harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\prod_{i=0}^{r-1}\frac{\Gamma_p(\langle(\frac{1}{2}-\frac{m}{q-1})p^i\rangle)\Gamma_p(\langle(\frac{1}{2}+\frac{m}{q-1})p^i\rangle)}{\Gamma_p(\langle\frac{p^i}{2}\rangle)\Gamma_p(\langle\frac{p^i}{2}\rangle)}=\overline{\omega}^m(-1).
\P\Big(\|v_I\|_2 \leq \frac{C}{\log^c n}\frac{m}{n}I \subset [n], |I|=m \Big) \leq \P\Big(\|v_I\|_2 \leq \frac{C}{\log^c n}\frac{m}{n}I \mathcal{E}\Big) + \P(\mathcal{E}^c) &\leq \P\left(\sum_{i=1}^m Y_i^2 \leq \frac{C}{\log^c n}\frac{m^2}{n}\right) + \exp(-cn).
\bar{q}_{\beta}(n,t)=\begin{cases*}E_{\beta, 1}\left(-\lambda t^{\beta}\right),\n=0, \displaystyle\sum_{\Omega_{n}}\prod_{j=1}^{n}\frac{\left(\rho^{j}\binom{r+j-1}{j}\right)^{x_{j}}}{x_{j}!}z_{n}!\left(\frac{\lambda t^{\beta}(1-\rho)^{r}}{1-(1-\rho)^{r}}\right)^{z_{n}}E_{\beta, z_{n}\beta +1}^{z_{n}+1}\left(-\lambda t^{\beta}\right),\n\ge1.\end{cases*}
\psi_{i,h(i)}^B(\alpha_{p,q})=\begin{cases}0 , \prod_{\ell=i+1}^{h(i)}\alpha_{p,\ell}, \prod_{\ell=i+1}^{h(i)}\alpha_{p,\ell}-\prod_{\ell=i+1}^{h(i)}\alpha_{q,\ell}.\end{cases}
p_n^{(\alpha,\beta)}(x,y)=y^n \mathcal{P}_n^{(\alpha,\beta)}(x/y),\,\,p_n^{(\alpha,\beta)}(x,1)=\mathcal{P}_n^{(\alpha,\beta)}(x), \,\,p_n^{(\alpha,\beta)}(0,y)=y^n.
u(t_0 + s) = \sum_{\substack{[\mathbf n] \in \mathcal Z t_0 + s < L \cdot \mathbf n \leq t_0 + s + L_{\max}}}\Theta_{[\mathbf n], t_0 + s}\mu (s + L \cdot (\mathbf n_0 - \mathbf n)) e_{j_0}.
C_{2g}=(t_{A_1}t_{A_2}\cdots t_{A_{2g}})^{4g+2}t_{a_{g+1}}^{-1}, C_{2g-1}=(t_{A_1}t_{A_2}\cdots t_{A_{2g-1}})^{2g}t_{a_g}^{-1}t_{a_g^\prime}^{-1},
g_{i j}= \left[ \begin{array}{l l l}{{1}}&{{0}}&{{0}}\\{{0}}&{{g_{2}}}&{{0}}\\{{0}}&{{0}}&{{g_{3}}}\\ \end{array}\right] \mathrm{~ a n d ~}h_{a b}= \left[ \begin{array}{l l}{{h_{4}}}&{{0}}\\{{0}}&{{h_{5}}}\\ \end{array}\right]
- \partial_{t}\langle{\cal O}_{k}{\cal O}_{k}\rangle = \langle P{\cal O}_{k}{\cal O}_{k}\rangle = t^{-{\frac{Q}{\alpha_{+}}}- 2{\frac{\beta_{k}^{+}}{\alpha_{+}}}- 1}\ \ \Rightarrow \ \, l a n g l e{\cal O}_{k}{\cal O}_{k}\rangle ={\frac{\alpha_{+}}{Q + 2 \beta_{k}^{+}}}\, t^{-{\frac{Q}{\alpha_{+}}}- 2{\frac{\beta_{k}^{+}}{\alpha_{+}}}}\ ,
\begin{cases}\mathbb{E}(B_{\alpha,\beta}^{2n+1}(t)) =0, \noalign{\vskip4pt}\mathbb{E}(B_{\alpha,\beta}^{2n}(t)) =\frac{(2n)!}{2^{n}\Gamma(\beta n+1)}t^{n\alpha}.\end{cases}
|(f_i \pm \varphi)(x)-(f_i\pm \varphi)(u)|=\left |g_i(x)-\left(g^{+}_i(u)-\frac12\pm \frac12\right) \right|= =\begin{cases}|g_i(x)-g^{+}_i(u)|\leq d(x,u) |g_i(x)-g^{+}_i(u)+1|=|g_i(x)-g^{+}_i(v)|\leq d(x,v)=d(x,u).\end{cases}
\lim_{n\to \infty}\frac{v_n(e^{\pi H_{v_n}f(u_n)}+e^{-\pi H_{v_n}f(u_n)}-2\cos(\pi P_{v_n}f(u_n))}{\sin(\pi P_{v_n}f(u_n))}&=\lim_{n\to \infty}\frac{\sqrt{(u_n-x+\delta)^2+v_n^2}}{\sqrt{1+(u_n-x)^2/v_n^2}}\frac{e^{\pi\mathcal{H}\varphi(x)}}{\frac{1}{\sqrt{1+(u_n-x)^2/v_n^2}}}=\delta e^{\pi\mathcal{H}\varphi(x)}\neq 0.
h_{\vec{z}}^{1^{k}}(\vec{x})=\sum_{\sigma \in \mathfrak{S}_{k}}\prod_{1\le i<j \le k}\!\frac{z_{\sigma(i)}-q^2 z_{\sigma(j)}}{z_{\sigma(i)}-z_{\sigma(j)}}\,\prod_{i=1}^{k}\left(\frac{z_{\sigma(i)}}{1+z_{\sigma(i)}}\right)^{x_{i}}\, (u_{1}\otimes \cdots \otimes u_{1}).
