harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
X=\left(\begin{array}{ccc}1 0 0 0 a 0 0 0 a^{-1}\end{array}\right),\,\,Y_w=\left(\begin{array}{ccc}1 0 v 0 1 1 0 0 1 \end{array}\right),\,\,Z_w=\left(\begin{array}{ccc}1 w u 0 b b^2+b+1 0 1 b+1 \end{array}\right)
\Lambda_{N+1}^{\infty}\Lambda_N^{N+1}(\omega,dy) =\left(\prod_{k=1}^{N-1}\frac{1}{k!}\right)\Delta_{N}(y)\int_{-\infty}^{y_1}\cdots \int_{y_N}^{\infty}\det \left(\phi_{\omega}^{(j-1)}(x_{N+2-i})\right)^{N+1}_{i,j=1}dx_1\cdots dx_{N+1}dy &=\left(\prod_{k=1}^{N-1}\frac{1}{k!}\right)\Delta_{N}(y)\det \left(\phi_{\omega}^{(j-1)}(y_{N+1-i})\right)^N_{i,j=1}dy.
\frac{1}{3}\frac{d}{dt}\int_{\Omega}n^{3}+\frac{8}{9}\int_{\Omega}|\nabla n^{\frac{3}{2}}|^{2}=\frac{4}{3}\int_{\Omega}n^{\frac{3}{2}}\nabla n^{\frac{3}{2}}\cdot( S(x,n,c) \cdot \nabla c ).
\{\sum_ir_i\mu_i\eta_i^*\ |\r_i\in R, \mu_i, \eta_i\in E^*, \mathrm{r}(\mu_i)=\mathrm{r}(\eta_i), w(\mu_i)=w(\eta_i)=s\}\,\cup \{\sum_ir_i\mu_i\eta_i^*\ |\r_i\in R, \mu_i, \eta_i\in E^*,\mathrm{r}(\mu_i)=\mathrm{r}(\eta_i)\in\mathrm{Sink}(E), w(\mu_i)=w(\eta_i)<s\}.
\int_{S^{d-1}}|\mathcal{F}(f\circ \theta_a^X)(Ru)|^2 du \leq \begin{dcases}M_1 R^{-d-1}\bigg(\int|(f\circ \theta^H )' (t)|dt\bigg)^2+O\big( a R^{-d-\frac{1}{2}}\big) M_2 a^2 R^{-d+1}\bigg(\int |f\circ \theta^H (t)| dt\bigg)^2 + O\big( a^3 R^{-d+\frac{3}{2}}\big).\end{dcases}
\sum_{j=1}^r\left\{\frac{km_j}{d}\right\}d_j=\sum_{j=1}^r\frac{s_jd_j}{d}=\sum_{j=1}^r\frac{km_jd_j-dn_jd_j}{d}=k-\sum_{j=1}^rn_jd_j.
\left(\frac{\lambda(t)}{\lambda(x)}\right)^{2}\left(\int_{\frac{4}{\lambda(x)^{2}}}^{\infty}\frac{d\omega}{\omega^{3/2}x^{8}\log^{b+2-4\alpha b}(x)}\right)^{1/2}\leq C \left(\frac{\lambda(t)}{\lambda(x)}\right)^{2}\frac{1}{x^{4}\log^{b+1-2\alpha b}(x)}
\mathrm{K e r}Q_{n}^{[ s ]}/ \mathrm{I m}Q_{n}^{[ s - 1 ]}= \Bigl \lbrace \begin{matrix}{0}&{f o r}&{s \not = 0}\\{{\cal H}_{n , n^{\prime}}}&{f o r}&{s = 0}\\ \end{matrix},
\Pr \left \{\ell(\hat{f}_{\rm det}(X)) > R \right \}& \leq 1 - \sum^{j^{*}}_{i=1}\Pr \{\hat{f}_{\rm det}(X) = w_i \}& = 1 - \left ( \sum^{j^{*}}_{i=1}\Pr \{X \in A_{D}(y_i) \}+ \gamma \right ) & \leq 1- ( (1-\gamma - \delta) + \gamma) = \delta,
\begin{array}{r l}{{( z_{0}; z_{1}, z_{2}) =}}&{{{\frac{1}{h}}\left({\frac{1 + x_{1}^{2}+ h^{2}}{2}}+ i \alpha_{0}( x_{1}) h^{3}; \right.}}\\{{}}&{{\left. x_{1}+ i \alpha_{1}( x_{1}) h^{3}, \; \;{\frac{i - i x_{1}^{2}- i h^{2}}{2}}+ \alpha_{2}( x_{1}) h^{3}\right)}}\\ \end{array}
\sum_{\lambda}sp_\lambda(X)s_\lambda(Y)=\frac{1}{1-XY}\prod_{1\leq k< l\leq n+1}(1-y_ky_l), \sum_{\lambda}o_\lambda(X)s_\lambda(Y)=\frac{1}{1-XY}\prod_{1\leq k\leq l\leq n+1}(1-y_ky_l),
q_1 = 2 -z + z \frac{M^{\prime \prime}(a_0,1,z) - M^\prime(a_0,1,z)}{M^\prime(a_0,1,z) - M(a_0,1,z)}+ z \frac{M^{\prime \prime}(a_0,1,z)}{M^\prime(a_0,1,z)}.
\psi_{i,p}^{-1}(g)\cdot_i \psi_{i,p}^{-1}(h) =\psi_{i,p}^{-1}(gh) &=\psi_{j,p}^{-1}(\phi_{ji}(p)(gh)) &=\psi_{j,p}^{-1}([\phi_{ji}(p)(g)][\phi_{ji}(p)(h)]) &=\psi_{j,p}^{-1}(\phi_{ji}(p)(g))\cdot_j\psi_{j,p}^{-1}(\phi_{ji}(p)(h))
\Big|\frac{\partial \xi^+}{\partial \xi}\Big|=\frac{\Big|I+ \frac{\xi}{|\xi|}\otimes\sigma\Big|}{8}& =\frac{|1+ \frac{\xi}{|\xi|}\cdot\sigma|}{8}=\frac{\cos^2(\theta/2)}{4}\ge \frac{1}{8}, \theta\in [0,\frac{\pi}{2}].
