harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
T_0 \le\begin{cases}B_4 \left(1+ \mu_0^{-1}\varepsilon^{-\frac{1}{\frac{1}{p-1}-\frac{n}{2}}}\right)^{\frac{1}{1+\beta}}(\beta \in (-1,1]), \exp \left( B_4\left( 1+ \mu_0^{-1}\varepsilon^{-\frac{1}{\frac{1}{p-1}-\frac{n}{2}}}\right)\right) (\beta = -1).\end{cases}
\tilde{H}(\hat{v}_1)=\biggl\{\hat{v}^{*}_2\in{\mathcal{S}_2}:\tilde{F}_2(i,\hat{v}_1(i),\hat{v}^{*}_2(i))=\inf_{v_2\in V_2(i)}\tilde{F}_2(i,\hat{v}_1(i),v_2)~\forall~i\in S\biggr\}.
\begin{aligned}\int_0^1 (v_x)^2(0,x)dx-\lambda\int_0^1\frac{v^2(0,x)}{a(x)b(x)}dx \le \int_0^1(v_x)^2(t,x)dx -\lambda\int_0^1\frac{v^2(t,x)}{a(x)b(x)}dx.\end{aligned}
\int_{M}\left( R-\frac{1}{2}Tor-\frac{1}{2}Tor^{\prime}\right) \left(\gamma ,\gamma \right) d\mu +\int_{M}\left\vert \gamma_{1,1}\right\vert^{2}d\mu -\int_{M}Tor\left( d_{b}u,\gamma \right) d\mu =0.
\begin{aligned}|p(M_nx)-\varphi_n(x)| \:\:\: \leq \:\:\: |p(M_nx)-\hat{\varphi}_n(x)|+|\hat{\varphi}_n(x)-\varphi_n(x)| \leq \:\:\: C\cdot C_{d,G}^{-2(1+\eta)}k^{-(1+2\eta)}\end{aligned}
\min_{j= \pm1}\Bigl| \frac{3}{4}(\sqrt{2}-1)^{\frac{4}{3}}\frac{N^2}{\xi_2'}+ (\sqrt{2}+1)^{\frac{2}{3}}\xi_2' -j \sqrt{3}(\sqrt{2}-1)^{\frac{1}{3}}N\Bigr|\leq 2^{23}{A'}^{-1}N.
\|s_{D}(p)\|_{v}=\begin{cases}{\bigg({\displaystyle \sum_{j=0}^{r}|\alpha_{j}\chi^{m_{j}}(p)|_{v}}\bigg)^{-1}}v \text{is Archimedean}, begin{align*}4mm]\Big(\displaystyle \max_{0\le j\le r}|\alpha_{j}\chi^{m_{j}}(p)|_{v}\Big)^{-1}v .\end{cases}
\epsilon\int_{0}^{1}\rho^{\epsilon}_x\bar{\sigma}_x+\int_{0}^{1}u^{\epsilon}_x\bar{\sigma}-\int_{0}^{1}\rho^{\epsilon}\bar{\sigma}_x+\lambda\int_{0}^{1}\rho^{\epsilon}\bar{\sigma}=\int_{0}^{1}f\bar{\sigma}.
t(\varphi_1,\dots,\varphi_{k})\cdot u_{\varepsilon_1,\dots,\varepsilon_k}=e^{{i\over 2}{\sum\limits_{j=1}^{k}\varepsilon_{k+1-j}\varphi_j}}\cdot u_{\varepsilon_1,\dots,\varepsilon_k},
\parallel T(r) \ge \dfrac{2}{3}\sum_{k=1}^3 \sum_{t\ne i}N_{(f^k,a_t)}^{[1]}(r)+o(T_f(r))\ge \dfrac{2}{3n}\sum_{k=1}^3 \sum_{t\ne i}N_{(f^k,a_t)}^{[n]}(r)+o(T_f(r)) &\ge\dfrac{4(q-1)}{9n(n+1)}T(r)+o(T_f(r)).
\int_B\left(\int_{\O_b}f(y)\d\mu_{\rho_{\O_b}}(y) \right)\d\mu_{\sigma}(b) =\int_M c(x)\left(\int_{\O_x}f(y)\d\mu_{\rho_{\O_b}}(y)\right)\d\mu_{\tau}(x) &=\int_M c(x)\left(\int_{s^{-1}(x)}f(t(g))\d\mu_{\overrightarrow{\rho}}(g)\right)d\mu_{\tau}(x) &=\int_M\left( \int_{s^{-1}(x)}c(s(g))f(t(g))\d\mu_{\overrightarrow{\rho}}(g)\right)\d\mu_{\tau}(x).
\begin{aligned}E_n =\int_{x^*}^{x_n}(f(x)-p_n(x))\dx=(-1)^{n+2}\frac{f^{(n+1)}(\xi)}{(n+1)!}h^{n+2}\int_0^{\frac{x_n-x^*}{h}}\mu(\mu-1)\cdots(\mu-n+1)(\mu-n)\d\mu\end{aligned}
\left. \begin{array}{l}{{\partial_{\phi}F^{a}= 0 , \, \, \, \, \, \, \, \, \, \, \, \, \, \, \partial_{\pi_{\phi}}G^{a}= 0 ,}}\\{{F^{b}\partial_{b}G^{a}- G^{b}\partial_{b}F^{a}+ \pi_{\phi}\partial_{\phi}G^{a}- 2 ( e^{\phi}- e^{- 2 \phi}) \partial_{\pi_{\phi}}F^{a}= 0 ,}}\\ \end{array}\right.
