harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f^{(\pm)}(\nu;k;x)=e^{\mp ikx}+e^{\mp ikx}\sum_{n=0}^{\infty}(\frac{1}{\pi})^{n+1}\int_{1}^{\infty}d\alpha_o ...\int_{1}^{\infty}d\alpha_n\frac{(\prod_{j=0}^{n-1}D(\nu;\alpha_j)e^{-2\alpha_j x})}{[\prod_{j=0}^{n-1}(\alpha_j+\alpha_{j+1})](\alpha_o\pm ik)}
{\cal L}={\cal L}_{0}( C_{\mu \nu \alpha}) +{\cal L}_{0}( B_{\mu \nu}) - 2 \sqrt{- 3 \Omega ( d - 3 )}C_{\mu \nu \alpha}D^{\mu}B^{\nu \alpha}- \frac{\Omega ( d - 3 )}{2}C^{\mu \nu \alpha}C_{\mu \nu \alpha}
\int_\Omega |u_1-u_2|dx \leq \int_\Omega(f_1-f_2)\varphi_0(u_1-u_2) dx + \int_\Omega (-g\circ u_1+g\circ u_2)sign(u_1-u_2)\varphi_0 dx & -\int_{\partial\Omega}\frac{\partial \varphi}{\partial{\bf{n}}_{L^{*}}}d|\eta_1-\eta_2|
\begin{array}{l c r}{{\eta_{j}^{+}= u_{\alpha}^{+}\eta_{j}^{\alpha}; \qquad \overline{{{\eta}}}_{j}^{+}= u_{\alpha}^{+}\overline{{{\eta}}}_{j}^{\alpha}}}\\{{\eta_{j}^{\alpha}\overline{{{\eta}}}_{\alpha j}= 1 ; \qquad \eta_{j}^{\alpha}{\eta}_{\alpha j}= \overline{{{\eta}}}_{j}^{\alpha}\overline{{{\eta}}}_{\alpha j}= 0}}\\ \end{array}
\theta_{\bot \parallel}= \frac{1}{y^{2}}\left( \partial_{\tau}\vec{x}\partial_{\sigma}\vec{x}+ \partial_{\tau}y \partial_{\sigma}y \right) = \frac{1}{a^{2}\tau}\left[ \vec{f}{\vec{c}}^{\prime}+ a a^{\prime}\right] + \frac{1}{a^{2}}\left( \vec{g}{\vec{c}}^{\prime}\right) + \ldots
\left( \begin{array}{c}\{M^f_{Aq}\}^{Y^1q}\{M^f_{Aq}\}^{Y^2q}\end{array}\right)=\left( \begin{array}{cc}0 i -i 0 \end{array}\right) \left( \begin{array}{c}\{M^b_{Aq}\}^{Y^1q}\{M^b_{Aq}\}^{Y^2q}\end{array}\right).
\begin{array}{rcl}G_{1i}= QL_iQ^\mathrm{T}/2^{n-s}= [P_1,\!P_2,\!\cdots,\!P_{2^{n-s}}]W_{[2^s,2^{n-s}]}TQ^\mathrm{T}/2^{n-s}= [P_1, P_2, \cdots, P_{2^{n-s}}](\textbf{1}_{2^{n-s}}\otimes I_{2^{s}})/{2^{n-s}},\end{array}
\Delta ( z^{2}){\cal F}( z ) = \mathrm{c o n s t .}\int_{0}^{\infty}\prod_{i}d y_{i}~ y_{i}^{n - 1}\prod_{i > j}( y_{i}- y_{j}) ~ e^{- \alpha \sum_{i}z_{i}^{2}/ 4 y_{i}- \beta \sum_{i}y_{i}- \gamma \sum_{i}\ln y_{i}}.
I(\tau) \; = \; \exp \left( \sum_{i,j=1}^M V_n(w_i - z_j) - \frac{1}{2}\sum_{i \neq j}^M V_n(w_i - w_j)- \frac{1}{2}\sum_{i \neq j}^M V_n(z_i - z_j) \right) \; .
\sum_{n_{j+1}=0}^\infty \dots\sum_{n_k=0}^\infty\left(1-z^{l+n_1+n_2+\dots+n_k-k}\right)\P(Y_1=n_1,\dots,Y_k=n_k) &=\P(Y_1=n_1,\dots,Y_{j}=n_j)-\sum_{n_{j+1}=0}^\infty\dots\sum_{n_k=0}^\infty z^{j+n_{j+1}+\dots+n_k-k}\P(Y_1=n_1,\dots,Y_k=n_k) &=:f_k(1, n_1, \dots, n_j)-f_k(z, n_1, \dots, n_j),
\begin{array}{r l}{{( b , \lambda , p_{2}) \longleftrightarrow ( c , \mu , p_{3}) ;}}&{{P \leftrightarrow R , s \leftrightarrow u \, ;}}\\{{( b , \lambda , p_{2}) \longleftrightarrow ( d , \nu , p_{4}) ;}}&{{P \leftrightarrow Q , s \leftrightarrow t \, .}}\\ \end{array}
H ( \eta ) ={\bf p}( T_{\eta}) ={\frac{1}{2}}L ( L - Q_{1}^{( 1 )}- 1 ) + \sum_{a = 1}^{n - 1}\sum_{j = 1}^{\ell}\sum_{\mu = 1}^{s_{j}^{( a )}}I_{j \mu}^{( a )}
f(x_{i})=\alpha (\theta_{i}+\theta_{3})(a+bx_{i})\eta^{\alpha-1}(x_{i})e^{-\eta^{\alpha}(x_{i})}\left( 1-e^{-\eta^{\alpha}(x_{i})}\right)^{\theta_{i}+\theta_{3}-1},\ \i=1,2.