V(0,z) =0, z\in[0,\infty), \frac{\partial V}{\partial r}(\gamma(\Omega),z) =0,z\in[0,\infty), \frac{\partial V}{\partial z}(r,0) =-d_s\int_{0}^{r}f^{\circledast}(\sigma)\,d\sigma,r\in(0,\gamma(\Omega)).
I_1 \leq o(1)+\frac{C_9}{nx}a_nQ'(a_ns)T(a_ns) &\leq o(1)+\frac{C_9}{nx}a_nQ'(a_n\rho)\frac{T(a_ns)}{T(a_n\rho)}\left(\frac{s}{\rho}\right)^{\max \{\Lambda, C_6 T(a_ns)\}-1}T(a_ns) &\leq o(1)+\frac{C_9}{nx}a_nQ'(a_n\rho)\sup_{y\in [0, \infty)}\left(\frac{s}{\rho}\right)^{\max \{\Lambda, C_6 y\}-1}y=o(1)\ \mbox{as}n\to\infty,
\Vert E_2(u) \Vert_{H^{\ell-1}}(t)=o(1)\Vert u \Vert_{H^{\ell+1}}(t)\ \textup{and}\ \Vert E_2(u) \Vert_{H^{\ell}}(t)=o(1)\Vert u \Vert_{H^{\ell+2}}(t).
(2) \quad{\mbox{\bf{i}}}^2={\mbox{\bf{k}}}^2={\mbox{\bf{1}}}, \quad{\mbox{\bf{j}}}^2= -{\mbox{\bf{1}}}, \quad{\mbox{\bf{i}}}{\mbox{\bf{j}}}={\mbox{\bf{k}}}, \quad{\mbox{\bf{j}}}{\mbox{\bf{k}}}={\mbox{\bf{i}}}, \quad{\mbox{\bf{k}}}{\mbox{\bf{i}}}=-{\mbox{\bf{j}}}.
\cdot{_3 F}_{2}\left\{\begin{array}{l}{{- n_{1}, - l + | m | , l + | m | + \delta_{1}+ \delta_{2}+ 1}}\\{{| m | + \delta_{1}+ 1 , - n + | m | + 1}}\\ \end{array}\biggr | 1 \right\}E_{n_{1}n_{2}}^{l m}
d s_{( 6 )}^{2}= -{\frac{h}{f_{w}f_{p}}}d t^{2}+{\frac{f_{p}}{f_{w}}}\left( d z -{\frac{r_{0}^{2}\sinh 2 \sigma_{p}}{2 r^{2}f_{p}}}d t \right)^{2}+{\frac{d r^{2}}{h}}+ r^{2}d \Omega_{3}^{2},
\begin{aligned}p(y) ={}\left(\frac{2}{\pi}\right)^{1/2}\biggl[1+\tau^{1/2}\left(\frac{8}{3}y^{3}-2y\right) & +\tau\left(\frac{32}{9}y^{6}-\frac{31}{3}y^{4}+\frac{15}{2}y^{2}-\frac{37}{48}\right)\biggr]e^{-2y^{2}}+O(\tau^{3/2})\end{aligned}
\frac{1}{F(z)}=\frac{1}{f_{0}}-\frac{f_{1}}{f_{0}^{2}}z+\frac{(f_{1}^{2}-f_{0}f_{2})}{f_{0}^{3}}z^{2}-\frac{f_{1}^{3}-2f_{0}f_{1}f_{2}+f_{0}^{2}f_{3}}{f_{0}^{4}}+\cdot\cdot\cdot.
E_{k,l}\left(\{\alpha^{-1}q^{m}+\alpha q^{-m}\}\right)=\frac{v_{m,k}v_{m,l}}{\|\textbf{\textit{v}}_{m}\|^{2}}=\left(1-\alpha^{-2}q^{2m}\right)\frac{\alpha^{2m}q^{-m(m+1)}}{(q;q)_{\infty}^{2}}f_{k}\left(\alpha^{-1}q^{m}\right)f_{l}\left(\alpha^{-1}q^{m}\right)\!,
{\rm Ind}^{(2n+1,\epsilon)}(D_a,S,U;R) =\frac{1}{2}{\rm Tr}\; S(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U)^{2n+2\epsilon}\chi_R^a =\frac{1}{2}{\rm Tr}\; SU[P_+,D_a](i[P_+,D_a])^{2n+2\epsilon}\chi_R^a.
-\div_y \Big(e^{\frac{|\Lambda+\nabla_y\widetilde{w}|^2}{2}+V}\big( (\Lambda+\nabla_y\widetilde{w}) (\Lambda+\nabla_y\widetilde{w})^T+I \big)\nabla_y v \Big) &\quad=\div_y \Big(e^{\frac{|\Lambda+\nabla_y\widetilde{w}|^2}{2}+V}V_{y_l}(\Lambda+\nabla_y\widetilde{w})\Big ),
\partial_t\epsilon-\partial_y(|D|\epsilon+\epsilon-p|V_0|^{p-1}\epsilon) &\qquad\qquad+\partial_y\big[|V_0+\epsilon|^{p-1}(V+\epsilon)-|V_0|^{p-1}V_0-p|V_0|^{p-1}\epsilon\big]+\Psi_{V_0}=0,
\Pi^f_{d,k,l,\Gamma_0, \Gamma_1, \Gamma_2}(x_{d,k,l, \Gamma_0, \Gamma_1, \Gamma_2}) = (T_{w_{1}}^{f}P_{w_{1}}^{f \:\perp}) \ldots (T_{w_{k}}^{f}P_{w_{k}}^{f \: \perp}) P_{V\Gamma_0}^f (T_{w_{k+l+1}}^{f}P_{w_{k+l+1}}^{f}) \ldots ( T_{w_{d}}^{f}P_{w_{d}}^{f}).