n_N \cdot \frac{\theta_N^2}{\sigma_N}= n_N \theta_N^{q_N}\leq 3^{q_N}n_N \varepsilon_N^{q_N}\leq 3(N-1)^{p_N-a_Nq_N}= 3(N-1)^{(3p_N-1)/6},
\liminf_{x\to x_{0},x\in I}\frac{p_{\nu_{1}\boxplus\nu_{2}}(x)}{p_{\mu_{1}\boxplus\mu_{2}}(x)}=\liminf_{x\to x_{0},x\in I}\frac{1}{\beta}\int_{\mathbb{R}}\frac{1+t^{2}}{|1-tG_{\mu_{1}}(F_{\rho_{1}}(x))|^{2}}\,d\sigma(t) \ge\frac{1}{\beta}\int_{\mathbb{R}}\frac{1+t^{2}}{|1-tG_{\mu_{1}}(F_{\rho_{1}}(x_{0}))|^{2}}\,d\sigma(t)>0,
\begin{aligned}\mathbb{E}\left[Y_{i}Y_{j}\right] =\mathbb{E}\left[(\omega_{t_{i}}-\omega_{s}+\xi_{s,t})(\omega_{t_{j}}-\omega_{s}+\xi_{s,t})\right] =\mathbb{E}\left[(\omega_{t_{i}}-\omega_{s})(\omega_{t_{j}}-\omega_{s})\right]+\mathbb{E}\left[\xi_{s,t}^{2}\right] =t_{i}-s+\gamma_{H}(t-s)^{2H}\geq\gamma_{H}(t-s)^{2H},\end{aligned}
\begin{array}{l}\mbox{$\overline n_{\Omega}(\nu,R)<\infty$ (so that $\alpha_{\Omega}(\nu,R)$ is defined as in 4)) and}\mbox{$d_{\Omega}(\nu,R)=[R/m_R(\alpha_1(\nu,R),\ldots,\alpha_{\Omega}(\nu,R)):R/m_R(\alpha_1(\nu,R),\ldots,\alpha_{\Omega-1}(\nu,R))]=\infty$.}\end{array}
\alpha^{\pm \pm}( n_{1}, n_{2}) = \sqrt{\frac{1}{f^{\pm \pm}( n_{1}, n_{2})}\frac{N_{k}}{4 \hbar \tau_{2}}}\, B i g [ - i \tau \hat{A}_{1}^{\pm \pm}( n_{1}, n_{2}) + i \hat{A}_{2}^{\pm \pm}( n_{1}, n_{2}) \Big ] \ \ \ ,
4 \pi \sqrt n ( n - 2 ){\frac{| \xi |}{\sqrt \lambda}}\left({\frac{6 | \xi |}{1 + 6 | \xi |}}\right)^{1 / 2}< \left({\frac{\phi}{M_{\mathrm{p}}}}\right)^{n - 2}< 8 \pi \sqrt{2 n}{\frac{| \xi |}{\sqrt \lambda}}\ ,
U_{\phi}= \frac{1}{V_{p + 1}}{\underbrace{J_{\phi}J_{\phi}}}= \frac{T_{p}^{2}}{8}V_{p + 1}\frac{\left[ n^{2}( 3 - p ) - 2 m^{2}\right]^{2}}{\Delta_{m , n}}{\underbrace{\phi \phi}}
\vec w :=\sum_{m=0}^{\infty}\frac{\lambda_u^{m}\cdot{m}^{d_u}}{\lambda^{m}}\frac{1}{\lambda_u^{m}\cdot{m}^{d_u}}M^{m}\vec u=\sum_{m=0}^{\infty}\frac{1}{\lambda^m}M^m\vec{u}.
\log \frac{1}{\tilde A}\geq \int_X (\varepsilon \psi_\varepsilon + \varepsilon M_\varepsilon) \frac{f \omega^n}{\tilde A}=\varepsilon M_\varepsilon + \int_X \varepsilon \psi_\varepsilon \frac{f \omega^n}{\tilde A}.
K(x) := \mathbb{P}(W_n +\sigma_n + V_n - \tau_n \leq x , W_n \leq D_n , W_n+ \sigma_n - \tau_n \leq 0) = \int_{0}^{\infty}dA(t) \int_{0^-}^{t}dF_n(u) \bar{G}(u)\int_{0}^{t-u}dB(s)V(x+t-u-s).
T_{n,i}(x;q)\ =\q^{\chi(i\ge n)}\Big( x\sum_{j=0}^{\left\lceil \tfrac{n-1}{n}i\right\rceil -1}T_{n-1,j}(x;q)+\sum_{j=\left\lceil \tfrac{n-1}{n}i\right\rceil}^{2n-3}T_{n-1,j}(x;q)\Big)\,,
\mu^G_+ = (\mu^G_{B_1^+}, \dots, \mu^G_{B_{m+2}^+}) =(\mu^+_{\le m+2}, \mu^+_{2,2}, \dots, \mu^+_{m+2,2}), \mu^G_- = (\mu^G_{B_1^-}, \dots, \mu^G_{B_{n-m}^-}) =(\mu^-_{\le n-m}, \mu^-_{2,2}, \dots, \mu^-_{n-m,2}).
|X(s)|^2 =2\big((\cosh s\cos\theta_1+\sinh s\cos\theta_2)^2+(\cosh s\sin\theta_1-\sinh s\sin\theta_2)^2 &\qquad+(\sinh s\cos\theta_1+\cosh s\cos\theta_2)^2+(-\sinh s\sin\theta_1+\cosh s\sin\theta_2)^2\big) &=2\big(2(\cosh^2 s+\sinh^2 s)+4\sinh s\cosh s\cos(\theta_1+\theta_2)\big) &=4\cosh 2s+4\sinh 2s\cos(\theta_1+\theta_2).
