harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{Q}_t(\omega):=\{Q\in\mathfrak{P}(\Omega):\Q \lll\mathcal{P}_t(\omega),\E_Q|\Delta S_t(\omega,\cdot)|<\infty,\E_Q[y\Delta S_t(\omega,\cdot)]\leq 0,\ \forall y\in\mathcal{H}_t(\omega)\}.
\mathbf{P}\left(L^\pm_{T_{\pm}(x)}(x)\geq r|x|^{\overline{\eta}_{\rm c}-2\gamma},A_1^\varepsilon\right)\leq \left(\frac{C|x|^{2-\beta-\gamma}}{r|x|^{\overline{\eta}_{\rm c}-2\gamma}|y|^{\beta(\overline{\eta}_{\rm c}-3\gamma)}}\right)^{r|x|^{-\gamma}/C}=\left(\frac{C}{r}|x|^\gamma\right)^{r|x|^{-\gamma}/C}.
H^{\phi}_{\nu}(f) = \frac{1}{2}{\rm log}\frac{2 \pi \alpha (1-\alpha)}{n}-\frac{log(\nu)}{2(1-\nu)}+ \frac{(\alpha-\gamma)^2}{2\alpha (1-\alpha)(1-\nu)}\left( \frac{1}{\nu}-1 \right) + O\left(\frac{1}{n}\right)
|w'_n (t) - w_{n-1}'(t)| = |m - w'_{n-1}(t)| = |w'_{n-1}(x_n) - w'_{n-1}(t)| \leq \|w_{n-1}''\|_{L^{\infty}(Y_n)}|t - x_n| & \leq K_n R_n.
H^{\alpha \beta \gamma \delta}= \nu_1 \overline{a}^{\alpha \beta}\, \overline{a}^{\gamma \delta}+ \frac{(1 - \nu_1)}{2}(\overline{a}^{\alpha \gamma}\overline{a}^{\beta \delta}+ \overline{a}^{\alpha \delta}\overline{a}^{\beta \gamma}) \, ,
III= -\frac{1}{2\lambda}\int_0^x \big(E_\lambda(x-2x_1)\Phi(x_1)P_\lambda(x_1)\big)'\,dx_1+ +\frac{1}{2\lambda}\int_0^x E_\lambda(x-2x_1)\big(\Phi(x_1)P_\lambda(x_1)\big)'\,dx_1.
\aligned\|f\|_{W\mathcal M^{p}(\varphi,w)}:= \sup_{B}\frac{\|f\chi_B\|_{L^{p,\infty}(w)}}{\varphi(B)^{1/p}}& = \sup_{B}\sup_{t>0}\frac{t (w(\{x\in B: |f(x)|>t\})^{1/p}}{\varphi(B)^{1/p}}<\infty,\endaligned
f_{-}^{n}= \frac{D_{0}^{n - 1}}{D_{0}^{n}}, \; f_{0}^{n}= \frac{\dot{D}_{0}^{n}}{D_{0}^{n}}+ \tau , \; f_{+}^{0}= \frac{D_{0}^{n + 1}}{D_{0}^{n}}.
\tilde{A}_{\mu}( x ) = 4 \pi \epsilon_{\mu \nu \rho \sigma}\int \delta \xi d s \omega ( \xi ( s ) ) W^{\rho \sigma}[ \xi | s ] \omega^{- 1}( \xi ( s ) ) \dot{\xi}^{\nu}\dot{\xi}^{- 2}\delta ( \xi ( s ) - x ) .
E'= E \cup \{vx_{(v,i)}: v\in V, x_{(v,i)}\in V_{v}\}&\cup \Big\{\{ux_{(u,v,i)},vx_{(u,v,i)}\}: \{u,v\}\in (V\times V)\setminus P, x_{(u,v,i)}\in V_{(u,v)}\Big\}& \cup \{xx': x,x'\in V'\setminus V\}.
\mbox{Tr}(B\,P_{[{\cal H\/}_k^{\perp}]}\,e^{-t\Delta_k})=\sum_{0{\leq}l{\leq}l_0}a_l(B\,P_{[{\cal H\/}_k^{\perp}]}\,|\,\Delta_k)\,t^{-l}+O(t^{\epsilon})
Q/Q(2) \downarrow X = 5 \otimes 1^{[1]}+ T(8) + 0^2, Q(2)/Q(3) \downarrow X = 5 \otimes 1^{[1]}+ T(8) + 0, Q(3)/Q(4) \downarrow X = 5 \otimes 1^{[1]}+ T(8) + 0, Q(4)/Q(5) \downarrow X = 0, Q(5)/Q(6) \downarrow X = 0.
T(g) = T(h), T(g') = T(h'), T(g^\Uparrow) = T(h^\Uparrow), T({g'}^\Uparrow) = T({h'}^\Uparrow), g^\Uparrow \leftrightarrow g'^\Uparrow, g \leftrightarrow g', h^\Uparrow \leftrightarrow{h'}^\Uparrow,
(\Upsilon \circ \pi_\lambda^\ast) (\tau) (\exp(x)) = \sum_{l \ge 0}(\pi_\lambda^\ast(\tau))(x^l)/l!({\rm by\the\definition\of}\ \Upsilon) &= \sum_{l \ge 0}\tau(x^l \cdot v_\lambda)/l!({\rm since}\ \pi_\lambda(x^l) = x^l \cdot v_\lambda) &= \tau(\exp(x) \cdot v_\lambda).
P^{-}= \frac{1}{2}\int d x^{-}d^{2}x^{\perp}: \left[ - \phi \partial^{\perp 2}\phi + \sum_{i = 1}^{2}\left( \bar{\psi}_{m}^{( i )}\frac{- \partial^{\perp 2}+ m^{2}}{i \partial^{+}}\gamma^{+}\psi_{m}^{( i )}+ 2 g \phi \bar{\psi}_{m}^{( i )}\psi_{m}^{( i )}+ g^{2}\bar{\psi}_{m}^{( i )}\phi \frac{\gamma^{+}}{i \partial^{+}}\phi \psi_{m}^{( i )}\right) \right] : \; .