S(u):=\frac{\displaystyle\int_\Omega\left[p\left(\frac{|\nabla u|}{K(u)}\right)^p+qa(x)\left(\frac{|\nabla u|}{K(u)}\right)^q\right]dx}{\displaystyle\int_\Omega\left[p\left(\frac{|u|}{k(u)}\right)^p+qa(x)\left(\frac{|u|}{k(u)}\right)^q\right]dx}.
C_{ij}^{ii}+C_{ij}^{jj}=0, \ \mbox{if}\i<j, C_{ij}^{ij}+C_{jk}^{jk}=C_{ik}^{ik}, \ \mbox{if}\i<j<k\ \mbox{and}\i\ne k, C_{ij}^{ij}+C_{ji}^{ji}=0, \ \mbox{if}\i<j<i, C_{ii}^{ii}=C_{xx}^{ii}, \ \mbox{for all}\x\in X.
G_0^{0(2)}*(um_1^{0(1)}(0)) = G_0^{0(1)}*(um_1^{0(1)}(0)) + c\langle m_1^{0(1)}(0),u\rangle &= G_0^{0(1)}*(um_1^{0(1)}(0)) & = m_1^{0(1)}(0)-1.
\sum_{r \in\mathbb{Z}}c^F_r(\widetilde{W_v^l},\widetilde{U_v^l})q_F^{-r/2}X^{-lr}= \omega_{\sigma^{(l)}}(-1)^{n-2}\gamma(X^l,\pi^{(l)},\sigma^{(l)},\psi^l)\sum_{r \in \mathbb{Z}}c^F_r(W_v^l,U_v^l)q_F^{r/2}X^{lr}.
|\mathfrak m_N^{B^2}(\xi)|\le Ce^{-\frac{c\kappa(d, N)^2}{100}\sum_{i=1}^d\sin^2(\pi\xi_i)}+Ce^{-\frac{c\kappa(d, N)^2}{100}\sum_{i=1}^d\cos^2(\pi\xi_i)}.
\dot{\alpha}_{( 0 )}^{\prime}- \dot{\gamma}_{( 0 )}\alpha_{( 0 )}^{\prime}- \dot{\alpha}_{( 0 )}( \beta_{( 0 )}^{\prime}- \alpha_{( 0 )}^{\prime}) = - \frac{1}{3}\dot{\phi}_{( 0 )}\phi_{( 0 )}^{\prime},
U_{e f f , T}( \phi ) = U_{c l}( \phi ) + \frac{T}{2}\sum_{n}\int \frac{d^{3}k}{( 2 \pi )^{3}}\ln \left[ \omega_{n}^{2}+{\bf \vec{k}}^{2}+ m_{p l}^{2}( \phi ) \right] , \nonumber
\hat S_i =\begin{bmatrix}I_p 0 0 \dots (\hat S_i)_{1,0}I_p 0 \cdots (\hat S_i)_{2,0}(\hat S_i)_{2,1}I_p \ddots & \ddots\end{bmatrix}, \hat H =\begin{bmatrix}\hat H_0 0 0 \cdots 0 \hat H_1 0 \ddots 0 0 \hat H_2 \ddots \vdots \vdots \ddots \ddots\end{bmatrix}, i =1,2.
\lambda_1^2=\dot{\lambda}+\lambda\frac{q_2q_3}{q_2^2},~~~~\lambda_1^1=\ddot{\lambda}+\frac{\lambda}{q_2^2}\left[q_2{\dot{q}}_3-\frac{q_2q_3}{q_2^2}(2q_2{\dot{q}}_2+q_2q_3)\right].
c_{\beta}= \begin{cases}{\langle 2\rangle_{v}}^{|\beta|-1}\quad\ \beta=[i,j]{\langle 2\rangle_{v}}^{|\beta|-1}\cdot \prod_{\ell=j}^{n-1}\big\{(v^{-4n+4\ell-2}-1)(v^{-4n+4\ell+6}-1)\big\}\quad\ \beta=[i,n,j] \end{cases}.
k^2 \int_{W_{k}}|u_{k}\overline{\partial}\xi|^2 |\partial \Phi|^2 e^{-k\varphi \circ \Phi}\omega'_{n}\lesssim k^2 \int_{\Phi(W_{k})}|u_{k}|^2 |\overline{\partial}\xi|^2 e^{-k\varphi}\omega_{n}= O(\frac{(\log k)^{2m}}{k^{m-2}}).
\|\Psi\|_{\mathcal H}^2= \int_0^\infty (\Psi^{(3)}(Z)^2 W(rZ)\:dZ + \int_0^\infty \Psi^2(Z) \left(\frac{W(rZ)}{Z^2}+ r \frac{W'(rZ)}{Z}\right) dZ,
\Phi^+_{m,\pm(j+\tfrac{1}{2})}:=\begin{pmatrix}i\Psi^{m}_{j\pm\tfrac{1}{2}}0\end{pmatrix}\Phi^-_{m,\pm(j+\tfrac{1}{2})}:=\begin{pmatrix}0 \Psi^{m}_{j\mp \tfrac{1}{2}}\end{pmatrix}
\lambda_{i}\frac{\partial W}{\partial\lambda_{i}}=C_{1}[-\frac{2}{3}J^{-5/3}\lambda_{i}\frac{\partial J}{\partial\lambda_{i}}(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2})+2J^{-2/3}\lambda_{i}^{2}]+2D_{1}(J-1)\lambda_{i}\frac{\partial J}{\partial\lambda_{i}}
J_t= J-\frac\partial{\partial x}\otimes d^c \left(\frac1 2 t^2 \left(\frac{1-\left|w \right|^2}{1+\left|w \right|^2}\right)^2 \right) -\frac\partial{\partial y}\otimes d^c \left(\frac1 2 t^2 \left(\frac{1-\left|w \right|^2}{1+\left|w \right|^2}\right)^2 \right)\circ J.