J_{1}( 0 , 0 , k , l ; m ) = C^{( 3 )}_{3}F_{2}\left( \left. \begin{matrix}{- k ,}&{- l ,}&{- k - l - \frac{D}{2},}\\{\quad \frac{- k - l}{2},}&{\quad \frac{1 - k - l}{2}}\\ \end{matrix}\right| \frac{t}{4 m^{2}}\right)
\lbrack (K+1)d\boldsymbol{x}^{k}]\boldsymbol{I}_{ij}^{+}\boldsymbol{P}_{k}^{\pm}=\pm \lbrack 2\boldsymbol{I}_{ij}^{+}\boldsymbol{P}_{k}^{\pm}\mp \frac{1}{2}(1+d\boldsymbol{x}^{ijk})],
\begin{aligned}\varepsilon^{-2N}b(x,\xi)\# \psi (x/h^\delta,\xi/h^\delta) &=\varepsilon^{-2N}\sum_{j=0}^{K-1}\frac{1}{j!}\frac{(ih)^j}{2^j}\sigma(D_x,D_\xi;D_y,D_\eta)^j (b(x,\xi) \psi (y/h^\delta,\eta/h^\delta))|_{y=x,\eta=\xi}+\tilde r, \end{aligned}
\widehat{\mathfrak{p}}= \mathfrak{t}[[z]] \oplus \bigoplus_{\beta \in R^{+}}\mathfrak{g}_{\beta}\otimes \mathbb{C}[[z]] \oplus \bigoplus_{\alpha \in R^* \subset R^{+}}\mathfrak{g}_{-\alpha}\otimes \mathbb{C}[[z]] \oplus\bigoplus_{\delta \notin R^*}\mathfrak{g}_{-\delta}\otimes z\mathbb{C}[[z]]
\delta \Psi_{1 1}= e^{- E}\left( \begin{array}{c}{-{\frac{1}{2 l}}\operatorname{s i n}{\frac{\pi x^{1 1}}{2 l}}\epsilon_{1}}{{\frac{1}{2 l}}\operatorname{c o s}{\frac{\pi x^{1 1}}{2 l}}\epsilon_{2}}\end{array}\right)
\min_{\nu\in\Delta(S_{II})}\ \max_{\mu\in\Delta(S_{I})}U_{I}(\lambda,\mu,\nu)\,=\, U_{I}\bigl(\lambda,\mu^{*}\!,\nu^{*}\bigr)\,=\! \max_{\mu\in\Delta(S_{I})}\ \min_{\nu\in\Delta(S_{II})}U_{I}(\lambda,\mu,\nu).
\frac{z_i(\alpha_i+y)R(z_i)}{\alpha_i(z_i+y)\prod_{j=1}^{N}(z_j-\alpha_i)}=(-1)^N\frac{(z_{N+1}-\alpha_i)}{\alpha_i}\iff (\alpha_i+y) z_i R(z_i)=-(z_i+y)P(\alpha_i)
\mathbb{E}\left[ \sup_{t\in\lbrack0,T]}\left\vert \int_{0}^{t}\Pi_{\tau\varepsilon}(s)ds\right\vert^{2}+\sup_{t\in\lbrack0,T]}\left\vert\sum\limits_{i=1}^{d}\int_{0}^{t}\Lambda_{\tau\varepsilon}^{i}(s)dW_{i}(s)\right\vert^{2}\right] \leq C\varepsilon^{3}.
\lim_{\epsilon\to 0}\frac{(1+\epsilon)F(v)-F((1+\epsilon)v)}{\epsilon}= \lim_{\epsilon\to 0}\frac{\epsilon F(v)-[F(v+\epsilon v)-F(v)]}{\epsilon}= F(v)-v [\nabla F(v)]^T\succeq 2c_1\mathbf{1}
{x r (x - a) - x^2 r' \over 2 r^2 r'}+{r' y^2 - r y^2 \over 2 r^2 r'}= \left [{x - a \over r'}-{x \over r}\right ]{x \over 2 r}+{r' y^2 - r y^2 \over 2 r^2 r'}.
\phi_{\ell(x)}(t) =\exp\left\{\frac{\alpha x}{2}\left(\frac{\frac{1}{2x}e^{it}}{1-\Big(1-\frac{1}{2x}\Big)e^{it}}-1\right)\right\}=\exp\left\{{\alpha x^2}\frac{(e^{it}-1)}{2x-(2x-1)e^{it}}\right\}.
\displaystyle - \frac{1}{2 6 8 8 0}( 1 + \xi )^{2}( 8 0 7 7 0 + 1 5 5 5 5 0 \xi + 7 9 5 2 1 \xi^{2}+ 3 4 7 2 \xi^{3}) \zeta ( 3 )
\frac{2}{3}\frac{a^{2}b^{2}}{b^{2}-a^{2}}(R_{J}(a^{2},b^{2},c^{2},a^{2})-R_{J}(a^{2},b^{2},c^{2},b^{2}))=\frac{2}{3}c^{2}R_{J}(a^{2},b^{2},c^{2},c^{2}).
\tau(g)\left(\tau(h)(z, \zeta)\right) = \tau(g)\cdot (T_{h}(z), \phi_h(z)\zeta) &= (T_{g}(T_{h}(z)), \phi_g(T_{h}(z))\phi_h(z)\zeta) &=(T_{gh}(z), (T_{h}^*\phi_g\cdot\phi_h)(z)\zeta) &=(T_{gh}(z), \phi_{gh}(z)\cdot\zeta)=\tau(gh)(z, \zeta).