{\cal D}^{(n)}_c(q,1;p)=\prod_{1\le i\le n}{\cal D}^{(1)}_c(q,1;p)_{x_i},{\cal D}^{(n)}_c(q,q;p)=c^{-n(n-1)/2}\prod_{1\le i<j\le n}\frac{1}{x_i^{-1}\theta_p\big(x_i x_j^{\pm 1}\big)}\prod_{1\le i\le n}{\cal D}^{(1)}_c(q,q;p)_{x_i}\prod_{1\le i<j\le n}x_i^{-1}\theta_p\big(x_ix_j^{\pm 1}\big).
0= 4\langle \tau_{H}, \alpha_{1}\rangle \{(2m_{2}+m_{1}+m_{3})(1-(\cot \langle 2\alpha_{1}, H\rangle )^{2}) &+n_{3}(1-(\tan \langle 2\alpha_{1}, H\rangle )^{2})-(2m_{2}+m_{1})\}(\alpha_{1}+\alpha_{2}).
\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial v}(X_t, \dot X_t, t) = (\dot \beta_t + \dot \gamma_t) \frac{\partial\mathcal{L}}{\partial v}(X_t, \dot X_t, t)+ \mu e^{\beta_t + \gamma_t}\frac{d}{dt}\nabla h(Z_t) - \mu e^{\beta_t + \gamma_t}\frac{d}{dt}\nabla h(X_t).
A = - \frac{\kappa^2}{4}\, \frac{p^{(1)}_{11}p^{(2)}_{11}}{\left( p^{(1)}_\perp - p^{(4)}_\perp\right)^2}\left( \frac{p^{(1)}_\perp}{p^{(1)}_{11}}- \frac{p^{(2)}_\perp}{p^{(2)}_{11}}\right)^4
e_{ij}= \big\langle \xi(x_i - x_j) + \varphi(x_i) - \varphi(x_j), \hat{b}_{ij}\big\rangle = l_{ij}\hat{b}_{ij}^T \xi \hat{b}_{ij}+ \big\langle \varphi(x_i) - \varphi(x_j), \hat{b}_{ij}\big\rangle.
q_{k,\ell}(t,h) = \sum_{\substack{i_1+i_2 +i_3 = k-1 i_1,i_2,i_3 \ge 0}}\frac{\alpha_{i_1,k}(\log t)^{i_3}}{i_{2}! i_{3}!}\frac{1}{2 \pi i}\int_{C_2}\zeta^{\ell}(s_2+1) \mathcal{D}_{k,\ell}^{(i_2,0)}(0,s_2) (t+h)^{s_2}ds_2.
( \lambda - \lambda_{0}) [ \frac{1}{u}- i \int \frac{d^{d}{k}}{( 2 \pi )^{d}}( \frac{1}{( k^{2}- \lambda ) ( k^{2}- \lambda_{0})}- \frac{1}{( k^{2}- \mu )^{2}}) ] = ( \vec{\phi}^{2}- v^{2})
\begin{aligned}\varepsilon\int_0^t{|(\Delta^h_k(\varepsilon w^2\partial_yu)_\Theta, \Delta^h_k(\varepsilon w^1)_\Theta)_{L^2}|dt'}\lesssim\varepsilon d_k^22^{-k}||(\varepsilon w^1)_\Theta||^2_{\tilde L^2_{t, \zeta'(t)}(B^1)}&\end{aligned}
\int_{0}^{x}\hat{f}_2(x+c-z)k_q(-z)dz=\int_{0}^{x}W^{(p+q)}(x+c-z)W^{(q)}(z)dz &-(\Phi(p+q)-\Phi(q))\int_{0}^{x}e^{\Phi(p+q)(x-z)}W^{(q)}(z)dz\int_{0}^{c}e^{\Phi(q)(c-z)}W^{(p+q)}(z)dz,
\tau_{ijk}^2 = 1 - \frac{r_{ijk}^{-\alpha}}{\sum_{l \in \Phi_{\mathrm{P}}}r_{ilk}^{-\alpha}}&= \frac{1}{1 + \frac{r_{ijk}^{-\alpha}}{\sum_{l \in \Phi_{\mathrm{P}}\setminus \{j\}}r_{ilk}^{-\alpha}}}
a_{3}=a_{2}+b\sqrt{a_{2}^{2}}=((a+b\sqrt{a^{2}})+b\sqrt{(a+b\sqrt{a^{2}})^{2}})+b\sqrt{((a+b\sqrt{a^{2}})+b\sqrt{(a+b\sqrt{a^{2}})^{2}})^{2}}
Q_{\pm}= - \frac{| 2 \alpha \widetilde{N_{\pm}}|^{4}}{\langle n \rangle_{\pm}}e^{- 2 | \alpha |^{2}\sin^{2}\varphi}\sin^{2}\varphi \left( \cos^{2}\varphi \, e^{- 2 | \alpha |^{2}\sin^{2}\varphi}\pm \cos ( | \alpha |^{2}\sin 2 \varphi + 2 \varphi ) \right) .
\begin{aligned}Z_\nu = \zeta^2 \big(w_\nu + (\kappa - Ad)\tilde{w}_\nu - A\tilde{w}\big) &= \zeta^2 \big((\kappa - Ad - 1)\tilde{w}_\nu - A\tilde{w}\big) &\leq \zeta^2 (|\tilde{w}_\nu| - A\theta |\nabla v|^2).\end{aligned}
\|u-P_{\lambda_0}u\|_{H^1}\leq 2(\|(I_E-\hat{P}_{\lambda_0})(u(0))\|+\|(I_E-\tilde{P}_{\lambda_0})(u(1))\|) &= 2(\|(I_E-\hat{P}_{\lambda_0})\hat{P}_\lambda(u(0))\|+\|(I_E-\tilde{P}_{\lambda_0})\tilde{P}_\lambda(u(1))\|) &\leq 2(\|(I_E-\hat{P}_{\lambda_0})\hat{P}_\lambda\|\|u(0)\|+\|(I_E-\tilde{P}_{\lambda_0})\tilde{P}_\lambda\|\|u(1)\|),
-a_1^{26}=a_4^{26}=a_2^{27}=a_1^{37}=-a_4^{37}=:a_1,a_4^{26}=a_3^{36}, &-a_2^{26}=-a_1^{36}=a_4^{36}=a_2^{37}=:a_2, &-a_3^{26}=-a_1^{27}=a_4^{27}=a_3^{37}=:a_3.