\begin{cases}\|\nabla u+(\nabla u)^t\|_{L^1(0, T_0; L^\infty)}+\|\nabla d\|_{L^2(0,T_0; L^\infty)}=+\infty,{\rm{or}}\ \|\rho\|_{L^\infty(0,T_0; L^\infty)}+\|\nabla d\|_{L^3(0,T_0; L^\infty)}=+\infty\ ({\rm{if}}\ 7\mu>9\lambda).\end{cases}
\left(\frac{{\rm det'}(\Delta_{v})}{{\rm det}(\Delta^{(0)}_{v})}\right)^{-\frac{1}{2}}\left(\frac{{\rm det}(\Delta_{A})}{{\rm det}(\Delta^{(0)}_{A})}\right)^{-1}\left(\frac{{\rm det}(\Delta_{c})}{{\rm det}(\Delta^{(0)}_{c})}\right)^{+1}= \left(\frac{{\rm det'}(\Delta_{-})}{{\rm det}(\Delta^{(0)})}\right)^{-1}
w_{g}w_{h}= v_{a}v_{r}^{*}v_{b}v_{s}^{*}=v_{a}v_{cg^{-1}r^{-1}}v_{cg^{-1}b^{-1}}^{*}v_{s}^{*}~~(\textrm{by Eq. \ref{isometries}}) =v_{cg^{-1}r^{-1}a}v_{cg^{-1}b^{-1}s}^{*}=v_{ca^{-1}a}v_{cg^{-1}h^{-1}}^{*}=v_{c}v_{c(hg)^{-1}}^{*}
y_k= \mathbf{h}_{kB}^H({\bf x}_B+{\bf e}_B)+\sum_{j=1}^{J}{\bf h}_{kj}^H ({\bf x}_j+{\bf e}_j)+w_k +n_k, \forall k\in\mathcal{C},, z_j={\bf g}_{jB}^H ({\bf x}_B+{\bf e}_B)+\sum_{\substack{i=1 i\neq j}}^{J}{\bf g}_{ji}^H ({\bf x}_i+{\bf e}_i)+{\bf g}_{jj}^H ({\bf x}_j +{\bf e}_j) +w^{'}_j+ n^{'}_j, \forall j\in\mathcal{D},,
I\left(W_{k:K}; \mathbb{H}, Q_{1:N}^{[k-1]}, A_{1:N}^{[k-1]}|W_{1:k-1}, Z \right) &= I\left(W_{k:K}; \mathbb{H}, Q_{1:N}^{[k-1]}, A_{1:N}^{[k-1]}, W_{k-1}|W_{1:k-2}, Z \right) - I \left(W_{k:K}; W_{k-1}|W_{1:k-2}, Z \right)
|\mathbb{E}h(W) - \chi^2_{(m-1)}h | \leq \frac{4}{(m+1)n}\bigg(\sum_{j=1}^m\frac{1}{\sqrt{p_j}}\bigg)^2\{19\|h\|+366\|h'\|+2016\|h''\| &\quad+5264\|h^{(3)}\|+106965\|h^{(4)}\|+302922\|h^{(5)}\|\}.
R^{\prime}(\rho) = C_\infty \kappa^{\gamma_g}\Big( \frac{(1 - 2\beta \gamma_g) \rho^2 + 2X(0) (1-\beta \gamma_g) \rho + 1 + X(0)^2}{[ 1 + (X(0) + \rho)^2]^{\beta \gamma_g + 1}}\Big ).
T =TP_{M_1}+TP_{M_2}+TP_{M_3}& =\alpha P_{M_1}+TP_{M_2}+TP_{M_3}& =\alpha I-(\alpha P_{M_2}-TP_{M_2})+(TP_{M_3}-\alpha P_{M_3}) \left[\because I=P_{M_1}+P_{M_2}+P_{M_3}\right].
\mbox{RHS of (\ref{general-5})}=\sum_{j\geq 0}\sum_{k,l\geq 0}c_e(k,l;q^j s)x^{2k+2l}{(-d/c,qs/ab,s,qs^2/a^2c^2;q)_{j}\over (q,q^{2}s^2/abcd,q^{1/2}s/ac,-q^{1/2}s/ac;q)_{j}}(qx/d)^j &\times{}_4\phi_3\left[{q^{-j},-b/a,qs/cd,-q^{-j}ac/s\atop-qs/ac,-q^{-j+1}c/d,q^{-j}ab/s};q,q\right].
Q(\zeta) =\frac{1}{4}\left(\frac{1-\mu_1^2}{\zeta^2}+\frac{1-\mu_2^2}{(\zeta-1)^2}+\frac{1-\mu_1^2 - \mu_2^2 +\mu_\infty^2}{\zeta(1-\zeta)}\right).
\begin{array}{c c c}\hline{\hline}&{}&{}\\{M 2^{e}\mapsto M 5^{m}}&{\qquad d * F_{4}= F_{4}\wedge F_{4}}&{\qquad d \star{\cal G}_{3}= d{\cal G}_{3}= - \hat{F}_{4}^{( 6 )}}\\{}&{}&{}\\ \hline{\hline}\\ \end{array}
\dim (h^{-1}(\mathcal{D}_i)\setminus (T^*\mathcal{U}\cap h^{-1}(\mathcal{D}_i))) = \dim ((\mathcal{Z}\cap h^{-1}(\mathcal{D}_i))\cup (T^*\mathcal{W}\cap h^{-1}(\mathcal{D}_i)) ) &\leq 2N - 3 &= \dim h^{-1}(\mathcal{D}_i) - 2
\hat{L}_0 = \{0\}\sqcup \bigsqcup_{m=1}^\infty \bigsqcup_{\gamma \in \Gamma/(\Gamma \cap N)}\{\gamma\cdot (0,0,0,m)\}\sqcup \bigsqcup_{m=1}^\infty \bigsqcup_{n=0}^{3m-1}\bigsqcup_{\gamma \in \Gamma}\{\gamma \cdot(0,0,3m,n)\}.