F_0(z)={}_6F_5\biggl(\begin{matrix}\frac12, \, \frac12, \, \frac12, \, \frac12, \, \frac12, \, \frac12 1, \, 1, \, 1, \, 1, \, 1 \end{matrix}\biggm|z\biggr)=\sum_{k=0}^\infty\frac{(\frac12)_k^6}{k!^6}z^k
D_k(\zeta)= \tau\left[\sum_{i=1}^k\frac{\|w_{i}(\zeta)\|^2}{\|w_{i+1}(\zeta)\|}\right]^{-1}\asymp\tau\left[\sum_{i=1}^ke^{\lambda^u(\delta_Q)(i-1)}\right]^{-1}\approx \tau e^{-\lambda^u(\delta_Q)k}.
\rho ( \eta ) \simeq \frac{1}{1 6 \pi^{2}\beta ( \beta - 1 ) a^{4}}\int_{\cal K}\mathrm{d}k \, k^{2}\left( \frac{\eta_{2}}{\eta_{1}}\right)^{2 \beta - 1}\left| \gamma_{2}\eta_{2}\right|^{2}\left| 1 + \frac{\gamma_{1}\eta_{1}}{1 - 2 \beta}\right|^{2}\omega ( k , \eta ) ,
+ \left[ - \frac{1}{| \mu | \varrho}K_{1}( | \mu | \varrho ) + \frac{1}{( \mu \varrho )^{2}}- \frac{\pi}{\mu^{2}}\delta^{2}( \vec{\varrho}) \right] \frac{( \vec{a}\cdot \vec{b})}{2 \pi},
\widetilde{\omega}^{- 2}= \omega^{- 2}\left[ 1 + \epsilon \eta \left( x^{i}, v \right) \right] \qquad \mathrm{~ a n d ~}\qquad \widetilde{B}= B \left( x^{i}, v \right) \left[ 1 + \epsilon \eta \left( x^{i}, v \right) \right]
\varphi ( x ) = \left( \begin{array}{c}{{\chi ( x )}}\\{{\xi ( x )}}\\ \end{array}\right) , \hspace{0 . 3 i n}\beta_{\mu}= - \left( \begin{array}{c c}{{0}}&{{\alpha_{\mu}}}\\{{\overline{{{\alpha}}}_{\mu}}}&{{0}}\\ \end{array}\right) .
0=\left\vert \frac{P}{\sqrt{c}}\right\vert^{2}-\left\vert \frac{Q}{\sqrt{c}}\right\vert^{2}=(\overline{a}b)\alpha+(a\overline{b})\overline{\alpha}+|b|^{2}|\alpha|^{2}-(\overline{d}e)\beta-(d\overline{e})\overline{\beta}-|e|^{2}|\beta|^{2}.
\varphi(e_i(z), f_j(w))=\frac{\delta_{i,j}}{q-q^{-1}}\cdot \delta\left(\frac{z}{w}\right),\ \varphi(\psi^-_i(z),\psi^+_j(w))=g_{a_{i,j}}(d^{m_{i,j}}z/w),
\{\Psi_i\}_{i\in\mathbb{N}}= \{\psi^+_i\, |\, i\in I_4\ \textup{and}\ \gamma^{+}_i>0\}\cup \{\psi^-_i\, |\, i\in I_4\ \textup{and}\ \gamma^{-}_i>0\}, \{\Psi_i\}_{i\in-\mathbb{N}}=\{\psi^+_i\, |\, i\in I_4\ \textup{and}\ \gamma^{+}_i<0\}\cup \{\psi^-_i\, |\, i\in I_4\ \textup{and}\ \gamma^{-}_i<0\},
\Upsilon(g * u) = \Upsilon(g ug^{-1}- g'g^{-1})=\Upsilon(g ug^{-1}) - \Upsilon(g'g^{-1}) &=\Omega(g) \Upsilon(u)\Omega(g)^{-1}- \Omega(g)'\Omega(g)^{-1}=\Omega(g) * \Upsilon(u).
j ( v , v^{\prime}) = \frac{\delta}{\delta v_{\mu}( x )}\mathrm{T r}\log{\cal M}|_{v_{\mu}( x ) = v , v_{\nu}( x ) = v^{\prime}}, \, \, \, \mu = 1 , 2 , 3 ; \, \, \nu = 5 , \ldots , 4 + D
g_{\mu \nu}= \left( \begin{array}{c c}{1}&{0}\\{0}&{\alpha^{2}\rho^{2}}\\ \end{array}\right) \b_{\mu \nu}={\frac{1}{{l_{B}}^{2}}}\left( \begin{array}{c c}{0}&{1}\\{- 1}&{0}\\ \end{array}\right)
\frac{\partial \sigma}{\partial u^{j}}=\begin{cases}X_{j}+ \dfrac{\partial u^{m-d}}{\partial u^{j}}X_{m-d}+ \dotsb + \dfrac{\partial u^{m-1}}{\partial u^{j}}X_{m-1}j \in \{1, \dotsc, m-d-1\}, X_{j}j \in \{m-d, \dotsc, m-1\},\end{cases}