V^{g^{4}}\left( L \right) = \frac{1}{2}g^{4}C_{A}C_{F}\left({- \frac{1}{L}}\right) \int_{\bf 0}^{\bf y}{d z^{i}}\int_{\bf 0}^{\bf y}{d z^{\prime j}}D_{i j}( z , z^{\prime}) .
\lambda_{n}^{( \ast )}= \left[ \operatorname*{l i m}_{\epsilon \rightarrow 0}\Xi_{n}( \epsilon ) \right]^{- 1}= \operatorname*{l i m}_{\epsilon \rightarrow 0}\left| \widehat{\eta}_{n}( \epsilon ) \right|^{- \epsilon / 2}\; ,
_{\phi}\rho^{\alpha}\circ_{\phi}\Pi^{\alpha}(x) =P_{\alpha}\circ_{\phi}\rho^{E}(i_{\alpha}\circ_{\phi}\Pi^{\alpha}(x))=P_{\alpha}\circ_{\phi}\rho^{E}(_{\phi}\Pi^{E}(i_{\alpha}(x))) &=P_{\alpha}(i_{\alpha}(x))=x.
u_{ij}=x_{i}x_{j}-q^{i-j-n}x_{j}x_{i}-(q^{n}-q^{-n})q^{i-j-r-rn}x_{i+r}x_{i-r}+p(1-q^{n})x_{\frac{i+j}{2}}x_{\frac{i+j}{2}}
{\mathcal N}_{\lambda, \mu}^{(k)}\big( u \vert q, s \big) =\prod\limits_{b \geq a \geq 1}\ \big( 1 - u q^{-\mu_a + \lambda_{b+1}}s^{-a+b}\big)^{\delta_{a - b - k \vert N}}\ \big( 1 - u q^{\lambda_{a}- \mu_b}s^{a-b-1}\big)^{\delta_{a - b + k + 1 \vert N}}
\sum_{\tau=0}^{n-s-2}\frac{1}{n}e^{n-\tau-1}_{i,g+\tau+1}+\left(1-\frac{1}{n}\right)\left[\sum_{\tau=0}^{n-s-2}\frac{1}{n^{n-\tau-s-1}}e^{s}_{i,g+\tau+1}\right] &+\frac{1}{n}\sum_{\tau=1}^{s-1}e_{i,g+n-1-s+\tau}^{s-\tau}-\left(1-\frac{1}{n}\right)e_{i,g}^{n-1}-\sum_{\tau=1}^{n-2}e_{i,g+\tau}^{n-1-\tau},
I_4 =\int_0^r -\frac{z}{2}(e_\sigma+\bar\psi)\cdot (e'_\sigma \phi +e'_\delta \psi){\rm{d}}z \leq \frac{1}{2}\Vert (\phi,\psi)\Vert_{L^2}\cdot \Vert z(e'_\delta,e'_\sigma)\Vert_{L^2}\cdot \left(\Vert e_\sigma\Vert_{L^\infty}+\Vert \bar\psi\Vert_{L^\infty}\right),
| f_\alpha (U | \bar U;x,t) | \le | f_\alpha - \bar f_\alpha - \nabla \bar f_\alpha (U-\bar U)) | &+ \left | \nabla \bar f_\alpha \nabla \bar A^{-1}\big (A(U,x,t) - \bar A - \nabla \bar A (U - \bar U) \big ) \right | &\le k_4 |A(U,x,t) - \bar A|^2 \, ,
\left({\bf w}_s^{(i)}-{\bf w}_t^{(i)}\right)^{(q)}=\int\limits_t^T\left(\sum_{j=0}^{q}\int\limits_t^s\phi_j(\tau)d\tau \cdot \phi_j(\tau)\right)d{\bf w}_{\tau}^{(i)}=\sum_{j=0}^{q}\int\limits_t^s\phi_j(\tau)d\tau \cdot \int\limits_t^T\phi_j(\tau)d{\bf w}_{\tau}^{(i)}.
\begin{array}{r c c c l}{G_{i \mu}}&{\to}&{G_{i \mu}^{\prime}}&{=}&{U_{i}\star G_{i \mu}\star U_{i}^{- 1}+ i g_{i}^{- 1}U_{i}\star \partial_{\mu}U_{i}^{- 1},}\\{\Phi}&{\to}&{\Phi^{\prime}}&{=}&{U_{1}\star \Phi \star U_{2}^{- 1},}\\ \end{array}
\frac{p_2+q_2}{p_1+q_1}<\frac{p_2+q_2'}{p_1+q_1'}\Leftrightarrow{p_1q_2+q_1'p_2<p_1q_2'+q_1p_2}&\Leftrightarrow{p_2(q_1'-q_1)<p_1(q_2'-q_2)}&\Leftrightarrow{\frac{p_2}{p_1}<\frac{q_2'-q_2}{q_1'-q_1}=\frac{q_2'}{q_1'}}.
