 2\int_{[0,t] \times [0,1]}{{1}}_{[0,r_0({\tilde{\boldsymbol{\zeta}}}(s)))}(y) \, \varphi_0^\varepsilon(s,y) ds\,dy.
\lim_{|\xi|_{p,q,r}+\theta^{\ell}\to\infty}\frac{|\partial_F\hat{\psi}|+|\partial_{\zeta}\hat{\psi}|^{\frac{p}{p-1}}+|\partial_w\hat{\psi}|^{\frac{p}{p-2}}}{|\xi|_{p,q,r}+\theta^{\ell}}=0
\left[ p^\mu\partial_\mu+\vec{Q}\cdot \vec{F}_{\mu\nu}p^{\mu}\partial_p^\nu+p^{\mu}\vec{Q}\times \vec{A}_\mu \cdot{\partial \over \partial \vec{Q}}\right]f(x,p,\vec{Q})=C(x,p,\vec{Q})+S(x,p,\vec{Q})
\nabla f(\mathbf{x}^{k+1}) + \sqrt{\eta}\mathbf{E}_{}^T\boldsymbol{\nu}^k + \frac{\rho}{2}\mathbf{E}_{}^T\mathbf{E}_{}\mathbf{x}^k + \rho(\mathbf{D}+ \epsilon \mathbf{P})(\mathbf{x}^{k+1}-\mathbf{x}^k) = \mathbf{0},
\begin{array}{@{}l@{}}{\displaystyle w_{ij}= (u_i, u_j), \;r_i=\frac{1}{2}\big(|\Psi_{h}^{k-1}|^{2}+ |2\overline{\Psi}_{h}^{k}- \Psi_{h}^{k-1}|^{2},\, u_i\big),}begin{align*}2mm]{\displaystyle s_{ij}= B(\overline{\mathbf{A}}_{h}^{k};\, u_i,\,u_j), q_{ij}= \big( (V + \overline{\phi}_{h}^{k})u_i, \, u_j\big) i,j = 1,\cdots, N.}\end{array}
Z_{C}[ J , \zeta , \bar{\zeta}] = \int{\cal D}A_{\mu}{\cal D}\psi{\cal D}\bar{\psi}\delta \left( \partial_{\mu}A^{\mu}- C ( x ) \right) d e t{\cal M}_{L}e^{i S + i \int d x ( J^{\mu}A_{\mu}+ \bar{\psi}\zeta + \bar{\zeta}\psi )}.
\hat{A}_{-1}= \begin{pmatrix}A_{-2}A_{-1}\cr O A_{-2}\end{pmatrix}, \hat{A}_0 = \begin{pmatrix}A_0 A_1 \cr A_{-1}A_0 \end{pmatrix}, \hat{A}_1 = \begin{pmatrix}A_2 O \cr A_1 A_2 \end{pmatrix},
r_1 := \sqrt{\vec{q}_1^{\;2}}, \hspace{1em}r_2 := \sqrt{\vec{q}_2^{\;2}}, \hspace{1em}x := \cos <\hspace{-0.75em}\mbox{\raisebox{0.2ex}{\scriptsize )}}\;(\vec{q}_1, \vec{q}_2) \hspace{1cm}.
\limsup_{\genfrac{}{}{0pt}{1}{N\to\infty}{N\,\in\,\varphi^{-1}(n)}}\ |v_{N}|\,\cdot\prod_{j=b_{N}+1}^{b_{N+1}-1}|w_{j}|=\limsup_{k\to\infty}\,2^{\,\delta^{(k)}-\tau^{(k)}}=\infty .
\mathrm{(P3)}:\begin{cases}\underset{\mathbf u, \mathcal K_{new}}{\max}\ |\mathcal K_{new}| \ \|\mathbf u-\mathbf w_k\| \leq r, \ \forall k\in \mathcal K_{new}\cup \mathcal P_{prio}, & \ \mathcal K_{new}\subseteq \mathcal P_{sec},\end{cases}
\rho_{p,\ell}(\beta_h) \phi_w(g^{-1},u)= \int_{X_{p,\ell}}\phi_w\, d\mu_{p,\ell}&\;+O_{p,w,\ell,\delta,\theta,\epsilon}\Big(\|g\|_E^\theta h^{1-(\theta-4)\delta}+\|g\|_E h^{-1/2+3\delta+\epsilon}\Big).