S_2\approx \,e^{-2\alpha r_n}\sum_{k=r_n+1}^{n-1}(n-k)(1+\psi(k-r_n)) \leq(1+\psi(1))e^{-2\alpha r_n}\sum_{k=r_n+1}^{n-1}(n-k) \leq (1+\psi(1))e^{-2\alpha r_n}n^2 \leq\frac{1+\psi(1)}{(\log{n})^2}\leq\frac{1+\psi(1)}{\log{n}}
\begin{aligned}{\underline{y}_d}= \Big\lbrack \int_{\Omega}{y_d}\, \varphi_j \,d\boldsymbol{x}\Big\rbrack_{j=1,\dots,n},{\underline{u}_d}= \Big\lbrack \int_{\Omega}{u_d}\, \varphi_j \,d\boldsymbol{x}\Big\rbrack_{j=1,\dots,n}\mbox{and}\underline{f}= \Big\lbrack \int_{\Omega}f \, \varphi_j \,d\boldsymbol{x}\Big\rbrack_{j=1,\dots,n}.\end{aligned}
\tilde{J}^{\pm}(u)= \begin{pmatrix}1 0 e_{2}^{\pm}(u) \ddots \vdots \ddots * \ldots e_{n-1}^{\pm}(u) 1\end{pmatrix}\begin{pmatrix}k_{2}^{\pm}(u) 0 &\ddots & \ddots 0 k_{n}^{\pm}(u)\end{pmatrix}&\begin{pmatrix}1 f_{2}^{\pm}(u) \ldots * &\ddots & \ddots f_{n-1}^{\pm}(u) 0 1\end{pmatrix}
y_{d}[j] =\sqrt{h_{ud}[j]}x_{u}[j] \!+ \!n_{d}[j] =\alpha[i][j](\sqrt{h_{ud}[j]P_{s}[i]h_{su}[i]}x_{s}[i]\!+ \!\sqrt{h_{ud}[j]}n_{u}[i])\!+\! n_{d}[j],
D_{12}= \frac{s_2 (at_3)^{1-s_1}}{s_1+s_3-1}\int_1^{at_3}\frac{v-[v]-\frac{1}{2}}{v^{s_2+1}(at_3+v)^{s_3}}dv &+ \frac{s_2 s_3}{s_1+s_3-1}\int_1^{at_3}\frac{v-[v]-\frac{1}{2}}{v^{s_2}}\int_{at_3}^\infty \frac{1}{u^{s_1}(u+v)^{s_3+1}}dudv &=D_{121}+D_{122},
S_4\ll k\frac{B^k}{\phi(B)^k}\#\mathcal{A}(x)(\log{R})^{k-1}\mathfrak{S}_B(\mathcal{L})I_k\sum_{\substack{b\le \eta\log{x}L=an+b\notin\mathcal{L}}}\frac{\Delta_L}{\phi(\Delta_L)}\ll \eta k(\log{k})S_2.
\bullet \Vert E_3 \Vert_{L^{1}(S_2)}\leq C\Vert f \Vert_{X_1}, \textrm{for}q=n &\bullet\Vert E_3 \Vert_{L^{1}(S_2)}\leq \dfrac{C}{\vert n-q \vert^2}\Vert f \Vert_{X_1}, \textrm{for}q \neq n
< A_{2}| n > = \left( \frac{k \tau_{2}}{2 \pi^{2}| \tau |^{2}}\right)^{1 / 4}\, \frac{( - i )^{n}}{\sqrt{n !}}\, \left( \frac{i \bar{\tau}}{2 \tau_{2}}\right)^{n / 2}\, e^{\frac{k}{4 i \pi \tau}A_{2}^{2}}\, P_{n}\left( \sqrt{\frac{k}{2 i \pi \bar{\tau}}}\, A_{2}; \frac{i \bar{\tau}}{- i \tau}\right) \ \ \ .
\Vert \alpha_{2n+1}\Vert \leq\frac{2\lambda}{2n+1-2\kappa_{0}}\sum\limits_{i=0}^{2n}\Vert \alpha_{2n-i}\Vert \delta_{0}^{-i-1}, \Vert \alpha_{2n}\Vert \leq\frac{\lambda}{n}\sum\limits_{i=0}^{2n-1}\Vert \alpha_{2n-1-i}\Vert \delta_{0}^{-i-1}.
\left( \hat{A}_{\mu}^{a}\hat{A}_{\nu}^{b}\right) \left( \hat{A}^{b \mu}\hat{A}^{a \nu}\right) - \left( \left( \hat{A}_{\mu}^{a}\hat{A}_{\nu}^{b}\right) \hat{A}^{b \mu}\right) \hat{A}^{a \nu}= 0 .
\varphi \big{|}_S (\tau, u, v) = \frac{1}{\tau}e^{-\frac{2 \pi i m}{\tau}(u^2 - v^2)}\varphi \left( -\frac{1}{\tau}, \frac{u}{\tau}, \frac{v}{\tau}\right), \mbox{where}S = \left( \begin{matrix}0 -1 1 0 \end{matrix}\right).
\inf_{\scriptstyle u\in{\rm Dom}(Q_{m,\Omega}^N)\atop \scriptstyle u\neq 0}\frac{Q_{m,\Omega}^N[u]}{\int\limits_{\Omega}^{\vphantom{.}}|x|^{-2m}|u|^2\,dx}={\cal H}_m~\!,
\displaystyle \min_{z_0,\dots,z_T}\frac{1}{2}\left(\|z_0 - b\|^2_{(B^{\infty})^{-1}}+\sum_{t=1}^{T}\|z_t - \mathcal{M}_t(z_{t-1})\|^2_{Q_t^{-1}}+\sum_{t=0}^{T}\|y_t - \mathcal{H}_t(z_{t})\|^2_{R_t^{-1}}\right).