v_{i}^{j_{i}}=y_{i,[\kappa_{i}^{j_{i}}]}^{j_{i},d}-\sum_{\beta=0}^{\kappa_{i}^{j_{i}}-1}a_{i}^{j_{i},\beta}(y_{i,[\beta]}^{j_{i}}-y_{i,[\beta]}^{j_{i},d})\,,j_{i}=1,\ldots,m_{i}\,,i=1,\ldots,s
\lambda g_{\pm}+v\partial_{x}g_{\pm}\mp\beta_{x}\partial_{v}g_{\pm}\mp\phi_{x}v\mu_{\pm,+}^{\prime}=0,\v>0 \lambda g_{\pm}+v\partial_{x}g_{\pm}\mp\beta_{x}\partial_{v}g_{\pm}\mp\phi_{x}v\mu_{\pm,-}^{\prime}=0,\v<0
\sum_{i=1}^{3}(\beta_{i}^{2}-\beta_{i+1}^{2})(\beta_{i}-\delta_{1})^{3}(\beta_{i+1}-\delta_{1})^{3}- \sum_{i=1}^{3}(\beta_{i}^{2}-\beta_{i+1}^{2})(\beta_{i}-\delta_{2})^{3}(\beta_{i+1}-\delta_{2})^{3},
\left(\partial_{0}+a(n\mathrm{m}_1)\right)^{-1}\stackrel{\tau_{\textnormal{w}}}{\to}\left(\partial_{0}+\partial_{0}\sum_{j=1}^{\infty}\left(-\sum_{\ell=1}^{\infty}\left(-\partial_{0}^{-1}\right)^{\ell}\int_{0}^{1}a^{\ell}\right)^{j}\right)^{-1}.
\prod_{t=1}^{\eta}\left( 1- \frac{v_n^2}{s_{nt}^2}\right) \geq \prod_{t=1}^{\infty}\left(1-\frac{(1-2\varepsilon (-\varphi)^{-n})^2}{u_t^2}\right) \big(1-\mathcal{O}(\eta^{-1})\big)\big(1 - \mathcal{O}(\varphi^{n/5})\big).
\int{\prod\limits_{0\le t\le T}{\omega_J\left( t \right)}\exp \left({iJ\int_0^T{dt\left\{{i{{\xi^{*}\dot \xi-{\dot \xi}^{*}\xi}\over{1+\left| \xi\right|^2}}+{{1-\left| \xi \right|^2}\over{1+\left| \xi \right|^2}}}\right\}}}\right)}
u=12(4\pm\sqrt6)\frac{y^2+(18\pm8\sqrt6)t}{(y^2+10\lambda_\pm t)^2}x+|t|^{\nu+3/2}\exp\left(-\frac{y^2}{4t}\right)\frac{C_1y+C_2|t|^{1/2}}{(y^2+10\lambda_\pm t)^2}\phantom{u=}\times\mathop{\rm HeunC}\left(\frac52\lambda_\pm,-\frac12,-5,\frac58\lambda_\pm(4\nu+1),-\frac52\lambda_\pm\nu-\frac{59}8\mp\frac{29}8\sqrt6,\frac{-y^2}{10\lambda_\pm t}\right),
\mathbb{E}\Big[\underset{t_0\leq t\leq T}{\sup}|\phi^{N,i}(t,\boldsymbol{x}_t^*)-\Psi^{N,i}(t,\boldsymbol{x}_t^*)|^2\Big]\leq\frac{C}{N^2}\Big(1+\frac{1}{N}\underset{j=1}{\overset{N}\sum}\mathbb{E}[|\xi^i-\xi^j|^2]+\frac{1}{N^2}\underset{i,j=1}{\overset{N}{\sum}}\mathbb{E}[|\xi^{i}-\xi^{j}|^2]\Big).
\mathrm{Tr}_2(g)=\mathrm{Tr}^{GL_2}_{SL_2}(g) =\mathcal{I}\circ\mathrm{Tr}(g) =\mathcal{I}\circ\left(\dfrac{1}{2^{d-1}}\sum_{\epsilon_i\in R}\psi'(\epsilon_i^{-1})g\mid_{k,w}\epsilon_i\right)
\left\langle \frac{1}{2\overline{\alpha}t}\left(\begin{array}{cc}I_d -\exp(tS^T) -\exp(tS) I_d\end{array}\right)\left(\begin{array}{c}x \xi\end{array}\right), \left(\begin{array}{c}x \xi\end{array}\right)\right\rangle = \frac{1}{2\alpha t}\left|e^{tS}x-\xi\right|^2
\sum^{[{t}/{(2v_n)}]-1}_{j=0}2v_n\biggl(\frac{\xi^2_{k,j}(u_n)}{2u_n^2}-\frac{(\sinh(u_n^2c^k_j(u_n)))^2}{k_nu_n^2}\biggr)^2I_{\Omega(k,n, t)}=o_p(1).
S_m(\nu^{+})=m \cdot (\nu^{+}_1,\dots,\nu^{+}_{rm},0^{n-rm})+(1,\dots,n)=(\underbrace{m \nu^{+}_1 +1,\dots, m \nu^{+}_{rm}+ rm}_{rm},\underbrace{rm+1,\dots,n}_{n-rm}).
X(p_0)=\frac{i\Gamma}{2\pi}\,\left[\left(p_0^2-\frac{3m^2}{2}+\frac{m^4}{2p_0^2}\right)\,\log\left| \frac{p_0^2 -m^2}{m^2}\right|+\frac{m^2}{2}-\frac{5p_0^2}{4}\right]\,.