H_1:=|{\bf D^1}|\cos(\theta_1)={\bf D^1}\cdot E_1, H_2:={\bf D^1}|\sin(\theta_1)={\bf D^1}\cdot E_2, & L_1:=|{\bf D^2}|\cos(\theta_2)={\bf D^2}\cdot E_1, \quad\L_2:={\bf D^2}|\sin(\theta_2)={\bf D^2}\cdot E_2,
B = \begin{bmatrix*}[r] 2 -1 0 0 -1 -2 -1 -1 0 -1 2 0 0 -1 0 2 \end{bmatrix*}, B^{-1}= \begin{bmatrix*}[r] 1 1 1/2 1/2 1 2 1 1 1/2 1 1 1/2 1/2 1 1/2 1 \end{bmatrix*}
|H_n[f]| \le n!\sum_{1\le k_1<\dots<k_n}\frac{a_{k_1}\dotsb a_{k_n}}{k_1^2k_2^3\cdots k_n^{n+1}}\ll n!\sum_{k_1=1}^\infty\dotsi\sum_{k_n=1}^\infty\frac{1}{k_1^{1+\delta}k_2^{1+\delta}\dotsb k_n^{1+\delta}}\left(\prod_{i=1}^{n}i^{i-1}\right)^{-1}\hphantom{|H_n[f]|}{}\ll n!\exp\left(-\frac12 n^2\log n+O\big(n^2\big)\right),
b\tilde M_{j+1}^4\epsilon_{j+1}^{\frac{1}{2m}}=b^5\nu_{s_{k+1}}^{-4\kappa}\tilde M_j^{4\kappa}\epsilon_j^{\frac{4^j}{2m}}< \epsilon_j^{-\frac{\kappa}{m}}(b\tilde M_j^4\epsilon_j^{\frac{1}{2m}})^\kappa\epsilon_j^{\frac{4^j-\kappa}{2m}}<1.
\chi^{ac}+2\rho_H= \sum_{\alpha\in \Phi_{Q^u}\cup\Phi_s^+}\alpha\circ\mathcal{H}+\sum_{\alpha\in \Phi_{Q^u}\cup\Phi_s^+}\alpha\circ\mathcal{P}=\sum_{\alpha\in \Phi_{Q^u}\cup\Phi_s^+}\alpha.
h_i = \frac{z^{d_i}}{(1-z)^2}+ \frac{z^{d_{1}}+ \cdots + z^{d_{i-1}}+ z^{d_{i+1}}+ \cdots + z^{d_r}}{1-z}= \frac{z^{d_1}+ \cdots + z^{d_r}- (+ z^{d_{1}+1}+ \cdots + z^{d_{i-1}+1}+ z^{d_{i+1}+1}+ \cdots + z^{d_r+1})}{(1-z)^2}.
\langle E_{0}| A_{0}^{( 1 )}\cdots A_{0}^{( n )}| E_{0}\rangle = \langle \hat{E}_{0}| \hat{A}_{0}^{( 1 )}\cdots \hat{A}_{0}^{( n )}| \hat{E}_{0}\rangle \, \langle \check{A}_{0}^{( 1 )}\cdots \check{A}_{0}^{( n )}\rangle_{G},
\sum_{w \in W}\left[x^{n-w}\right] h_{\zeta^k}(x) = \sum_{w \in W}(-1)^w \sigma_w(\zeta^k) = \sum_{w \in W}(-1)^w \mathcal{F}_{\zeta}[\delta_w](k) = \mathcal{F}_\zeta\left[ \sum_{w \in W}(-1)^w \delta_w \right](k).
\sum \frac{1}{\omega}\equiv \sum_{- \infty}^{+ \infty}\, \!^{\prime}\frac{1}{\omega_{n}}\sim \frac{L}{2 \pi}2 \int_{0}^{\infty}\frac{1}{\omega ( k )}\left[ 1 + \frac{1}{L}\delta^{\prime}( k ) \right] d k
t_{a}^{\prime}=\frac{t_{a}-\frac{vx_{a}}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}},t_{b}^{\prime}=\frac{t_{b}-\frac{vx_{b}}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
f(\theta^{*},\omega^{*},t^{*})=f(\theta(t^{*}),\omega(t^{*}),t^{*})\leq ||f_0||_{L^{\infty}(\mathbb T \times [-M, M])}e^{KT_{\kappa}}e^{-K\kappa\delta(t^{*}-T_{\kappa})}
K_{I}^{\prime}( \theta ) = \ell^{\prime}\frac{\textrm{s n}\left[ \frac{\textbf{K}}{i \pi}\left( \theta - i \frac{\pi}{2}\right) \right]}{\textrm{c n}\left[ \frac{\textbf{K}}{i \pi}\left( \theta - i \frac{\pi}{2}\right) \right] \textrm{d n}\left[ \frac{\textbf{K}}{i \pi}\left( \theta - i \frac{\pi}{2}\right) \right]}.
\frac{1 + \frac{{\rm sgn}(a)}{\Gamma \left( \theta^+_{\chi}\right) \cos \left( \theta^+_{\chi}\right)}}{1 + \frac{1}{\tan \left( \theta^+_{\chi}\right)}}=\frac{\tan \left( \theta^+_{\chi}\right)}{1 + \tan \left( \theta^+_{\chi}\right)}+\frac{\tan \left( \theta^+_{\chi}\right)}{\ln \left( -\tan \left( \theta^+_{\chi}\right) \right)}\;,
\sum_{k=0}^{\infty}{'}(bk)^\nu K_\nu (bk)=\frac{\sqrt{\pi}}{b}2^\nu \Gamma (\nu +1/2)\sum_{n=0}^{\infty}{'}\left[ \left( \frac{2\pi n}{b}\right)^2+1\right]^{-\nu -1/2}
V(\mathcal{Q}(I))= \{(0,\,0,\,1,\,1,\,1,\,1,\,1),\,(0,\,1,\,1,\,1,\,1,\,1,\,0),\,(1,\,0,\,0,\,1,\,1,\,1,\,1),\, (1,\,1,\,0,\,0,\,1,\,1,\,1),\,(1,\,1,\,1,\,0,\,0,\,1,\,1),\,(1,\,1,\,1,\,1,\,0,\,0,\,1),\, (1,\,1,\,1,\,1,\,1,\,0,\,0),\,({1}/{2},\,{1}/{2},\,{1}/{2},\,{1}/{2},\,{1}/{2},\,{1}/{2},\,{1}/{2})\},
\sum_{n=0}^\infty F_{n,r}(x)\frac{t^n}{n!}=\sum_{n=0}^\infty \sum_{k=0}^nk!x^kS_r(n+r,k+r)\frac{t^n}{n!}&=\sum_{k=0}^\infty k!x^k\sum_{n=k}^\infty S_r(n+r,k+r)\frac{t^n}{n!}&=e^{rt}\sum_{k=0}^\infty x^k(e^t-1)^k &=\frac{e^{rt}}{1-x(e^t-1)}.