\bar{\cal K}= \left( \begin{array}{c c}{{- \frac{d^{2}}{d x^{2}}+ 4 - \frac{6}{\cosh^{2}x}}}&{{0}}\\{{0}}&{{- \frac{d^{2}}{d x^{2}}+ \sigma^{2}- \frac{2}{\cosh^{2}x}}}\\ \end{array}\right)
\Delta n\sum_{l=1}^\infty (2l+1)A_l=\left .\Delta n \lim_{\varepsilon \to 0}\left( \overline{\cal D}_{rr}-2\overline{\cal D}_{r\rho}+\overline{\cal D}_{\rho \rho}\right)\right |_{r=1+{\varepsilon}/{2}\atop \rho=1-{\varepsilon}/{2}}+1\,{,}
\mu[A_1, \dots, A_m](z) = \prod_{i=1}^{m}(1-\partial_i)\det(zI_d+\sum_{i=1}^{m}z_iA_i )\big|_{z_1=\dots=z_m=0}&= \prod_{i=1}^{m}(1-\partial_i)\det(\sum_{i=1}^{m}w_iA_i)\big|_{w_1=\dots=w_m=z}&= \prod_{i=1}^{m}(1-\partial_i) p(z,z, \dots, z),
\mu_x\big(\overline{(\mu_x)\cap W_S(x,G)}\big) \geq\limsup_{n\rightarrow\infty}\mu_x^n\big(\overline{(\mu_x)\cap W_S(x,G)}\big) &\geq\limsup_{n\rightarrow\infty}\mu_x^n\big((\mu_x^n)\cap W_S(x,G)\big) &= 1
\begin{pmatrix}0 a 0 0 0 b 0 0 0\end{pmatrix}=\begin{pmatrix}0 1 0 0 0 b 0 0 0\end{pmatrix}\begin{pmatrix}1 0 0 0 a 0 0 0 1\end{pmatrix}\begin{pmatrix}0 1 0 0 0 ab 0 0 0\end{pmatrix}=\begin{pmatrix}1 0 0 0 a 0 0 0 1\end{pmatrix}\begin{pmatrix}0 1 0 0 0 b 0 0 0\end{pmatrix}
\begin{aligned}2(\phi(\zeta_c^n-\zeta_c^{n-1}),\zeta_c^n)_{\mathcal{T}_h}\geq(\phi\zeta_c^n,\zeta_c^n)_{\mathcal{T}_h}-(\phi\zeta_c^{n-1},\zeta_c^{n-1})_{\mathcal{T}_h}. \end{aligned}
f(x_1,x_2,x_3,x_4,x_5,x_6) = (x_1,x_2,-x_3,-x_4,-x_5,-x_6), g(x_1,x_2,x_3,x_4,x_5,x_6) = f(x_1+\frac12,x_2,x_3,x_4,-x_5,-x_6) .
f(x) (xf'(x)+x^2f''(x)) =\left(\sum_{i=k}^l \frac{x^i}{i!}\right)\left(\sum_{i=k}^l \frac{x^i}{(i-1)!}+\frac{x^i}{(i-2)!}\right)= \left(\sum_{i=k}^l \frac{x^i}{i!}\right)\left(\sum_{i=k}^l \frac{ix^i}{(i-1)!}\right) \geq \left(\sum_{i=k}^l \frac{x^i}{(i-1)!}\right)^2=(x f'(x))^2.
Q_k=D^q_k\sum_{i=1}^{M}\frac{C^q_i}{\beta_{i,q}(\beta_{i,q}+\gamma_{k,q})}-\frac{D^q_k}{\gamma_{k,q}}, hat{P}_k=-\sum_{i=1}^{M}\frac{C^q_i}{\beta_{i,q}}\frac{D^q_k e^{\beta_{i,q}(b-y)}}{\beta_{i,q}+\gamma_{k,q}}.
\sigma_{c}^{\mathrm{n p}}= - \frac{2 N_{f}}{\pi^{2}}\int_{0}^{\Lambda_{\mathrm{n p}}}d p \frac{p^{2}\Sigma ( p )}{p^{2}+ \Sigma^{2}( p )}= \frac{N_{f}^{2}}{4}\Lambda_{\mathrm{n p}}^{2}\frac{d \Sigma}{d p}\bigg |_{p = \Lambda_{\mathrm{n p}}}.
\int_X -v d^cu \wedge d \Omega \wedge \Omega^{n-2}\leq B_\Omega C\left (\begin{aligned}\int_X -v du \wedge d^c u \wedge \omega \wedge \Omega^{n-2}+ + \int_X-v \omega^2 \wedge \Omega^{n-2}\end{aligned}\right)
\langle \prod_{i = 1}^{k}{\cal O}_{W_{i}}\rangle = \sum_{n = 0}^{\infty}\sum_{j | n}\sum_{m = 1}^{m_{n , j}}e^{- n t}\int_{\tilde{\cal M}_{0 , n , j , m}^{M}}\chi ( \nu_{j , m})
\frac{d}{dt}\frac{\delta \bar{L}}{\delta X}=\frac{\delta\bar{L}}{\delta\varphi}\circ \varphi^{-1}-\mathcal{L}_{X}\frac{\delta \bar{L}}{\delta X}-\left( div_{d^{n}q}X\right) \frac{\delta \bar{L}}{\delta X}.
\frac{d}{dx}\left(\frac{(x+1)^3}{(\log(x+1))^{3\beta-3}}g(x)\right) = \frac{(x+1)^2g(x)}{(\log(x+1))^{2\beta-2}}\left(\alpha\beta+\frac{3}{(\log(x+1))^{\beta-1}}-\frac{3(\beta-1)}{(\log(x+1))^\beta}\right)
\int_{\mathbb{S}^n}u \mathrm{d}A = -\int_{\mathbb{S}^n}\dfrac{n-j-1}{2}u^2 \mathbb{d}A - \int_{\mathbb{S}^n}\dfrac{j+1}{2n}|\nabla u|^2 \mathrm{d}A + O(\varepsilon) \|u\|_{L^2(\mathbb{S}^n)}^2 + O(\varepsilon) \|\nabla u\|_{L^2(\mathbb{S}^n)}^2.