D_{pert, s\eta^{i+1}}(\psi_i+s\mathfrak{c}^g_{i+1}-s\mathfrak{h}^g_{i+1}+s\mathfrak{e}^g_{i+1}+s\mathfrak{k}^b_{i+1,s}+s^2\mathfrak{k}^{\perp}_{i+1,s}) &=s^2\mathcal{A}+s^2\mathcal{B}+s\mathcal{C}:= \mathfrak{f}_{i+1}\mbox{}mod(\ker(D|_{L^2_1}))
|{k}^{l+1}z_l^j(B^j;k)-{k}^{l+1}u_l(B^j;k)|\leq C(\frac{B^jk}{1+|kB^j|})^{l+1}e^{|\Im kB^j|}E(B^j;k)=o(1),\,\mbox{as}l\rightarrow\infty,
d^0(m) = \sum_{s\in S}(sm-ms)e_s d^1(\sum_{s\in S}m_se_s) = \sum_{s_0,s_1\in S}(s_0m_{s_1}- m_{s_1}s_0 - s_1m_{s_0}+ m_{s_0}s_1 )e_{s_0,s_1}
\frac{\partial}{\partial t}\Theta(t,x) = \int_{\S^d}W_r\left( \sum_{i=1}^d m_i y_i + m_{d+1}x\right) ~\sin\left( \sum_{i=1}^d n_i \Theta(t,y_i) + n_{d+1}\Theta(t,x)\right) \ \d y
C_{p,g+1}^{(\ell)}=\sum_{j=1}^{p}(-1)^{j-1}\sum_{t=0}^{\lfloor\frac{p-j}{2}\rfloor}\binom{n-p+j+2t}{t}\sum_{\gamma=0}^{p-j-2t}2^{\gamma}\binom{\ell}{\gamma}\binom{n-\ell}{p-j-2t-\gamma}\binom{p-j-2t-\gamma}{p-j-t-g-1}.
g ( A d S ; s o l i t o n ) ={r^{2}}d t \otimes d t +{\frac{L^{2}}{r^{2}}}{( 1 -{\frac{{r_{0}}^{n}}{r^{n}}})^{- 1}}d r \otimes d r +{\frac{r^{2}}{L^{2}}}{( 1 -{\frac{{r_{0}}^{n}}{r^{n}}})}d \phi \otimes d \phi +{r^{2}}\sum_{i = 1}^{n - 2}\; d \theta_{i}\otimes d \theta_{i}.
N^{A B}= \bar{\epsilon}_{1}\Gamma^{A B C}D_{C}\epsilon_{2}+ \textstyle{\frac{1}{8}}\bar{\epsilon_{1}}\Gamma^{C_{1}C_{2}}\epsilon_{2}F_{\ \C_{1}C_{2}}^{A B}+ \textstyle{\frac{1}{9 6}}\bar{\epsilon}_{1}\Gamma^{A B C_{1}\cdots C_{4}}\epsilon_{2}F_{C_{1}\cdots C_{4}}.
\lim_{\lambda\to\infty}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{L^\Phi(\mathbb{R}^n)}=\left[\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{L^\Phi(\mathbb{R}^n)};
i_{\alpha\beta\gamma}(t)=Ti_{abc}(t)=\frac{2}{3}[\begin{matrix}1&-\frac{1}{2}&-\frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\ \end{matrix}][\begin{matrix}i_{a}(t)\\i_{b}(t)\\i_{c}(t)\end{matrix}]
\frac{1}{x}\frac{\partial \psi(x, S, Q)}{\partial x}= \frac{\sqrt{2S^2 W(2 \exp(2Q) S^2 x^2)}}{x}= \sqrt{2S^2 \cdot 2 \exp(2Q) S^2}\frac{\sqrt{W(y)}}{\sqrt{y}}\le 2 S^2 \exp(Q)
[\hat{\tilde{\rm G}}_{\pm}^a(n), \hat{\tilde{\rm G}}_{\pm}^b(m)]_{-}=ie_{\pm}\varepsilon_{abc}\hat{\tilde{\rm G}}_{\pm}^c(n+m)+i{\alpha}_{\pm}e_{\pm}^3 \varepsilon_{abc}A_{1,\pm}^c(n+m)\pm e_{\pm}^2 n \delta_{n,-m}\delta^{ab}(1 \pm \frac{4\pi}{\rm L}\alpha_{\pm}).
\begin{aligned}\left|\int_{\Omega}F(x,u_j^0(x)) dx\right| \le |J_{\lambda_k}(u_j)|+ \bigg|\dfrac{1}{2}\mathcal{B}_{\alpha}(u_j^+,u_j^+)+ \dfrac{1}{2}\mathcal{B}_{\alpha}(u_j^-,u_j^-) & -\dfrac{\lambda_k}{2}\int_{\Omega}\left(|u_j^+(x)|^2+|u_j^-(x)|^2 \right) dx\ - \int_{\Omega}\left(F(x,u_j(x))- F(x,u_j^0(x)) \right)dx \bigg|\end{aligned}
\sum_{k=0}^{n-1}1 \cdot \sqrt{\frac{1}{n^2 \pi^2}\int_{\frac{k+1/2}{n}}^{\frac{k+3/2}{n}}f'(x)^2 \, dx}\leq \sqrt{n}\sqrt{\sum_{k=0}^{n-1}\frac{1}{n^2 \pi^2}\int_{\frac{k+1/2}{n}}^{\frac{k+3/2}{n}}f'(x)^2 \, dx}= \sqrt{n}\sqrt{\frac{1}{n^2 \pi^2}\int_{0}^{1}f'(x)^2 \, dx}
p_{AR}(a,r) =\sum_{b}p_{A,B}(a,b)p_{R|B}(r|b) &=\epsilon\, p_A(a)p_{R|B}(r|\mathtt{e})+(1-\epsilon)p_A(a)p_{R|B}(r|a) &=p_A(a)q_{R}(r)\lambda(r)\!+\!p_A(a)\big(q_{R|A}(r|a)\!-\!q_R(r)\lambda(r)\big) &=p_A(a)q_{R|A}(r|a)
F_{x_{1}x_{2}x_{3}}^{\Delta_{1}\Delta_{2}\Delta_{3}}= \frac{1}{| x_{1 2}|^{\Delta_{1}+ \Delta_{2}- \Delta_{3}}\, | x_{2 3}|^{\Delta_{2}+ \Delta_{3}- \Delta_{1}}\, | x_{3 1}|^{\Delta_{3}+ \Delta_{1}- \Delta_{2}}}.