S ( \{\xi \}) = \det ( \Phi_{i j}) = \prod_{i = 1}^{D}{\frac{h_{i}( \{\xi \})}{f_{i}( \xi_{i})}}\ , \M_{i}( \xi_{1}, \dots , \xi_{i - 1}, \xi_{i + 1}, \dots , \xi_{D}) ={\frac{\partial S}{\partial \Phi_{i 1}}}={\frac{S ( \{\xi \})}{h_{i}^{2}( \{\xi \})}}\enspace ,
\frac{d}{dt}[X^*(t,\omega)-\overline{X}^*(t)]=A(t)[X^*(t,\omega)-\overline{X}^*(t)]+B(t)R^{-1}(t)B^T(t)[\Upsilon(t,\omega)-\overline{\Upsilon}(t)], &\frac{d}{dt}[\Upsilon(t,\omega)-\overline{\Upsilon}(t)]= (Q_x(t)+Q_m(t))[X^*(t,\omega)-\overline{X}^*(t)]-A^T(t)[\Upsilon(t,\omega)-\overline{\Upsilon}(t)].
(1) \alpha_k(0)=0 = \gamma_{k}, (2) \displaystyle\sum_{\ell=0}^{k-1}\dfrac{\alpha_k(\ell)}{4^\ell}+\sum_{\ell=k}^{2k}\dfrac{\beta_k(\ell)}{4^\ell}+\dfrac{\gamma_k}{2^{4k+3}}=0, (3) \displaystyle\sum_{\ell=0}^{k-1}(-1)^\ell\dfrac{\alpha_k(\ell)}{4^{\ell}}+\sum_{\ell=k}^{2k}(-1)^\ell\dfrac{\beta_k(\ell)}{4^\ell}-\dfrac{\gamma_k}{2^{4k+2}}=0.
(d_{n-1},d_{n-2},\ldots,d_0) = (r_1, r_1 + r_2, \ldots, r_1+r_2 +\cdots + r_n), (d_{-n+1},d_{-n+2},\ldots,d_0) = (c_1, c_1 + c_2, \ldots, c_1+c_2 +\cdots + c_n).
(T^{\dag}_{E,F})^2T^{\sharp}=((T^{\dag}_{E,F})^2T^{\sharp})TT^{\sharp}=T^{\dag}_{E,F}T^{\sharp}T^{\dag}_{E,F}TT^{\sharp}=T^{\dag}_{E,F}(T^{\sharp})^2TT^{\dag}_{E,F}TT^{\sharp}=T^{\dag}_{E,F}(T^{\sharp})^2.
\left\Vert y\sup_{\left(y,\frac{(n+1)^2}{n^2}y\right)}\frac{{\rm d}}{{\rm d}x}\left( \left( \begin{matrix}\Psi_{12}(x) 0 \Psi_{22}(x) 0\end{matrix}\right)e^{-2x^{1/2}}\right)\right\Vert_{L^2(0,\infty)}= O(1),
\Lambda_{m a g n}^{1 0 - 2 F}\tilde{Y}^{1 1}\lambda^{5}\tilde{q}^{2 F - 2}\tilde{\bar{q}}^{2 F - 2}q^{* \; 3}\bar{q}^{* \; 3}M_{1}\frac{h}{\mu^{2}}.
w_{np}^{(j)}= \frac{1}{k_nk_p}\int_{t_{n-1}}^{t_n}\int_{t_{p-1}}^{t_p}\omega_{\alpha_j}(t-\vartheta) d\vartheta dt = \frac{\lambda_{n,p}^{(j)}-\lambda_{n-1,p}^{(j)}}{k_n k_p \Gamma(\alpha_j+2)},
M - F M F^* =\begin{bmatrix}-G_q B_q^* G_p B_p^* e_1 e_1^* 0\end{bmatrix}=\begin{bmatrix}-G_q G_p 0 0 0 e_1\end{bmatrix}\begin{bmatrix}B_q^* 0 0 B_p^* e_1^* 0\end{bmatrix}.
\begin{array}{ll}R_{\mbox{cr}}=0\, ; D=2\, , begin{align*}5mm] \displaystyle R_{\mbox{cr}}=\frac{9}{2}m_{0}^{2}\, ; D=3\, , begin{align*}5mm] R_{\mbox{cr}}=12 m_{0}^{2}\, ; D=4\, .\end{array}
\bigg \langle \sum_{j}( H_{\infty}W ( n_{j}) ) \prod_{i \not = j}W ( n_{i}) +{\frac{1}{N^{2}}}\sum_{j}\sum_{i \not = j}n_{i}n_{j}W ( n_{i}+ n_{j}) \prod_{k \not = i , j}W ( n_{k}) \bigg \rangle = 0
h(x) \le \Bigl(g_5(x)+\frac{2}{6^6}(3^6 x - 304)(27-2^6x)\Bigr)\Bigr|_{x=\frac{88}{211}}+ \frac1{10\cdot 6^6}- \eta &=-\frac{56411054147338655852222279153714125703}{6316397576863411724254503567359990235792000}<0,
P_{2(k + n-j)}(E_4,E_6) = \sum_{2a+3b = k+n-j}c_{ab}^j E_4^aE_6^b = A^{k+n-j}\left(\sum_{2a+3b = k+n-j}\frac{c_{ab}^j}{(1-K)^{a+b}}\right).