\frac{\tilde \mu}{2}\|x^*-v_N\|^2\leq \frac{\tilde\mu}{2}\|x^*-v_0\|^2+\frac{\rho M^2N(N+3)}{4}+\frac{NrM^2}{2}+\sum^N_{i=1}\frac{\delta_i}{a_i}&+2\sum_{i=1}^N\frac{\varepsilon_i}{a_i^2}+\sqrt{2\tilde\mu}\sum_{i=1}^N \|x^*-v_{i}\|\cdot\sqrt{\frac{\delta_i}{a_i}}.
u_{\theta}(r,\theta)=\frac{2}{3}(\frac{R^{3}}{2r^{3}}+1)B_{1}V_{1}(cos\theta)+\sum_{n=2}^{\infty}\frac{1}{2}(n\frac{R^{n+2}}{r^{n+2}}+(2-n)\frac{R^{n}}{r^{n}})B_{n}V_{n}(cos\theta).
\sum_{m,r,s,j,k \atop 2m+2j+r+s=p}a_{kmrsj}Q^m\phi_k^{r,s,j}+ \sum_{m,r,s,j \atop 2m+2j+r+s+2=p}b_{mrsj}Q^m\widetilde\phi^{r,s,j}=0
p^{2}=\frac{M_{o}^{4}+M_{d}^{4}+M_{p}^{4}-2M_{o}^{2}M_{d}^{2}-2M_{o}^{2}M_{p}^{2}-2M_{d}^{2}M_{p}^{2}}{4M_{o}^{2}}c^{2}
=i_{0}e^{-\frac{R}{L}t}+\frac{E}{2jL}L^{-1}[\frac{1}{s+\frac{R}{L}}(\frac{1}{\frac{R}{L}-j\omega}-\frac{1}{\frac{R}{L}+j\omega})+\frac{1}{s-j\omega}\frac{1}{\frac{R}{L}+j\omega}-\frac{1}{s+j\omega}\frac{1}{\frac{R}{L}-j\omega}]
L_{d}G\left(p\|q\right)=\sum_{i}\frac{T_{i}}{a-b}\left[\left(\frac{T_{i}}{p_{i}}\right)^{a-1}-\left(\frac{T_{i}}{p_{i}}\right)^{b-1}\right]+\left(1-\alpha\right)\left(p_{i}-q_{i}\right)
({3}/{4}){\textstyle\sum_{j=k+m+1}^{\nu+m+1}}\,\Xi_j\le({1}/{4}){\textstyle\sum_{j=k}^{k+m}}\,\Xi_j+ba^{-1}{\textstyle\sum_{j=k}^{k+m}}\varphi\big(\Phi(x^{\ell(j)})\!-\!\Phi^*\big).
F_{>s}(x):= \begin{dcases*}\exp\bigg\{\sum_{k>s}c_kx^k\bigg\}, $\mathsf{F}^{(n)}=\mathsf{S}^{(n)}$ \exp\bigg\{\sum_{j\ge 1}\sum_{k>s}c_kx^{jk}/j\bigg\}, $\mathsf{F}^{(n)}=\mathsf{G}^{(n)}$\end{dcases*}.
g\left(\sum^{j-1}_{i=1}v_i+\sum^{j}_{i=1}\frac{1}{2^{i+1}}d_1\right)=g\left(\sum^{j}_{i=1}v_i+\sum^{j}_{i=1}\frac{1}{2^{i+1}}d_1\right)=1+2^{-j},~j=1,\dots,l,
\begin{cases}-\varepsilon^2(K_H(x)\nabla u)= \left(u-\left( \frac{\alpha|x|^2}{2}+\beta\right)\ln\frac{1}{\varepsilon}\right)^{p}_+,\ \ x\in B_{R^*}(0), u=0,\ \ x\in\partial B_{R^*}(0),\end{cases}
\sum_{r=1}^{\lfloor c_s n d^{-\frac{s}{s-2}}\rfloor}\binom{n}{r}(erp)^{\frac{rs}{2}}\leq \sum_{r=1}^{\lfloor c_s n d^{-\frac{s}{s-2}}\rfloor}\left(\frac{en}{r}\cdot(erp)^{\frac{s}{2}}\right)^r= \sum_{r=1}^{\lfloor c_s n d^{-\frac{s}{s-2}}\rfloor}(e^{\frac{s}{2}+1}r^{\frac{s-2}{2}}np^{\frac{s}{2}})^r.