G_{\phi}^{A B}( p ) = \delta^{A B}G_{\phi}( p ) ={\delta^{A B}}{p^{- 2}}\ \ \mathrm{a n d}\ \ \G_{\psi}( p ) ={b}M^{2 \Delta}C ( \Delta ){p^{- 2 ( \mu - 2 + \Delta )}}
a^{4132}=a^{1432},a_{1}^{4132}=a_{1}^{1432},a_{2}^{4132}=a_{2}^{1432},a_{3}^{4132}=a_{3}^{1432},a_{4}^{4132}=a_{4}^{1432}, a_{12}^{4132}=a_{12}^{1432},a_{13}^{4132}=a_{13}^{1432},a_{14}^{4132}=a_{14}^{1432},a_{24}^{4132}=a_{24}^{1432},a_{34}^{4132}=a_{34}^{1432}, a_{124}^{4132}=a_{124}^{1432},a_{134}^{4132}=a_{134}^{1432}
\chi(\rho)^{1/2}\, \chi\Big(1-\rho -\frac{2\pi i \kappa}{\log T}\Big)^{1/2}= \Big(\frac{\gamma}{2\pi}\Big)^{\pi i \kappa/\log T}\bigg(1+O\Big(\frac{1}{T}\Big)\bigg) &= e^{\pi i \kappa}\, \Big( 1 + O\big((\log T)^{-1}\big) \Big)
\lim_{n\rightarrow \infty}n^{\alpha+\frac 12}\int_0^{\tau}(t-s)^\alpha\sigma'(X_s) \left( \int^{\eta_n(s)}_0 \psi_{n,2}(u,s)\, \sigma(X_{\eta_n(u)})\, dW_u \right) ds=0.
M_{C M}= \left( \begin{array}{l l l}{{\mathcal{A}_{C M}}}&{{\mathcal{B}_{C M}}}&{{\mathcal{E}_{C M}}}\\{{\mathcal{B}_{C M}}}&{{\mathcal{A}_{C M}}}&{{\mathcal{E}_{C M}}}\\{{( \mathcal{E}_{C M})^{t}}}&{{( \mathcal{E}_{C M})^{t}}}&{{\mathcal{G}_{C M}}}\\ \end{array}\right) ,
(1-2xt)\frac{d}{dt}R^+(x;t) =2x(1-2x)\frac{d}{dx}R^+(x;t)+R^+(x;t)+R^-(x;t), (1-2xt)\frac{d}{dt}R^-(x;t) =2x(1-2x)\frac{d}{dx}R^-(x;t)+xR^+(x;t)+2xR^-(x;t).
|\mbox{{\rm ID}}_{1}|=|\mbox{{\rm RCA}}^{*}_{0}+(\Pi^{1}_{1}\mbox{{\rm -CA}})^{-}|=|\mbox{{\rm RCA}}_{0}+(\Pi^{1}_{1}\mbox{{\rm -CA}})^{-}|=\vartheta(\varepsilon_{\Omega+1}).
\Gamma ( G ) = S ( G ) \Gamma ( N / 2 ) [ \frac{g^{2}}{( 4 \pi )^{5 / 2}}]^{N}\int [ d x ]_{3 N}U^{- 5 / 2}( \phi / U + \sum x_{i}M_{i}^{2})^{- N / 2}.
\P_{\theta}\{\theta^*(\alpha,Y)\leqslant\theta|Y\in B_2(\alpha)\}= \P_{\theta}\{F(Y;\theta)\leqslant\alpha|Y\in B_2(\alpha)\}= \P_{\theta}\bigg\{F_2(Y;\theta)\leqslant \frac{\alpha-F(y_1;\theta)}{F(y_2;\theta)-F(y_1;\theta)}\,\Big|\,Y\in B_2(\alpha)\bigg\}= \frac{\alpha-F(y_1;\theta)}{F(y_2;\theta)-F(y_1;\theta)},
S_{a|x|^2}\big((x,y,t),(x',y',t')\big) = S_{a_n |x|^2}\big((x,y,t_n),(x',y',t_n')\big) \delta_0[a_n(s-s') - b(t_n-t_n')] &= \frac{2 a_n \delta_0[a_n(s-s') - b(t_n-t_n')]}{\bigl(\pi a_n[(x-x')^{2}+ (y-y')^{2}] - 2\pi i[(t_n-t_n')+a_n(x+x')(y-y')]\bigr)^{2}}
\zeta_{i}= - w_{i}^{[ b e r n ]}+ ( \partial_{i}\ln | \varpi^{- 1}\Omega_{\lbrack 0 ]}| ) ~ \left( \ln | \Omega_{\lbrack 1 ]}| \right)^{\ast}+ \left( \Omega_{\lbrack 1 ]}^{\ast}\right)^{- 1}\partial_{i}\Omega_{\lbrack 1 ]},
P(S_n=m\mid x_1^n) = \frac{f_{m}(x_n)\sum\limits_{i=1}^{M}p_{i, m}P(S_{n-1}=i \mid x_1^{n-1})}{\sum\limits_{j = 1}^{M}f_j(x_n)\sum\limits_{i=1}^{M}p_{i, j}P(S_{n-1}=i \mid x_1^{n-1})},
F(h)= \inf\limits_{\lambda\in \left[\tau_1, \tau_2\right]}\left\{\int_0^{\lambda}\left(h_2(r)-\frac{h_1(r)}{h_1(\lambda)}h_2(\lambda)\right)^2dr+\int_{\lambda}^1\left(h_2(r)-\frac{1 -h_1(r)}{1-h_1(\lambda)}h_2(\lambda)\right)^2dr\right\}^{\frac{1}{2}}
\hat{\sigma_{\overline{x}_{w}}^{2}}=\frac{n}{(n-1)(n\overline{w})^{2}}[\sum(w_{i}x_{i}-\overline{w}\overline{x}_{w})^{2}-2\overline{x}_{w}\sum(w_{i}-\overline{w})(w_{i}x_{i}-\overline{w}\overline{x}_{w})+\overline{x}_{w}^{2}\sum(w_{i}-\overline{w})^{2}]
\int_{0}^{2}(1-t)G_{k}(t)dt=0 \implies\frac{\alpha_{0}(k+1)!}{2C_{k}}-\frac{2\alpha_{1}(k+1)ka_{k}^{2}}{(2\pi)^{2}}+\tilde{C_{k}}=-\frac{b_{k}^{2}2\alpha_{1}(k+1)k}{(2\pi)^{2}}\frac{\int_{0}^{2}(1-t)(1+t)^{k-2}dt}{\int_{0}^{2}(1-t)(1+t)^{k}dt}.