( \partial_{\bar{l}}A_{0}^{z}) ( \partial_{l}A_{0}^{\bar{z}}) - ( \partial_{\bar{l}}A_{0}^{\bar{z}}) ( \partial_{l}A_{0}^{z}) = \frac{1}{2}{\cal L}_{0}= \frac{4}{\cosh^{2}( l )}\, .
\sin^{2}\theta g_{2}((\nabla\pi_*)(U_{2},V_{2}),\pi_*Z) =g_{2}(\nabla\pi_*(U_{2},\omega\phi V_{2}),\pi_*Z)-g_{1}(\mathcal{T}_{U_{2}}\omega V_{2},\mathcal{B}Z) &+g_{2}(\nabla\pi_*(U_{2},\omega V_{2}),\pi_*(\mathcal{C}Z))-g_{1}(V_{2},\phi U_{2})\eta(Z)
{\cal N}_{M}[{\cal H}_{\vec{x}}] ={\frac{1}{2}}\partial_{t}\phi_{x}\partial_{t}\phi_{x}+{\frac{1}{2}}\partial_{\vec{x}}\phi_{\vec{x}}\partial_{\vec{x}}\phi_{\vec{x}}+{\cal N}_{M}[ V ( \phi_{\vec{x}}) ] -{\frac{1}{2}}I_{0}[ M^{2}] +{\frac{1}{4}}M^{2}I_{1}[ M^{2}] \; ,
\begin{array}{ll}2 u_{\bar{z}z}= -e^{2u}+e^{-2u}(|\xi_1|^2 - |\xi_2|^2), e^{-2u}( \xi_{1 \bar{z}}+ \xi_2 \bar{\sigma}) = ( h_z + \sigma h ), e^{-2u}( \xi_{2 \bar{z}}+ \xi_1 \bar{\sigma})= (h_z + \sigma h ), Im ( \sigma_{\bar{z}}) = e^{-2u}Im (\xi_1 \bar{\xi}_2). \end{array}
\mathcal{J}^i(\alpha^i,\boldsymbol{\alpha}^{-i}):=\frac{1}{2}\mathbb{E}\Big\{\int_0^T\Big[Q\left(x^i_t+l(\nu^{N,i}_{\boldsymbol{x}_t})\right)^2+R\left(\alpha^i_t+h(\mu^{N,i}_{\boldsymbol{\alpha}_t})\right)^2\Big]dt+G\left(x^i_T+g(\nu^{N,i}_{\boldsymbol{x}_T})\right)^2\Big\},
S^{loc}f(x)=Sf(x)-S^{glob}f(x) &=S(f\chi_{4B_j}(y_1))(x)+S(f\chi_{(4B_j)^c}(y_1))-\int_{\mathbb{R}^d}\int_Y \chi_{N_2^c}S(x,y)(x_1,y_1)f(y)\, d\mu(y_2)\,dy_1 &=S(f\chi_{4B_j}(y_1))(x)+\int_{\mathbb{R}^d}\int_Y \chi_{(4B_j)^c}(y_1)-\chi_{N_2^c}(x_1,y_1))S(x,y)f(y)\, d\mu(y_2)\,dy_1.
1 >\frac{8}{27}(\frac{1}{b_{13}+x_1}+\frac{1}{b_{12}+x_1}) &\ge \frac{8}{27}\frac{(1+1)^2}{b_{13}+b_{23}+2x_1}&=\frac{32}{27}\frac{1}{1+\frac{1}{2}x_1}\ge \frac{32}{27}\frac{1}{1+\frac{1}{11}}=\frac{88}{81},
\left|r\frac{d}{dr}\left[C(z,w)\right]\right|\lesssim \frac{1}{|1-z\overline{w}|^2}\frac{1}{|1-zw|^{2\lambda}}\asymp\frac{|1-z\overline{w}|^{-2}}{(|1-z\overline{w}|+|1-zw|)^{2\lambda}},z=re^{i\theta}.
\delta \, S'_{CS}= \frac \ell{8 \pi G}\int_{|r|=\bar r}Tr[\partial_t g_1 \, g_1^{-1}\, \delta \zeta -\partial_t g_2 \, g_2^{-1}\, \delta \tilde \zeta \, ] |_{\varphi=0}\, dt \qquad ,
\sum_{i\geq n}i t^{i-1}= \frac{nt^{n-1}-(n-1)t^n}{(1-t)^2}, \sum_{i\geq n}i(i+1) t^i = \frac{t\bigl(n(n-1)t^{n+1}-2(n^2-1)t^n+n(n+1)t^{n-1}\bigr)}{(1-t)^{3}}, \sum_{i\geq n}i^2 t^{i}= \frac{(n-1)^2t^{n+2}-(2n^2-2n-1)t^{n+1}+n^2t^n}{(1-t)^3}.