\left \|\sum\limits_{j=N}^{\infty}\frac{\lambda_j \chi_{Q_j}}{\omega(Q_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}+\left \|\sum\limits_{j=N}^{\infty}\frac{\mu_j \chi_{P_j}}{\omega(P_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}}\to0,\as\N\to\infty.
P\int_0^{\infty{e}^{{i}\varphi}}\frac{t^{n-1}\,{e}^{at}}{z\,{e}^{t}-1}\,{d}t = z^{-1}\int_0^\infty\frac{t^{n-1}\,{e}^{-(1-a)t}}{1-z^{-1}\,{e}^{-t}}\,{d}t -{\rm sgn}(\varphi)\, i\,\pi\,(-\ln z)^{n-1}\,z^{-a}.
g = \left[ \left( 1 - \frac{2m}{r}\right)^{-1}- \left( 1 - \frac{2m}{r}\right) f_r^2 \right] dr^2 - 2\left( 1 - \frac{2m}{r}\right) f_r f_a dr du^a & + \left[r^2 \tilde{\sigma}_{ab}- \left( 1 - \frac{2m}{r}\right)f_a f_b \right] du^a du^b.
x=\frac{C_1^2-C_1+C_2}{2 C_1},y=\frac{C_1^2+C_1-C_2}{2 C_1}, &\sum_{k=0}^{\infty}C_k z^k=\frac{1-\frac{(1-(x+1) z) (y z+1)}{(1-x z) (1-(1-y) z)}}{z}=\frac{C_1 z}{\left(1-\frac{\left(-C_1^2+C_1+C_2\right) z}{2 C_1}\right) \left(1-\frac{\left(C_1^2-C_1+C_2\right) z}{2 C_1}\right)}
\left( \frac{C_{n,k_{1}}- (n-1)\mathbb{I}_{k_{1}=2}- \mu_{k_{1}}}{\sqrt{2k_{1}}}, \ldots, \frac{C_{n,k_{l}}-\mu_{k_{l}}}{\sqrt{2k_{l}}}\right) \stackrel{d}{\to}N_{l}(0, I_{l})
\pi(x) \cdot \pi_*^{d(x)}(f) (y) = \pi_*^{d(x)}(f) \left( \left(\pi(x) \right)^{-1}y\right) & = \sum_{z \in \pi^{-1}\left(\left(\pi(x) \right)^{-1}y\right)}f(z) & = \sum_{z \in x^{-1}\cdot \pi^{-1}\left( y \right)}f(z) & = \sum_{z' \in \pi^{-1}\left( y \right)}f(x^{-1}z') & = \pi_*^{r(x)}(x \cdot f) (y)
\P_i\{Y_1-i=-2\}=(q-\varphi(i))(q-\varphi(i-1)), &\P_i\{Y_1-i=0\}=(q-\varphi(i))(p+\varphi(i-1))+(p+\varphi(i))(q-\varphi(i+1)), &\P_i\{Y_1-i=2\}=(p+\varphi(i))(p+\varphi(i+1)),
D_E\mathfrak{w}_t= \big(O(r^2)\mathcal{L}_1(\mathfrak{v}_t)+O(r)\mathcal{L}_0(\mathfrak{v}_t))+\big(O(r^2)\mathcal{L}_1^*(\mathfrak{v})+O(r)\mathcal{L}_0^*(\mathfrak{v})) := f^*.
u(u(S_2)) = \frac{\beta^2}{4}\frac{1-\tfrac32 D}{2-\beta}- \frac{(2+{\beta})D}{4}> \frac{\beta^2}{4(2-\beta)}\bigg(1- \frac{3\beta^2}{\beta^2+8}- \frac{2(4-\beta^2)}{\beta^2+8}\bigg)=0\, .
\|\tilde \phi_{k+1}[T]- (p, 0)\|_{H^1_x\times L^2_x}= \|\left(\phi_k[T]- (p, 0)\right)^p\|_{H^1_x\times L^2_x}\lesssim \varepsilon \|\phi_k[T]- (p, 0)\|_{H^1_x\times L^2_x},