\langle \nabla_{\varphi}\ell,\psi \rangle=\nabla_{\varphi}\ell(\psi)=A_u(\psi)(x)p+ \psi(x)\frac{\partial H}{\partial y}(t,x,y,\varphi,u,z,p,q,r)|_{y=\varphi(x)}; \psi \in \mathcal{D}.
\chi [{\cal M}] ( y , \bar{y}) = \sum_{q_{\mathrm{L}}, q_{\mathrm{R}}= 0}^{\hat{c}}( - 1 )^{q_{\mathrm{L}}+ q_{\mathrm{R}}}h_{q_{\mathrm{L}}, \hat{c}- q_{\mathrm{R}}}y^{q_{\mathrm{L}}}\bar{y}^{q_{\mathrm{R}}}\, ,
\sigma_n^t(x_1,\dots,x_n) := t( \sigma_{\lceil n/2 \rceil}^t(x_1,\dots,x_{\lceil n/2 \rceil}), \sigma_{\lfloor n/2 \rfloor}^t(x_{\lceil n/2 \rceil +1},\dots,x_n) )
\int_{t_0}^0 \int_{\partial\Omega_k}|\partial_tu_k|^2ds\;dt =\int_{t_0}^0\int_{\partial\Omega_k}|d\pi_N(u_k)\partial_{\nu_k}u_k|^2ds\;dt =\int_{t_k+r_kt_0}^{t_k}\int_{\partial B}|u_t|^2d\phi\;dt \le\int_{t_k+r_kt_0}^{\infty}\int_{\partial B}|u_t|^2d\phi\;dt\to 0
\int d^{2}x \left\{2 \chi_{\ell}^{\dagger}\left( \partial_{z}\partial_{\bar{z}}+ \partial_{\bar{z}}\partial_{z}\right) \phi_{\ell}- 2 \phi_{\ell}^{\dagger}\left( \partial_{z}\partial_{\bar{z}}+ \partial_{\bar{z}}\partial_{z}\right) \chi_{\ell}\right\}= - 4 \sin \delta_{\ell}.
\int_{\epsilon}^\xi\frac{dX}{2\sqrt{X(X-\lambda)}}=\int_\epsilon^{\xi}\frac{dX}{2X}+\frac12\sum_{n=1}^{\infty}\frac{(\frac12)_n}{n!}\int_\epsilon^{\xi}\left(\frac{\lambda^ndX}{X^{n+1}}\right)
N:=M+1, \ \textup{for the expansion of}I^{\frac{n}{2}}_{y}\circ E^{\frac{n}{2}}_{x,M}f^{-\alpha}, &N':=2(M+1), \ \textup{for the expansion of}E^{\frac{n}{2}}_{y,M}\circ E^{\frac{n}{2}}_{x,M}f^{-\alpha}.
\widetilde\nabla_X P(Y,\xi)= X(P(Y,\xi))-P(\nabla_XY,\xi)-P(Y,\nabla^\perp_X\xi) = X\langle Y,\xi \rangle-\left\langle\nabla_XY,\xi\right\rangle-\langle Y,\nabla^\perp_X\xi\rangle.
\underset{\left\{\pmb{F}_k\right\}_{k=1}^K,\left\{\pmb{W}_k\right\}_{k=1}^{K}}{}\,\,\,\,\,\qquad\qquad\Omega \,\,\,\,\, \pmb{S}_k \succeq \pmb{0},\,\,\, \sigma_{min}(\pmb{S}_k) \geq \epsilon,
\frac{1}{\sqrt{1-([\eta]_{m}-2^{-j})^2}}=\frac{1}{\sqrt{1-(\frac{\eta}{\sqrt{2m+2}}+R-2^{-j})^2}}&=\frac{1}{\sqrt{1-2^{-2j}}}\frac{1}{\sqrt{1+2^{1-j}\frac{\eta}{(1-2^{-2j})\sqrt{2m+2}}+2^{1-j}R-2R\frac{\eta}{\sqrt{2m+2}}}}&=\frac{1}{\sqrt{1-2^{-2j}}}\left(1-2^{-j}\frac{\eta}{(1-2^{-2j})\sqrt{2m+2}}+R'\right),
\begin{aligned}|J_{\rho_t}- J_{\infty}| \le\int_{t}^{\infty}\bigg|\frac{d}{dr}J_{\rho_r}\bigg|dr &\le C H(\rho_{0}|M_{\Omega_{\rho_{0}}})\int_{t}^{\infty}e^{-\int_{0}^{r}B(s)ds}dr.\end{aligned}
{\cal{L}}_{\Lambda}=\frac{1}{2}(\partial \phi )^{2}-\frac{1}{2}m^{2}\phi^{2}-\frac{\lambda}{4!}\phi^{4}+\frac{1}{2}\delta Z_{1}(\partial \phi )^{2}-\frac{1}{2}\delta m^{2}\phi^{2}-\frac{\delta \lambda}{4!}\phi^{4}.
|I_4| \le \dfrac{2^{N+ps}}{|x|^{N+ps}}\int_{B_{\frac{|x|}2}(0)}|\Upsilon(y)|^{p-1}dy \le \dfrac{C(N,s,p)}{|x|^{N+ps}}\int_{B_{\frac{|x|}2}(0)}|\Upsilon(y)|^{p-1}dy =\dfrac{C(N,s,p)}{|x|^{N+ps}}I_4'.