\int \, (\, \mathfrak{J}_L ((\mathfrak{J}^L f)_x ), f_x \,) \, d\mu_{\mathfrak{G}/H}= \int \, (\, \mathfrak{J}_L ((\mathfrak{J}^L f)_{h\cdot q}), f_{h\cdot q}\,) \, d\mu_{\mathfrak{G}/H}(q) =\int \, (f_q , f_q \,) \, d\mu_{\mathfrak{G}/H}(q),
\vartheta_{\sigma}^{( n )}( w | \zeta ) = \Theta_{x^{2}}\left( - \sigma \prod_{a \in A}w_{a}^{- 1}\prod_{j = 1}^{2 n}\zeta_{j}\right) \prod_{j = 1}^{2 n}z_{j}^{- n}\prod_{a , b \in A \atop a < b}w_{a}^{- 1}\Theta_{x^{2}}( w_{a}/ w_{b}) .
g^n(x+1) - \lambda_n -\frac{1}{(1+\rho_n)^2}(\eta^n(x+1)-\rho_n) = g^n(x+1)-g^{n,\ell}(x) & + g^{n,\ell}(x) - \psi^{n,\ell}(x) & + \psi^{n,\ell}(x)-\lambda_n-\frac{1}{(1+\rho_n)^2}(\eta^{n,\ell}(x)-\rho_n) & + \frac{1}{(1+\rho_n)^2}\big(\eta^{n,\ell}(x)-\eta^n(x+1)\big),
{}\begin{array}{l}\bar{g}(u) = 2\frac{v}{u}- u \ln (e^v-1) \bar{f}(u) = \frac{v}{2^{3/2}\pi u}\left ( (e^{v}-1) - \frac{1}{2}u^2 e^{v}\right)^{-1/2}\frac{v^2}{u^2}= \int_0^{v}\frac{t\, dt}{1-e^{-t}}\end{array}
\quad\mathbb{P}[\bigcap_{m=1,l=1}^{M,L}\{(U_m,V_l)\notin \mathcal{F}\}] &\le\mathbb{P}[(U,V)\notin\mathcal{F}]+\mathbb{P}[\imath_{U;V}(U;V)\ge\log ML-\gamma] &\quad+\frac{\exp(\gamma)}{\max\{M,L\}}+e^{-\frac{1}{2}\exp(\gamma)}.
\sum_{j=1}^n\left(P^i_j-P^1_j\right)\log\sum_{i'=1}^m\widetilde\lambda_{i'}P^{i'}_j+\epsilon\log\frac{1+\varepsilon\widetilde\lambda_i}{1-\varepsilon\widetilde\lambda_i}-\epsilon\log\frac{1+\varepsilon\widetilde\lambda_1}{1-\varepsilon\widetilde\lambda_1}=-H(P^i)+H(P^1),\i=2,\cdots,m.
\mathrm{d}_p\ell_t (u) =\mathrm{d}_p\ell_t (Z.p) =\tfrac{\mathrm{d}\,}{\mathrm{d}s}_0\ell_t (\mathrm{Exp}(sZ)p) &= \tfrac{\mathrm{d}\,}{\mathrm{d}s}_0 L_t(\mathrm{Exp}(sZ))p = h_t(Z).p = \mathrm{e}^{-tB}u.
\aligned{\widetilde{\mathcal T}}{}^i_{jk}=L^i_{\underline{jk}}-\omega^i_{jk}, \endaligned \displaybreak[0] \aligned{\widetilde{\mathcal W}}{}^i_{jmn}= R^i_{jmn}-\omega{}^i_{jm|n}+ \omega{}^i_{jn|m}+ \omega{}^\alpha_{jm}\omega{}^i_{\alpha n}-\omega{}^\alpha_{jn}\omega{}^i_{\alpha m}, \endaligned
\frac{d}{dt}\Bigl|_{t=0}|z|^{2(\nu-\rho')(H_0)}|x|^{2(\nu-\rho')(H_0)}= - 2(\nu-\rho')(H_0) (y,x), &\frac{d}{dt}\Bigl|_{t=0}r_z r_x= 2 (y^t x - x^t y), &\frac{d}{dt}\Bigl|_{t=0}F\Bigl(\frac{z}{|z|^2}\Bigr)= \bigl(2 (x,y)x - y|x|^2\bigr) (\nabla_{y}F)(x).
\det(M') = \mu \vert \alpha \vert (-1)^{\vert \alpha \vert -1}+ \tfrac{1}{2 \mu c_\alpha}\lambda \vert \alpha \vert (-1)^{\vert \alpha \vert -1}&= |\alpha|(-1)^{\vert \alpha \vert -1}\left( \mu + \tfrac{\lambda}{2 \mu c_\alpha}\right)
V(\mathcal{T}):=\textit{CR}^1_0(\mathcal{T}):=\{v\in P_1(\mathcal{T}):\ v \textup{mid}(F)F\in\mathcal{F}(\Omega) &v(\textup{mid}(F))=0F\in\mathcal{F}(\partial\Omega) \}.
\sum_{1\le j\le n}\int_{M_j}\int_{\partial D_j}|f(w_j,\hat w_j)|^2\rho|dw_j|d\mu_j(\hat w_j) \le 2\liminf_{r\rightarrow1-0}\frac{\int_{\{z\in M:2\psi(z)\ge\log r\}}|g|^2\hat\rho}{-\log r}= 2\liminf_{r\rightarrow1-0}\frac{\int_{\{z\in M:2\psi(z)\ge\log r\}}|g|^2\hat\rho}{1-r}< +\infty.
d e g [ ( T^{+})^{n_{+}}( T^{0})^{n_{0}}( J )^{\overline{{n}}}( T^{-})^{n_{-}}{Q_{1}}^{m_{1}}{Q_{2}}^{m_{2}}{\overline{{Q}}_{1}}^{\overline{{m}}_{1}}{\overline{{Q}}_{2}}^{\overline{{m}}_{2}}] \ =
s_h({\boldsymbol{w}}_h,{\boldsymbol{u}},{\boldsymbol{y}}_h,{\boldsymbol{v}}) := \sum_{M\in\mathcal M_h}\tau_M (\kappa_M(({\boldsymbol{w}}_M\cdot\nabla){\boldsymbol{u}}), \kappa_M(({\boldsymbol{y}}_M\cdot \nabla){\boldsymbol{v}}))_M t({\boldsymbol{u}},{\boldsymbol{v}}) := \gamma (\nabla \cdot{\boldsymbol{u}}, \nabla \cdot{\boldsymbol{v}}).