\nu W_{l_i}W_{l_j}\sum_{e\in E^+}\Big(|A_i(e)||A_j(e)|+|B_i(e)||B_j(e)|-|A_i(e)||B_j(e)|-|B_i(e)||A_j(e)|\Big) &= \nu W_{l_i}W_{l_j}\sum_{e\in E^+}\Big((|A_i(e)|-|B_i(e)|)(|A_j(e)|-|B_j(e)|)\Big)=\nu W_{l_i}W_{l_j}\sum_{e\in E^+}t_i(e)t_j(e),
\varphi_{a\to b}^t(z)=\begin{cases}\tilde{h}(z)\cdot h_b\circ A^t_{a\to b}(z), \mbox{if $z\in Q$ and $T^t_f\circ \kappa_a(z)\in \kappa_b(R)$;}0, \mbox{otherwise.}\end{cases}
\lambda_{112}\lambda_{qrr}[a_{(1\,r)},a_{(2\,q)}]_c= -3\xi^2\lambda_{112}\lambda_{122}\lambda_{qrr}\chi_2(g_{(3\,r)})a_{(3\,r)}a_{(3\,q)}&+\chi_2(g_{(1\,r)})sa_{(2\,p)}a_{(1\,p)}g_{qqr}.
\gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}[=\sqrt{\frac{c^{2}}{c^{2}-v^{2}}}=\frac{c}{\sqrt{c^{2}-v^{2}}}]=\frac{1}{\sqrt{1-\beta^{2}}}=\frac{dt}{d\tau},
\sum_{A=1}^N\left(Tw_A,Tw_A \right)_W= \sum_{A=1}^N\int_M h_V\left(ev(Tw_A),ev(Tw_A) \right)dv = \int_M \sum_{A=1}^N h_V\left(ev(Tw_A),ev(Tw_A) \right)dv,
\int_{0}^{\infty}\frac{d t}{t^{1 + s}}\left[ \alpha \frac{C^{2}}{t^{2}}+ \beta \frac{C}{t}\right] e^{- C / t}= s ( \alpha s + \alpha + \beta ) \int_{0}^{\infty}\frac{d t}{t^{1 + s}}e^{- C / t}
2i\chi_D(\xi_n)= \frac{\big(1-\alpha_N^{0}(\xi_n)\big)\big(1-\beta_N^{0}(\xi_n)\big)}{1-\alpha_N^{0}(\xi_n)\beta_N^{0}(\xi_n)}2i\sin \theta_N(\xi_n)+ &+ i (-1)^{n+1}\big(\hat\varphi_1(-n)+\hat\varphi_2(n)\big)+ &+ \frac{2(-1)^{n+1}}{(2i\pi\xi_n)^{N+1}}\int_0^1\varphi_1r_{N+1}\,dt + \frac{\ell^2_n}{n^{N+1}}.
0 =L(e_{ii})e_{kl}-e_{kl}L(e_{ii})+e_{ii}L(e_{kl})-L(e_{kl})e_{ii}+L(e_{ki}) &=\left(C_{kk}^{ii}-C_{ii}^{ii}\right)e_{ki}+\left(C_{ik}^{ii}+C_{ik}^{kk}\right)e_{ii}.
|\Delta_{\frac{-1}{2\pi R}}^1 \nabla_{u_0}g_a(A_\theta^t(y))| \leq \begin{cases}M_6 a^{-2}R^{-1}M_7 + M_8 a^{-1}R^{-1}| (f\circ \theta^H )'(a^{-1}t)|) \end{cases}
\lambda_i(t) =|t|-\frac{x_i^k}{\|\vec x^k\|_\infty}t\ge 0, \,\,\,\forall\, i\in\{1,\ldots,n\}, \|\vec s\|_1-r^k =\|\vec s\|_1-\frac{(\vec x^k, \vec s)}{\|\vec x^k\|_\infty}= \sum_{i=1}^n \lambda_i(s_i),
I_{\pm\infty}\!\mkern2mu :=\mkern2mu\left\{i: [[y_{i-}^{\textsc{g}}, y_{i+}^{\textsc{g}}]]=\left<-\infty, +\infty\right>\right\}, I_{\cap}\mkern1mu :=\mkern2mu\left\{i: [[y_{i-}^{\textsc{g}}, y_{i+}^{\textsc{g}}]]=[y_{i-}^{\textsc{g}}, +\infty]\cap[-\infty, y_{i+}^{\textsc{g}}]\right\}.
\int_{G^{4s+1}}\left(\sum_{\gamma\in{\widehat G}, \gamma\ne{\widehat e}}\,\frac{|{\widehat f}(\gamma)|^2}{\displaystyle\sum_{j=1}^{4s+1}|{\widehat \lambda_{b_j,d_{b_j},n}}(\gamma)|^2}\right)\,\prod_{j=1}^md\mu_G(b_j)\le \frac{M}{2^{6s+1}\pi^{4s+1}}\sum_{\gamma\in{\widehat G}}|{\widehat f}(\gamma)|^2,
\left[ F,\pi_{a_{2k+1}}\right] =\left[ F,\nu_{a_{2k+1}}^{a_{2j}}\right]\gamma_{a_{2j}}+\left[ F,\gamma_{a_{2j}}\right] \nu_{a_{2k+1}}^{a_{2j}}\approx 0,\;k=0,\cdots ,\Lambda ,
S_{W Z}^{N K K}= \mu_{6}\int_{R^{6 + 1}}\mathrm{T r}\left( i_{\hat{k}}{\hat{N}}^{( 8 )}+ l_{p}^{2}i_{\hat{k}}{\hat{\tilde{C}}}\wedge{\hat{\cal F}}+ l_{p}^{4}{\hat{C}}_{{\hat{\mu}}{\hat{\nu}}{\hat{\rho}}}D{\hat{X}}^{\hat{\mu}}D{\hat{X}}^{\hat{\nu}}D{\hat{X}}^{\hat{\rho}}\wedge{\hat{\cal F}}\wedge{\hat{\cal F}}+ \dots \right) \, .