\#\mathop{\mathrm{supp}}V_{F,\lambda}^{(>)}\ =\ \#\bigg\{x\in \omega ({\underline{r}})\bigg|\V_{F,\varepsilon}(x)\geq (1-\varepsilon^{\prime})\frac{\eta}{\lambda}\bigg\}\ =\F\big((1-\varepsilon^{\prime})^{-1}\lambda \big).
A_{4 F}\sim \frac{\Gamma ( \frac{1}{2}s - \frac{1}{8}Q^{2}) \Gamma ( \frac{1}{2}t - \frac{1}{8}Q^{2})}{\Gamma ( \frac{1}{2}s + \frac{1}{2}t - \frac{1}{4}Q^{2})}
\partial_{M}( \sqrt{- g}g^{M N}\partial_{N}\phi ) + \frac{\alpha}{2 ( q + 1 ) !}\sqrt{- g}e^{- \alpha \phi}F^{2}+ \frac{\beta}{2 ({\tilde{d}}+ 1 ) !}\sqrt{- g}e^{- \beta \phi}G^{2}= 0 .
\psi_{\lambda,\ell}(x)= \lambda^{-\frac{1-d+\ell}{2}}\psi(\lambda^{1-\frac{1}{d-\ell}}x_1, \dots, \lambda^{1-\frac{d-\ell}{d-\ell}}x_{d-\ell},x_{d-\ell+1},\dots,x_d),
\|f\|_{\dot{H}^{N,b}}:=\big\{\sum_{k\in\mathbb{Z}}\|P_kf\|^2_{L^2}(2^{2Nk}+2^{2kb})\big\}^{1/2}, &\|f\|_{\dot{W}^{N,b}}:=\sum_{k\in\mathbb{Z}}\|P_kf\|_{L^\infty}(2^{Nk}+2^{bk}), &\|f\|_{\widetilde{W}^{N}}:=\|f\|_{L^\infty}+\sum_{k\geq 0}2^{Nk}\|P_kf\|_{L^\infty}.
\left[ \hat X_{1,q},\;\hat P_{1,q}\right] = i\hbar - (1-q^2)\left(\frac{i\hbar}{2(\lambda_1 +\lambda_2)^2}\right) \left( \lambda_1 K_2 \tilde A_2^\dagger \tilde A_2 + \lambda_2 K_1 \tilde A_1^\dagger \tilde A_1 \right);
F\left( x_{1}\right) -F\left( x_{1}+r^{\ast}\right)=\int_{x_{1}}^{x_{1}+r^{\ast}}\left\vert F^{\prime}\left( x_{1}\right)\right\vert dx \geq \int_{x_{1}}^{x_{1}+r_{1}}\left\vert F^{\prime}\left(x_{1}\right) \right\vert dx &\geq \left\vert F^{\prime}\left(x_{1}+r_{1}\right) \right\vert \cdot r_{1}\gtrsim 1.
(-1)^n(ga(v)(T-\mu)^{-1})^{n}u &=I_n\left((-1)^ng^n\overline{v(k_1)\dots v(k_n)}u(k_1,\dots,k_n)\prod_{\ell=1}^{n}( T^{(\ell)}(k_1,\dots,k_\ell) -\lambda )^{-1}\right)
\begin{array}{ll}x_1 x_2 = 0 \,, \hbox{(double four-fold)}x_1 x_2 + x_3 x_4 = 0 \,, \hbox{(double surface)}x_1 x_2 + x_3 x_4 + x_5 x_6 = 0 \,, \hbox{(double point).}\end{array}
v_N^{x,Sp(2)}= l \otimes 1 + l^{\ast}\otimes 1 + l^{\ast}\otimes l + l \otimes l^{\ast}, v_N^{y,Sp(2)}= ll^{\ast}\otimes 1 + 1 \otimes l + 1 \otimes l^{\ast}+ l^2 \otimes l^{\ast}+ (l^{\ast})^2 \otimes l,
\begin{cases}\phi_r''-\Delta \phi_r +\sum_{s=1}^p\widehat \alpha_{rs}\phi_s=0 \hbox{in}(0, +\infty)\times \Omega, \phi_r=0 \hbox{on}(0, +\infty)\times \Gamma_0, \partial_\nu\phi_r +\sum_{s=1}^p\widehat \beta_{rs}\phi_s=0 \hbox{on}(0, +\infty)\times \Gamma_1,\end{cases}
\alpha^s = a_0^{s-4}(a_0a_1^{sn_1}a_2^{sn_2}\cdots a_m^{sn_m})a_0^3a_{m+1}^{sn_{m+1}}\in A \intertext{for all $s \ge 4$ by the inductive hypothesis, and}\beta^s = p^{s-3}(pa_1^{sn_1}a_2^{sn_2}\cdots a_m^{sn_m})p^2a_{m+1}^{sn_{m+1}}\in A
1 \geq C_N = \frac{\prod \limits_{k=0}^N c(\theta_k) \cdot p'(x_k)}{\prod \limits_{k=0}^N c(\theta_k)(1-x_k)}= \prod \limits_{k=0}^N \frac{1 - 2x_k}{1 - x_k}= \prod \limits_{k=0}^N (1 - \frac{x_k}{1 - x_k}) \geq \prod \limits_{k=0}^N (1 - \gamma x_k) = C_N^* > 0,
\left[{\frac{1}{\alpha}}{\frac{d^{2}}{d r^{2}}}+{\frac{2}{\alpha}}{\frac{1}{r}}{\frac{d}{d r}}-{\frac{2}{\alpha}}{\frac{1}{r^{2}}}\right] b_{0}+ M \left({\frac{1}{\alpha}}- 1 \right){\frac{d a_{0}}{d r}}= M^{2}b_{0},
{\frac{\sqrt{2}\vartheta_{2}^{2}\widetilde \vartheta_{3}}{\vartheta_{3}}}+ \sqrt{\vartheta_{3}^{2}\widetilde \vartheta_{3}^{2}+ \vartheta_{2}^{2}\widetilde \vartheta_{4}^{2}- \vartheta_{4}^{2}\widetilde \vartheta_{2}^{2}}= 0 .