\frac{\partial}{\partial r_{k}}\equiv(\frac{\partial}{\partial x_{k}},\frac{\partial}{\partial y_{k}},\frac{\partial}{\partial z_{k}}),\frac{\partial}{\partial\dot{r}_{k}}\equiv(\frac{\partial}{\partial\dot{x}_{k}},\frac{\partial}{\partial\dot{y}_{k}},\frac{\partial}{\partial\dot{z}_{k}})
\Lambda_{0}:=\sum_{m = 0}^{s}\|\bar{z}\|_{H^{s - m}(\Omega)}^{2}+(1 + T) \sup_{0 \leq t \leq T}\big\|\bar{D}^{s - 2}\bar{f}(t, \cdot)\big\|_{L^{2}(\Omega)}+ T^{1/2}\int_{0}^{T}\|\partial_{t}^{s - 1}\bar{f}(t, \cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}t.
[(1+\mu_1)(1+\mu_2) - (1+\mu_3)]^2 = [\underbrace{\mu_2(1+\mu_1)}_{\geq 0}+ \underbrace{\mu_1 - \mu_3}_{\geq 0}]^2 \geq (\mu_1 - \mu_3)^2 \geq \mu_1^2 + \mu_3^2
\kappa{\left|\frac{\partial\zeta}{\partial s}\right|}^2+\frac{\partial\zeta}{\partial t}\frac{\partial^2\zeta}{\partial t^2}+\frac{\partial\zeta}{\partial s}\frac{\partial^2\zeta}{\partial t\partial s}-\frac{1}{2}\frac{\partial}{\partial t}|\nabla\zeta|^2=0
J(r_1K_1,\dots,r_kK_k) = \sum_{{m_1,\dots,m_k=j}\atop{m_1+\dots +m_k=(k-1)d+j}}^d r_1^{m_1}\cdots r_k^{m_k}\varphi^{(j)}_{m_1,\dots,m_k}(K_1,\dots,K_k),
\int_{t_1}^{t_2}\sum_{i=m}^n \sum_{j=1}^i(jg_{i+1}-jg_i-g_j)V_{i,j}\psi_i(s)\psi_j(s)ds \to \int_{t_1}^{t_2}\sum_{i=m}^{\infty}\sum_{j=1}^i(jg_{i+1}-jg_i-g_j)V_{i,j}\psi_i(s)\psi_j(s)ds
k^{2}( r , l , E ) = ( 1 -{\frac{2 M}{r}}+{\frac{Q^{2}}{r^{2}}})^{- 1}[ ( 1 -{\frac{2 M}{r}}+{\frac{Q^{2}}{r^{2}}})^{- 1}E^{2}-{\frac{l ( l + 1 )}{r^{2}}}- m^{2}] .
\sum_{|I|=n}|\det(\{u_i\}_{i\in I})| \sum_{|I|=n-2}|\det(u,v,(u_i)_{i\in I})| & \le \sum_{|I|=n-1}|\det(u,(u_i)_{i\in I})|\sum_{|I|=n-1}|\det(v,(u_i)_{i\in I})|.
d_{r}\omega_{h_{k}....h_{1},j}^{i}\overset{def}{=}\frac{\partial \omega_{h_{k}....h_{1},j}^{i}}{\partial x^{r}}-\Gamma_{ar}^{i}\omega_{h_{k}....h_{1},j}^{a}+\Gamma_{jr}^{a}\omega_{h_{k}....h_{1},a}^{i}
{4 M^{2}}\, \left[ D \epsilon^{\beta \gamma \kappa \lambda}\partial_{\nu}{\partial_{\beta}}T_{( \kappa \lambda )}{}^{\nu}+ 2 a \partial_{\nu}{\partial_{\beta}}T^{( \beta \gamma ) \nu}\right] = e^{2}\, \partial_{\nu}\left({\partial}^{\gamma}\lambda^{\nu}-{\partial}^{\nu}\lambda^{\gamma}\right) .
\overline{M2}_{2k}(n) - \overline{N2}_{2k}(n) =\sum_{j=1}^k (2j)! S^*(k,j)(\overline{\mu 2}_{2j}(n) - \overline{\eta 2}_{2j}(n)) &\ge\overline{\mu 2}_{2}(n) - \overline{\eta 2}_{2}(n) &> 0 ,
\bar{f}(y^n,\bar{x}) = g(y^n,\bar{x})+ \lambda^{-1}\langle \nabla_{1}d(y^n,\bar{x}),\bar{x}-y^n\rangle & \leq - [g (\bar{x},y^n)+\lambda^{-1}\langle \nabla_{1}d(y^n,\bar{x}),y^n-\bar{x}\rangle].