\left[ \partial_{\mu}\partial^{\mu}- \partial_{y}^{2}+ e^{- 2 \sigma}\left\{m^{2}+ \left( \frac{3}{1 6}- \xi \right) R \right\}\right] \tilde{\varphi}+ e^{- \frac{7}{2}\sigma}V^{\prime}( e^{\frac{3}{2}\sigma}\tilde{\varphi}) = 0 .
\sum_{i=\ell}^L T_{{\rm SVC},n,i}^*= \begin{cases}\frac{1}{D_1(\tau_C)}\sqrt{\frac{a_{n}b_{n,1}D_2(\tau_C)}{v^*s_1+\eta_n^*}}-\frac{D_2(\tau_C)}{D_1(\tau_C)}, \ell=1 \frac{1}{D_1(\tau_C)}\sqrt{\frac{a_{n}b_{n,\ell}D_2(\tau_C)}{v^*s_\ell}}-\frac{D_2(\tau_C)}{D_1(\tau_C)}, \ell \in\{2,\cdots,L\}\end{cases}.
\rho^{k+1}= \left({\arg \min}_{\rho}\left\{||\mathbf{y}- \mathbf{A}\rho||_2^2 + \lambda/2 ||\rho- \mathbf{z}^k +\mathbf{u}^k||_2^2\right\}\right), \mathbf{z}^{k+1}= \Pi_{\mathcal{C}}(\rho^{k+1}+ \mathbf{u}^k), \mathbf{u}^{k+1}= \mathbf{u}^k + (\rho^{k+1}- \mathbf{z}^{k+1}).
[N_T(x+a),N_T(y+b)]_{\varrho}= [T(a)+0,T(b)+0]_{\varrho}=[T(a),T(b)], &N_T\big([N_T(x+a),y+b]_{\varrho}-[N_T(y+b),x+a]_{\varrho}-N_T([x+a,y+b]_{\varrho})\big) & =N_T \big(([T(a),y]+\varrho(T(a))b)-([T(b),x]+\varrho(T(b))a)-(0+T(\varrho(x)b-\varrho(y)a))\big) & = T(\varrho(T(a))b-\varrho(T(b))a)
\sup_{\mathcal{T}\geq 1}|\frac{1}{\mathcal{T}}\int_0^{\mathcal{T}}\tilde G(\mu^{N,M}(t),z^{N,M}(t))dt - \frac{1}{\mathcal{T}}\int_0^{\mathcal{T}}\tilde G(\bar \mu^{N,M}(t),\bar z^{N,M}(t))dt|\leq \kappa_3(\epsilon), \ \ \ \ \ \ \ \lim_{\epsilon\rightarrow 0}\kappa_3(\epsilon) =0.
E_{M,k}(X) := -\sum_{j=1}^{2g}\sum_{\ell=1}^{k}\frac{Z(\gamma_{j,\ell}^{-1})}{k\gamma_{j,\ell}^{1-k}Z'(\gamma_j^{-1})}\frac{\gamma_{j,\ell}}{\gamma_{j,\ell}-1}e^{iX(\theta(\gamma_j)+2\pi\ell)/k}.
\delta_{\kappa_{1}}p_{1}^{\mu}= 2 \bar{\kappa}_{1}\left( \begin{array}{c}{{\lambda_{2}^{2}\gamma^{\mu}\, \gamma \cdot p_{1}\ - \gamma \cdot \lambda_{2}\, \, \gamma \cdot p_{1}\, \lambda_{2}^{\mu}}}\\{{+ \lambda_{2}\cdot p_{1}\, \, \gamma \cdot \lambda_{2}\, \, \gamma^{\mu}\,}}\\ \end{array}\right) \dot{\theta}_{1}.
\begin{aligned}\min_{\mathbf{F}_R, \mathbf{F}_B}~ ||\mathbf{F}^\star - \mathbf{F}_R\mathbf{F}_B||_F s.t.~ ||\mathbf{F}_R\mathbf{F}_B||_F^2 = N_s, \mathbf{F}_R \in \mathcal{F}_R.\end{aligned}
\psi_F(x)+ \psi^*_A(x)=\sum_{\vec k,s}\frac{1}{\sqrt{V}}e^{i[\vec{k}\cdot \vec{x}-\hbar^{-1}E(\vec k)t]}u_s{(\vec{k})}c_s(\vec k) + \sum_{\vec k,s}\frac{1}{\sqrt{V}}e^{-i[\vec{k}\cdot \vec{x}-\hbar^{-1}E(\vec k)t]}v^*_s{(\vec{k})}d^\dag_s(\vec k),
I = \frac{1}{16 \pi (8\pi^6)}\int d^{10}x \sqrt{-g^{(10)}}\left( e^{-2 \phi^{(10)}}R^{(10)}+ 4 e^{-2 \phi^{(10)}}(D \phi^{(10)})^2 - \frac{1}{12}H^{(10) 2}\right) ,
\sum_{2r_0+r_1=d_1+1}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}\frac{(d_2+2)!}{2^{r_0}(d_2-d_1+1)!}{n_1 \choose d_1+1}{n_2 \choose d_2+2}.
\delta \left( \zeta^{+}( 0 )^{\prime}- \zeta^{-}( 0 )^{\prime}\right) = - 2 \zeta^{+}( \delta \phi , 0 ) - 2 \zeta^{-}( \delta \phi , 0 ) + 2 \zeta^{\tilde{F}+ F}( \delta \phi , 0 ) .