\zeta \left( \begin{array}{r c r}{{s_{1}}}&{{\ldots}}&{{s_{k}}}\\{{\sigma_{1}}}&{{\ldots}}&{{\sigma_{k}}}\\ \end{array}\right) : = \sum_{n_{j}> n_{j + 1}> 0}\quad \prod_{j = 1}^{k}\frac{\sigma_{j}^{n_{j}}}{n_{j}^{s_{j}}}
\mathrm{Tr}(M_0 \otimes M_1 \otimes \cdots \otimes M_n) := \sum_{i_0, \cdots, i_n}(M_0)^{i_0}_{i_1}\otimes (M_1)^{i_1}_{i_2}\otimes \cdots \otimes (M_n)^{i_n}_{i_0}.
\begin{aligned}Res_i^{\Lambda}:Rep(\mathcal{R}_{n+1}^{\Lambda}) \mapsto Rep(\mathcal{R}_{n}^{\Lambda}) N \mapsto Ne_{n,i}Ind_i^{\Lambda}:Rep(\mathcal{R}_{n}^{\Lambda}) \mapsto Rep(\mathcal{R}_{n+1}^{\Lambda}) M \mapsto M\otimes_{\mathcal{R}_{n}^{\Lambda}}e_{n,i}\mathcal{R}_{n+1}^{\Lambda}. \end{aligned}
\sup_{f\in L^1(\Gamma_-,d\xi)\atop \|f\|_{L^1(\Gamma_-,d\xi)}=1}\big|\int_{\Gamma_+}\phi(z,v)j_+KJf(z,v) d\xi(z,v)\big|\le C_1e^{2\|\tau\sigma_-\|_\infty}\|k\|_{\infty}\eta^{n-1},
T_{1,C_1}(V=0) = \frac{q^{2g+1}}{9! \pi \zeta_q(2)}\Bigg[ \sum_{j=0}^{\alpha-1}\sum_{m=0}^{2g}\frac{\sin (2 \pi \theta_0(2m-j))}{2m-j}Q(2g-m) + \sum_{j=0}^{\alpha-1}\sum_{m=1}^{2g}\frac{\sin (2 \pi \theta_0(2m+j))}{2m+j}Q(2g-m) \Bigg] +O(q^{2g+1}\alpha),
\begin{array}{c c}{{\theta^{i}= \left( \begin{array}{c}{{\vartheta_{-}^{a}}}\\{{( \bar{\vartheta}_{a}^{-})^{t}}}\\ \end{array}\right) ~ ~}}&{{~ ~ \tilde{\theta}^{i}= \left( \begin{array}{c}{{\vartheta_{+}^{a}}}\\{{( \bar{\vartheta}_{a}^{+})^{t}}}\\ \end{array}\right)}}\\ \end{array}
t \frac{du^*}{dt}= (\lambda^*(t)+\lambda_1^*(t))u^* \mbox{on $(t_{\xi},T]$}, t \frac{dq^*}{dt}= (\lambda^*(t)+\lambda_1^*(t)+c^*(t))q^* + \gamma^*(t)u^* \mbox{on $(t_{\xi},T]$}.
m(A,B)=[\begin{matrix}1&x_{1}a_{12}&x_{1}a_{13}&\cdot\cdot\cdot&x_{1}a_{1n}\\ 0&1&x_{2}a_{23}&\cdot\cdot\cdot&x_{2}a_{2n}\\ \vdots&\vdots&\vdots&&\vdots\\ 0&0&0&\cdot\cdot\cdot&1\end{matrix}].