{\frac{1}{i \beta}}\int_{A^{\prime}}{\frac{d z}{\sin{\frac{\pi}{\beta}}z}}~{\frac{\cos \frac z 2}{\sin^{4}\frac z 2}}= -{\frac{1}{1 8 0}}( \gamma^{2}- 1 ) ( 7 \gamma^{2}+ 1 7 ) ,
_{v_{4,1}}= \frac{-\partial_{2}v_{4,c}(s,|x+y|)}{|x+y|^{2}(s-t)}\left((x+y)\cdot y\right)\left(\hat{x}\cdot(x+y)\right) &+\frac{v_{4,c}(s,|x+y|)}{(s-t)|x+y|}\left(-y\cdot \hat{x}+ \frac{\left(\hat{x}\cdot (x+y)\right)\left(y\cdot(y+x)\right)}{|x+y|^{2}}\right)
T_n^* = \sqrt{\frac{n_1n_2}{n}}\sum_{k=1}^2 \sum_{i=1}^{2n_k}\int_{t_1}^{t_2}\rho(u) w_{2n;i}^{(k)}(Z_{2n;i}^{(k)}(u) - \overline{Z}_{2n}(u)) \d u,
\left(^{U[\vec \varphi]}a_1\right)^a \cong a_1^a(x) - \epsilon^{abc}\, \varphi^b(x) \, a_1^c(x) + \partial_1 \varphi^a(x) = a_1^a(x) - D_1^{ab}[a_1] \varphi^b(x) \; ,
\mathfrak{m}_{\delta_{1}\delta_{2}\delta_{3}}^{a_{1}a_{2}a_{3}}=\left(Y(\mathcal{I}_{\delta_{1}}^{a_{1}})_{\delta_{3}\delta_{2}}^{a_{2}a_{3}}+1\right)\cdot\left(Y(\mathcal{I})_{\delta_{2}\delta_{1}}^{a_{1}a_{2}}+1\right)\cdot\left(Y(\mathcal{I})_{\delta_{3}\delta_{1}}^{a_{1}a_{3}}+1\right)
\tilde\beta_0(a,b,c)=(\sigma_2\sigma_{1,3}\sigma_2) \sigma_2^{a-1}\sigma_1^{b-1}\sigma_3^{c-1}\quad\tilde\beta(a,b,c)=(\sigma_2\sigma_{1,3}\sigma_2)^3 \sigma_2^{a-1}\sigma_1^{b+1}\sigma_3^{c+1},
M_{n}(0) \geq cR(M_{n}(0))=R(M_{1}(0)) \iota_{N(A)^\bot}^\ast M_{n}^{'}(0)\iota_{N(A)}\left(\iota_{N(A)}^\ast M_{n}^{'}(0)\iota_{N(A)}\right)^{-1}=\iota_{N(A)^\bot}^\ast M_{n}^{'}(0)^{*}\iota_{N(A)}\left(\iota_{N(A)}^\ast M_{n}^{'}(0)^{*}\iota_{N(A)}\right)^{-1},
\begin{aligned}e^{k_E (\tau_1 - t)}(\frac{1}{2}b_M k_M k_E (T - \tau_1)^2 + b_E k_E (T - \tau_1) + b_E) > b_M k_M (T - \tau_1) + b_M k_M (\tau_1 - t) + b_E. \end{aligned}
\| \varphi\|_{\mathrm{HK}}^{(m)}:= V^{(m)}(\varphi) + \sum_{i=1}^m \|\varphi_{\delta,i}\|_{\mathrm{HK}}^{(m-1)}+ \sum_{i=1}^m \| \varphi_{T,i}\|_{\mathrm{HK}}^{(m-1)},
g_\mu := \arg\max_{\psi \in \mathcal{H}}\left\{- \frac{\mathrm{d}}{\mathrm{d}\gamma}\mathrm{KL}\left(T_{\gamma}{\#}\mu \| \pi \right)\Big|_{\gamma=0}\quad\|\psi\|_{\mathcal{H}}\leq 1\right\}.
H = \frac{1}{2 M}\sum_{m = 1}^{D - 1}\sum_{i , j = 1}^{N}( p_{i j}^{m})^{2}+ M^{5}\sum_{m , n = 1}^{D - 1}\sum_{i , j = 1}^{N}| [ X^{m}, X^{n}]_{i j}|^{2}\ .
\forall\;0\le k<n,y_{k}=\sqrt[m]{\mbox{Prod}_{\mathbf{P}_{k}}\left(\mathbf{x}^{\top^{\left(m-1\right)}},\mathbf{x}^{\top^{\left(m-2\right)}},\cdots,\mathbf{x}^{\top^{j}},\cdots,\mathbf{x}^{\top^{1}},\mathbf{x}^{\top^{0}}\right)}
\mathbf{S}^\mathrm{pg}\mathbf{u}^\mathrm{pg}= \mathbf{f},\mathbf{S}^\mathrm{c}\mathbf{u}^\mathrm{c}= \mathbf{f},\mathbf{S}^\mathrm{nn}\mathbf{u}^\mathrm{nn}= \mathbf{f},
\min\{(\mu\psi)^*(x_1) + \varphi^*(x_2) + \iota_{\{b\}}^*(x_3) : B^\top x_1 + W^\top x_2+ K^\top x_3 = 0, x_1\in\mathbb{R}^{2d}, x_2\in\mathbb{R}^q, x_3\in\mathbb{R}^p\}.
\|(u_1^n, u_2^n)\|_{\dot H \times \dot H}^2 =\|(v_1^n, v_2^n)\|_{\dot H \times \dot H}^2+ \|(u_1, u_2)\|_{\dot H \times \dot H}^2+o_n(1), \|(u_1^n, u_2^n)\|_{q}^q = \|(v_1^n, v_2^n)\|_{q}^q + \|(u_1, u_2)\|_{q}^q + o_n(1) \ \ \ \ 1 \leq q < \infty.
1 \frac{1}{2}+ 2 \frac{1}{3}+ 3 \frac{1}{4}+ 4 \frac{1}{5}+ 5 \frac{1}{6}- 1 \frac{1}{2}- 1 \frac{1}{3}- 1 \frac{1}{4}- 1 \frac{1}{5}- 1 \frac{1}{6}
{\frac{\partial}{\partial u}}[ p - p_{i}p^{i}/ r_{1}^{2}- \gamma^{2}( u ) / r_{1}^{2}r_{5}^{2}] \ll [ p - p_{i}p^{i}/ r_{1}^{2}- \gamma^{2}/ r_{1}^{2}r_{5}^{2}]^{3 / 2},
\qquad\qquad\qquad\frac{(\frac{1}{6})_k(\frac{5}{6})_k}{(1)_{k}^2}=\frac{\binom{3k}{k}\binom{6k}{3k}}{432^k}, begin{align*}1mm] H_k(-\tfrac{1}{6})+H_k(-\tfrac{5}{6})=6H_{6k}-3H_{3k}-2H_{2k}+H_k,
R(z,\lambda)=\sum_{i=1}^nE_{i,i}\otimes E_{i,i}+\sum_{i\neq j}a(z,\lambda_i-\lambda_j)E_{i,i}\otimes E_{j,j}+\sum_{i\neq j}b(z,\lambda_i-\lambda_j)E_{i,j}\otimes E_{j,i}.