P(Y_j=y_j) = \sum_{\boldsymbol{y}_{-j}}P(\boldsymbol{Y}=\boldsymbol{y}), & = \sum_{n \geq y_j}P(\vert \boldsymbol{Y}\vert = n) \sum_{\boldsymbol{y}_{-j}}P_{\vert \boldsymbol{Y}\vert = n}(\boldsymbol{Y}=\boldsymbol{y}), P(Y_j=y_j) = \sum_{n \geq y_j}P(\vert \boldsymbol{Y}\vert = n) P_{\vert \boldsymbol{Y}\vert = n}(Y_j =y_j),
Q(X,\tau) ={\rm Ind}((D\otimes[\triangle(X)\otimes \Theta_1(T_{C}X)+2^k\Theta_2(T_{C}X)+2^k\Theta_3(T_{C}X)])_+) &=\int_X\widehat{A}(TX)[{\rm ch}(\triangle(X)){\rm ch}(\Theta_1(T_{C}X))+2^k{\rm ch}(\Theta_2(T_{C}X))+2^k{\rm ch}(\Theta_3(T_{C}X))].
\begin{aligned}\tau^s \sum_{j\le j_0}2^{j(\bar{s}+s\alpha)}\psi_j \Vert \nabla \dot S_{j-1}u(\tau)\Vert_{L^{\infty}}X_j(\tau) \le C Z_{s,\bar{s}}(\tau) \Big( \sup_{j\le j_0}\|\nabla \dot S_{j-1}u(\tau)\|_{L^\infty}\psi_j \Big).\end{aligned}
a ~\stackrel{(\ref{2.17})}{=}a(1-t)z^{-1}a=a(1-t)e^{-1}(ez^{-1})a ~\stackrel{(\ref{2.20})}{=}a(1-t)e^{-1}[ea(1-t)z^{-2}+(z^{\ast})^{-2}et]a ~=a^2(1-t)z^{-2}a\in a^2R.
\rho_{1}^{*}\psi_{1}^{*}(i)=\biggl[\sum_{j\in S}\psi_{1}^{*}(j)\pi_{ij}(v_1^{*}(i),\tilde{v}_{2}(i))+c_1(i,v_1^{*}(i),\tilde{v}_{2}(i))\psi_{1}^{*}(i)\biggr],
\int_{0}^{t}(t-s)^{\beta - \beta/\alpha - 1}s^{\beta - \beta/\alpha}ds =\int_{0}^{1}t^{\beta - \beta/\alpha - 1}(1-z)^{\beta - \beta/\alpha - 1}z^{\beta - \beta/\alpha}t^{\beta - \beta/\alpha + 1}dz =t^{2\beta - 2\beta/\alpha}B(\beta - \beta/\alpha + 1, \beta - \beta/\alpha).
\lim_{h\to 0}II_h = \int_0^T \iint\limits_{\mathbb{R}^N\times\mathbb{R}^N}\frac{H\left(x,y,\frac{(u_1+u_2)(x,t)}{2}-\frac{(u_1+u_2)(y,t)}{2}\right)}{|x-y|^N}\,dx\,dy\,dt.
\lambda^{2}(-\frac{1}{2}\frac{\partial^{2}}{\partial X^{2}}+ \frac{iQ_{B}}{2}\frac{\partial}{\partial Y}+ \frac{1}{2}\frac{\partial^{2}}{\partial Y^{2}}+e^{\sqrt{\frac{2}{|\kappa |}}(X+Y)}){\cal T}=\frac{N}{24(x^{+})^{2}}{\cal T}
\left\{j_{\alpha}^{0}\left( t , \sigma \right) , j_{m}^{0}\left( t , \sigma^{\prime}\right) \right\}\approx \left( e m b^{*}S_{\alpha m}\left( \Delta \right) \right)_{r}\cdot \Lambda^{r}\cdot \delta \left( \sigma - \sigma^{\prime}\right) \quad ,
\mathbf{x}^{k+1}= (\mathbf{I}- \frac{\xi \rho}{2}\mathbf{E}_{}^T\mathbf{E}_{})\mathbf{x}^k - \xi \nabla f(\mathbf{x}^{k+1}) - \sum_{t=0}^k (\frac{\rho \eta \xi}{2})\mathbf{E}_{}^T\mathbf{E}_{}\mathbf{x}^t.
\log L(s, \rho) = \log \left(\prod_{v}\det(1-\rho(\mathrm{Frob}_{v}) \mathrm{N}(v)^{-s})^{-1}\right) & = \sum_{\ell = 1}^{\infty}\frac{1}{\ell}\left(\sum_{\deg(v) \mid \ell}\deg(v) \mathrm{tr}\left(\rho(\mathrm{Frob}_{v})^{\frac{\ell}{\deg(v)}}\right) \right) q^{-\ell s}.
Q^{(k-1)}= (2 - Q^{(k)}) \left\{1 - \frac{2 (a_k+b_k - 1) Q^{(k)}\left[r Q^{(k)}_r - 2 a_k + (a_k+b_k) Q^{(k)}\right]}{\left[r Q^{(k)}_r - 2 a_k + (a_k+b_k) Q^{(k)}\right]^2 +r^2 Q^{(k)2}(2 - Q^{(k)})^2}\right\},
T(z) = r^{\frac{\sigma_{3}}{4}}G_{0}(rz;r\vec{x},\vec{s})(rz)^{\frac{\alpha}{2}\sigma_{3}}\begin{pmatrix}1 s_{1}h(rz) 0 1\end{pmatrix}e^{-\sqrt{r}g(z)\sigma_{3}}.
\quad\:\:\sum_{k=0}^{\infty}\frac{\binom{2k}{k}^5}{(-2^{12})^k}\Big\{(20k^2+8k+1)\big[8H_{2k}^{(2)}-3H_{k}^{(2)}\big]+4\Big\}=\frac{8}{3}, begin{align*}1mm] \sum_{k=0}^{\infty}\frac{\binom{2k}{k}^5}{(-2^{20})^k}\Big\{(820k^2+180k+13)\big[11H_{2k}^{(2)}-3H_{k}^{(2)}\big]+43\Big\}=\frac{128}{3}.