\begin{aligned}V_{rr}(r, p) = 2\beta r^{-3}|p|^2+a(1-\beta)^{\alpha-1}\beta(1-\alpha)(\beta-\alpha\beta-1)r^{\beta(1-\alpha)-2}|p|^\alpha &=r^{-3}|p|^\alpha \left(2\beta|p|^{2-\alpha}+a(1-\beta)^{\alpha-1}(\beta-\alpha\beta) (\beta-\alpha\beta-1)r^{\beta(1-\alpha)+1}\right).\end{aligned}
f ( x ) = \int_{- \infty}^{\infty}\frac{d \xi}{2 \pi}\mathrm{e}^{i ( 1 - x ) \, \xi}\langle \widetilde{H}( p ) | W_{\scriptstyle \Pi}({v \! \cdot \! n}\, \xi \mu - i 0 ) | \widetilde{H}( p ) \rangle \, ,
\delta{\cal I}_{U(1)}=-\lim_{\beta\rightarrow 0}2^{N/2-1}\int [dX]\;\frac{e^{-(X+X)^2/2\beta}}{(2\pi\beta)^{(D-1)/2}}=-\frac{2^{N/2-1}}{2^{D-1}}=-\frac{1}{4},
f(\{x_{i,r}\}_{i\in I}^{1\leq r\leq k_{i}})=0\ \x_{i,s_{1}}=x_{i,s_{2}}+d_{i}\hbar=\cdots=x_{i,s_{1-a_{ij}}}-d_{i}a_{ij}\hbar=x_{j,r}-\frac{d_{i}a_{ij}}{2}\hbar
\beta^\infty = \frac{2}{a}\sqrt{\left( 1 - \frac{2(m-1)(1-a(N+1))}{N}\right)^2 + \frac{4(m-1)ma}{N}(1-a(N+1))}& + \frac{2}{a}\left( 1 - \frac{2(m-1)(1-a(N+1))}{N}\right)
[L_t,f_r] =L_t\sum_{z\in Y}f(z)\chi_z-(-1)^{\partial L\partial f}\sum_{y\in Y}f(y)\chi_yL_t &=\sum_{y,z\in Y}\chi_yL_t\chi_z(f(z)-f(y))=\sum_{\substack{y,z\in Y d(y,z)\leq 2r+p_t}}\chi_yL_t\chi_z(f(z)-f(y))
\mathbb{P}(t;\partial_t) =2^{p-1}\partial_t^p+\sum_{n=2}^pu_n(t)\partial_t^{p-n}\ , \mathbb{Q}(t;\partial_t) =\beta_{p,p'}\left(2^{p'-1}\partial_t^{p'}+\sum_{n=2}^{p'}v_n(t)\partial_t^{p'-n}\right)\ ,
\left( \begin{array}{c}{{0}}\\{{g^{\prime}\gamma_{d}^{\alpha}B_{\alpha}\psi_{l L}}}\\ \end{array}\right) + \frac{M_{i}+ \eta_{i}}{\sqrt{2}}\left( \begin{array}{c}{{\psi_{1 R}}}\\{{\psi_{1 R}}}\\ \end{array}\right) .
\widehat{\cal R}= ( 1 + 2 \widehat{\cal N}_{0}) \widehat{\cal N}_{\perp}( 2 - \widehat{\cal N}_{\perp}) , \qquad \widehat{\cal C}_{+}= ( \widehat{\cal S}- \widehat{\cal R}) ( 1 + 2 \widehat{\cal N}_{0}) .
\Gamma_+^{-1}A^{(r)}C_r - \alpha_r C_3 \Gamma_+^{-1}A^{(r)}=C \delta_{r3}- \frac{1}{2}\frac{\alpha}{1+\mu \alpha k}C U_3^{-1}Y Y^{\rm T}A^{(r)}C U_r^{-1}.
(\Delta_{\psi(t)}X)^{2}=\frac{t^{2}}{m^{2}}(\Delta_{\psi_{0}}P)^{2}+\frac{2t}{m}(\langle\frac{1}{2}(XP+PX)\rangle_{\psi_{0}}-\langle X\rangle_{\psi_{0}}\langle P\rangle_{\psi_{0}})+(\Delta_{\psi_{0}}X)^{2},
\mathrm{H}_{3 1}^{F}= - \frac{1}{4}{\cal F}_{3 1}\left( 9 \alpha_{3 1}+ 6 \beta_{3 1}\right) \int d^{3}{\vec{x}}i \pi ({\vec{x}}) \tau_{a}\partial_{a}{\xi}( \vec{x}) \left( \frac{\ell_{P}}{{\cal L}}\right)^{2 \left( 1 + \Upsilon \right)}
2J_1 (\tau) = \int_{U_1(\tau)}e^{2\rho\tau}\left[(u_r^2 +u_t^2)e^{2\tau}(\cosh^2 s + \sinh^2 s) + 4 u_r u_t e^{2\tau}\sinh s \cosh s \right] d\mu + \int_{U_1 (\tau)}e^{2\rho\tau}\left[2\rho u u_r e^{\tau}\sinh s + 2\rho u u_t e^{\tau}\cosh s\right] d\mu + \frac{1}{3}\int_{U_1 (\tau)}e^{3\tau}|u|^6 d\mu.
\Gamma_{4}( \varphi_{c}) = - \mathrm{l n}\int{\cal D}\delta \varphi{\cal D}\delta \varphi^{\dagger}{\cal D}v{\cal D}\Lambda \mathrm{e x p}\{- \int d^{4}x \tilde{\cal L}( \varphi_{c}+ \delta \varphi ) \}.
I(M_1, M_2; Y_{2, 0}^T)=I(M_2; Y_{2, 0}^T)+I(M_1; Y_{2, 0}^T|M_2) \geq I(M_2; Y_{2, 0}^T) + \frac{snr_2}{snr_1}I(M_1; Y_{1, 0}^T|M_2),
q^{\alpha^{2}/2}(-abq^{-\alpha};q)_{\infty}=\frac{1}{\sqrt{2\pi\log q^{-1}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{x^{2}}{\log q^{2}}+i\alpha x\right)}{\left(abq^{-1/2}e^{-ix};q\right)_{\infty}}dx
f'(h_1)(\xi) &= \sum_jf(g_j)(\pi(\xi)|_{X_j}) &=\sum_j[\sum_{g}\theta(g, g_j)(g(\pi(\xi)|_{X_j}))-\sum_{\substack{g, \bar{g}g\bar{g}=g_j}}\theta(g, \bar{g})(\pi(\xi)|_{X_j})+\sum_{g}\theta(g_j, g)(\pi(\xi)|_{X_j})].
\lambda^{}_{t}=\sum_{n=0}^{\infty}\lambda^{2n}_{t}&=\overline{\gamma}_{0}+\sum_{n=1}^{\infty}\int_{0}^{t}h(t-s)N^{2n-1}(ds) &=\overline{\gamma}_{0}+\int_{0}^{t}h(t-s)N^{}(ds).
I_d(\mathbf{u}) =\frac 14\sum_{i=1}^d \|u_i\|^2_{\lambda_i}=\frac 14\Big( \sum_{i=1}^d \mu_i|u_i|_{4}^{4}+2\sum_{i<j}b_{ij}|u_iu_j|_2^2 \Big).