\delta_{{\rm shift}}\int_{\Sigma_{g}}\sqrt{g}= -\int_{\Sigma_{g}}\frac{1}{2}\sqrt{g}\, tr \, (r^{\gamma}_{\mu}g_{\gamma\nu}+ r^{\gamma}_{\nu}g_{\gamma\mu}) = -\int_{\Sigma_{g}}\sqrt{g}\, r^{\mu}_{\mu}.
E_{c_{1}}(\theta_{0}):=\left\{\theta\in l_{2}:\mathop{\sum}_{i>\lfloor c_{1}(B/\varepsilon)^{2/(2\alpha_{0}+1)}\rfloor}(\theta_{i}-\theta_{0,i})^{2}/\varepsilon^{2}\leq (B/\varepsilon)^{2/(2\alpha_{0}+1)}\right\}.
0< ~\big(p(\hat{x},\hat{t})-2\big)\big\langle X\xi_x,\xi_x\big \rangle-\big(p(\hat{y},\hat{t})-2\big)\big\langle Y\xi_y,\xi_y\big \rangle &+\sum_{i=1}^n\lambda_i\big(X-Y\big)+2\langle \mu,\delta \hat{x}+\delta \hat{y}\rangle -r\alpha/2,
a_j = \binom{j+n-1}{n-1}- \ell \sum^r_{i=1}\binom{j + d_i - d+ n-1}{n-1}+ (r-1) \ell \binom{j}{d}& + \sum_{k=2}^\ell (-1)^k \binom{\ell}{k}\binom{j - k (d - d_1)+ n-1}{n-1}begin{align*}1ex] + \binom{\ell}{2}\binom{j + 2 d_2 - 2 d + n-1}{n-1}+ \ell (\ell -1) \binom{j + d_1 + d_2 - 2 d+ n-1}{n-1}.
1 = \sigma_{-1}(N^\omega_{\partial M})(t,\xi')\sigma_1(\Lambda_{g,F}^\omega)(t,\xi') + S^{-1}_{1,0}\implies \sigma_{-1}(N^\omega_{\partial M})(t,\xi') = \frac{\chi(\xi')}{\sigma_1(\Lambda_{g,F}^\omega)(t,\xi')}.
\rho_{n,p}= \Big[{\frac{p(2j^{(4)}+1-p)(2j^{(0)}+1-p)(2j^{(4)}+2j^{(0)}+2-p)(N-n-p+2j^{(3)}+2)(N-n-p+1)}{(2j^{(0)}+2j^{(4)}+2-2p)^2(2j^{(0)}+2j^{(4)}+1-2p)(2j^{(0)}+2j^{(4)}+3-2p)}}\qquad\times{(p-n-N+2j^{(1)}+2j^{(2)})(n-p-N +2j^{(0)}+2j^{(4)}+1)}\Big]^{\frac12}\,,
{\cal{\pi}}_{\mathrm{b}, 0 i}( p_{0}= 0 ; T ) = - \frac{\epsilon_{0 i j}p^{j}}{1 6 \pi \kappa}\frac{\tau^{2}}{T}\left[ 2 m^{2}\log ( M / T ) + ( m^{2}- \xi \kappa^{2}) F \right]
* j_{F 1}= \frac{\tilde{f}2 \kappa_{1 0}T_{F 1}}{2}\delta^{( 6 )}( x ) \wedge_{i = 1}^{6}d x_{i}\wedge \left( \delta ( x_{9}- \hat{x}_{9}( \rho ) ) \frac{d \psi}{2 \pi}\wedge d x_{9}+ \delta ( \rho - \hat{\rho}( x_{9}) ) d \rho \wedge \frac{d \psi}{2 \pi}\right) ,
\delta \left( y\right) =\sup_{a,b\in G}d\left( T_{a}y,T_{b}y\right) \leq\sup_{a,b\in G}(d\left( T_{a}y,T_{a}x\right) +d\left( T_{a}x,T_{b}x\right)+d\left( T_{b}x,T_{b}y\right) ) & \leq 2Ld\left( x,y\right) +\delta \left( x\right) <\infty .
{\mathbf Q}_{\rm CFIV}= \frac{8}{\pi(k+2)}({\cal G}_{\frac{k}{2}}(\frac{1}{2},\frac{1}{2},\-\frac{k}{2}-1 )-{\cal G}_{\frac{k}{2}}(\frac{1}{2},-\frac{k+1}{2},0)-{\cal G}_{\frac{k}{2}}(\frac{1}{2}, \frac{k+3}{2},0)).
\mathbb E_x\left[\frac{1}{n}\sum_{i=0}^{n-1}h\circ X_i \mid \tau > n\right] = \frac{\displaystyle \frac{1}{n}\sum_{i=0}^{n-1}\frac{1}{\lambda^i}\mathcal P^i\left(h(\cdot) \frac{1}{\lambda^{n-i}}\mathcal P^{n-i}(\cdot,M) \right)(x)}{\displaystyle \frac{1}{\lambda^n}\mathcal P^n(x,M)}.
{S_B^{00}}(\bar{\psi},{\psi_0},\hat{g})=-{{d-26}\over{12}}\oint{d}\hat{s}d{\hat{s}'}\bar{\psi}(\xi){\hat{K}_{D}}(\xi,\xi')\bar{\psi}(\xi')-{{d-26}\over{12}}{\chi_c}{\psi_0}.
- \frac{1}{- \mu a}( 1- \frac{r_+}{r})^{-1}( 1- \frac{r_-}{r})^{- \frac{a- \chi^2 ( 1/a - 3/2 )}{a + \chi^2 /2}}r^{\frac{2}{a}- 4}dr^2- r^{\frac{1-a}{a}}(1- \frac{r_-}{r})^{\frac{1-a}{a}\frac{\chi^2}{a + \chi^2 /2}}d \Omega
-2\mu\langle v,R_{h}v\rangle =2\mu\langle v,v- R_{h}(v)\rangle-2\mu |v|_{H}^{2}\le 2\mu |v-R_{h}(v)|_{L^2}^2-{3\mu\over 2}|v|_{H}^{2}\le \nu\|v\|_{V}^{2}-{3\mu\over 2}|v|_{H}^{2}.
\sum_{v_r\in I_{\ell}}\min\{2^{\ell+1},|\csc(t/(2(v_r))|\}\le (1+ t/(\pi 2^{\ell+2}u_0))(2(B+1)2^{\ell+1}&+ B2^{\ell+2}\log(2^{\ell+1}))\ll t(\ell+1)/u_0.
\Omega ( a_{0}, a_{1}; b_{0}, b_{1}) = \left\langle{\frac{\mathrm{D e t}( 1 - a_{1}U ) \mathrm{D e t}( 1 - b_{1}\bar{U})}{\mathrm{D e t}( 1 - a_{0}U ) \mathrm{D e t}( 1 - b_{0}\bar{U})}}\right\rangle .
\Phi_{1}^{-}\Phi_{1}^{+}= \Phi_{2}^{-}\Phi_{2}^{+}= \frac{\alpha}{\alpha^{2}- \beta^{2}}, ~ ~ \Phi_{1}^{-}\Phi_{2}^{+}= \Phi_{2}^{-}\Phi_{1}^{+}= \frac{\beta}{\alpha^{2}- \beta^{2}}.
\sum_{j:z_{j}\in S\left( t\right)}\mu\left( j\right) \leq C\beta\left( 0,t\right)^{-1}\approx C\left( \log\frac{1}{\left(1-\left\vert z_{k}\right\vert^{2}\right)^{\beta}}\right)^{-1}& \approx C\left( \log\frac{1}{1-\left\vert z_{k}\right\vert^{2}}\right)^{-1}=C\mu\left( z_{k}\right) ,
\mbox{$\hat P_{k,s}\equiv\hat G_{k,s}\hat\Pi^{(q)}_{k,s}\hat G^*_{k,s}\mod O(k^{-\infty})$ on $D_0$}, \mbox{$\hat P_{k,s}\equiv A_k\hat\Pi^{(q)}_{k,s}A^*_{k}\mod O(k^{-\infty})$ on $D_0$},
E_{term}=\frac{\partial\theta}{\partial n_{\perp}}=\frac{\partial^{2}C/\partial n_{\perp}^{2}}{\partial C/\partial n}=\frac{C_{yy}C_{x}^{2}-2C_{xy}C_{x}C_{y}+C_{xx}C_{y}^{2}}{(C_{x}^{2}+C_{y}^{2})^{3/2}}
G_{j}(\lambda,b,\Omega,f_{1},f_{2})(w):=\mbox{Im}\left\lbrace\Big(\Omega\Phi_{j}(w)+S(\lambda,\Phi_{2},\Phi_{j})(w)-S(\lambda,\Phi_{1},\Phi_{j})(w)\Big)\overline{w}\overline{\Phi_{j}'(w)}\right\rbrace,
\zeta(t) = (u(t),u_t(t),\delta(t),\delta_t(t)) = \left( v(t),v_t(t),\gamma(t),\gamma_t(t) \right) + \left( w(t),w_t(t),\theta(t),\theta_t(t) \right) & =: \xi(t) + \chi(t)
g_j(z) = \frac{\dot{b}(a^\ast +\sigma^\ast z)^{j-1}\ddot{b}(a^\ast+\sigma^\ast z) \phi(z)}{\int \dot{b}(a^\ast +\sigma^\ast z)^{j-1}\ddot{b}(a^\ast+\sigma^\ast z) \phi(z)d z}, j=1, \ldots, M
\mathbb{E}[Z_t(m)^2] = \sigma_m^2 := \sum_{e^{W+C+m-1}< k \le e^{W+C+m}}\frac{1}{2 k e^{2 k \sigma}}\mathbb{E}[Z(m) Z_t(m)] = \rho_{m,t}\sigma_m^2 := \sum_{e^{W+C+m-1}< k \le e^{W+C+m}}\frac{\cos(k t)}{2 k e^{2 k \sigma}}
\sum_{\substack{a_1+\cdots+a_p=a a_1,\ldots,a_{p-k}\geq0 a_{p-k+1},\ldots,a_p\geq1}}\binom{a}{a_1,\ldots,a_p}=\sum_{j=0}^{k}(-1)^j\binom{k}{j}(p-j)^a.
\left(\begin{array}{c}Y_c(y) \mu_c(y) \lambda_c(y)\end{array}\right):=\left(\begin{array}{c}{\rm D}\theta_c(F(x_c(y))+Y/c)^* \mu+ch(x_c(y)) \Pi_{K^*}(\lambda-c g(x_c(y)))\end{array}\right) \, .
\boldsymbol{u}_{\varepsilon ,\lambda}=0\mbox{on}\partial \Omega \mbox{and}f_{\varepsilon ,\lambda}(t,x,v)=f_{\varepsilon ,\lambda}(t,x,v^{\ast})\mbox{for}x\in \partial \Omega \mbox{with}v\cdot \nu (x)<0
S(l,0,v;p)=\frac{1}{l^{1/2+v}}+\frac{2\pi i^{2k}}{(2\pi)^{1-2v}}\frac{\Gamma(k-v)}{\Gamma(k+v)}\frac{l^{v-1/2}}{p^{1+2v}}+O(V_p(0,v,k)).
\sum_{k=0}^\infty \sum_{n=1}^\infty \frac{(2n)^{k-1}(2x)^{2n}}{{2n \choose n}}\frac{t^k}{k!}= \frac{x}{(1-x^2)^{1/2}}\left(x \sqrt{1-x^2}P \left(x^2, \frac{t}{1-x^2}\right) + \arcsin(x) Q \left(x^2, \frac{t}{1-x^2}\right) \right).