\sum_{a=1}^n\langle\phi,\omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi\rangle =\sum_{a=1}^n g_Q(\nabla_{\bar V_a}H^{0,1}, V_a) |V_a\lrcorner\,\phi|^2.
F\;= g(g\varphi - 3g_{\xi})\bigg(-U_{\xi^3}+2\varphi U_{\xi^2}-3W_{\xi^2}+\varphi W_{\xi}+\varphi_{\xi}W \qquad\qquad\qquad+U_{\xi}(2\lambda-\varphi^2+3\varphi_{\xi})+U(2\lambda_{\xi}-2\varphi\varphi_{\xi}+\varphi_{\xi^2})\bigg).
\lambda_{0}^{\prime}= \lambda_{0}+ \sum_{i = 1}^{N}( \alpha_{i}- 1 ) \log | \omega_{i}c + d | , \quad \displaystyle \omega_{i}^{\prime}= \frac{a \omega_{i}+ b}{c \omega_{i}+ d}, \qquad \alpha_{i}^{\prime}= \alpha_{i}\, .
\mathcal{A}^{\hat{\varphi}}_{(\alpha_0,\alpha_1)}(u,\eta) =\int_{S^1}\lambda(\dot{u}(t))\,dt-\int_{\frac12}^{1}L_t(u(t))\,dt &=\eta+\int_{S^1}\left(\lambda(X_L(u))-L_t(u)\right)\,dt &=\eta
\mathrm{Im}Z([0,x]) =\sum_\alpha \alpha X([a_-(\alpha),x_\alpha]) &>\sum_\beta \beta X([a_-(\beta+\delta),x_{\beta+\delta}]) &=\sum_\beta \beta X([a_-(\beta+\delta)\wedge b_+(\beta),b_+(\beta)]) &=\sum_\beta \beta (X([b_-(\beta),b_+(\beta)])-X([b_-(\beta),y_\beta])) &=0-\mathrm{Im}Z([0,y])
B_{n m}= \left( \begin{array}{c c}{0}&{- 1}\\{1}&{h}\\ \end{array}\right) , \qquad B^{n m}= \left( \begin{array}{c c}{h}&{1}\\{- 1}&{0}\\ \end{array}\right) , \quad B^{n m}B_{n m}= - 2 .
S_{e f f}= \int \! d^{4}x ~ [ - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}+ \frac{1}{2}m^{2}( A_{\mu}+ \partial_{\mu}\rho )^{2}- A^{\mu}\partial_{\mu}B + \frac{1}{2}\alpha ( B )^{2}+ \dot{{\overline{{{\cal P}}}}}\frac{1}{\partial_{i}\partial^{i}}\dot{{\cal P}}-{\overline{{{\cal P}}}}{\cal P}]
-\int\limits_X \xi \wedge \partial\overline{\rho_\omega^{2,\,0}}\wedge \omega_{n-2}= \int\limits_X \xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \partial \omega_{n-2}- \int\limits_X \partial\xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \omega_{n-2}.
\frac{(\nu;q)_y}{(\mu;q)_y}\, \sum_{r=0}^{y}(-\mu)^r q^{mr}q^{\frac{r(r-1)}{2}}\frac{(q;q)_y}{(q;q)_r\, (q;q)_{y-r}}S_{y-r,m}= \sum_{r=0}^{y}(-\nu q^m)^r q^{\frac{r(r-1)}{2}}\frac{(q;q)_y}{(q;q)_r\, (q;q)_{y-r}}.
\frac{\partial_\alpha}{|z_\alpha|}z_{tt}=-(\frac{\partial_\alpha}{|z_\alpha|})^2P e^{i\theta}+i\frac{\partial_\alpha a}{|z_\alpha|}e^{i\theta}-\frac{\partial_\alpha P}{|z_\alpha|}e^{i\theta}\frac{i\partial_\alpha\theta}{|z_\alpha|}+iae^{i\theta}\frac{i\partial_\alpha\theta}{|z_\alpha|}.
\|z_\ast^T X\|_1 = \|(z_0^T+z_1^T)X_S\|_1 +\|(z_0^T+z_1^T)X_{S^c}\|_1 &\ge \|z_0^T X_S\|_1 - \|z_1^TX_S\|_1 +\|z_1^T X\|_1 - \|z_1^T X_{S}\|_1 &= \|z_0^T X\|_1 + (\|z_1^T X\|_1 - 2\|z_1^T X_{S}\|_1).
\frac{\delta(n+1)}{duH_{Y}(u)}N_{ramF}(r,s) \geq \frac{1}{d}\sum_{j=1}^{q}[N_{f}(r,s;Q_{j})-N_{f}^{[n_{u}]}(r,s;Q_{j})] &-\frac{\delta(2 n+1)(n+1)D}{du}\sum_{j=1}^{q}N_{f}(r,s;Q_{j}) .
u=\arg\min_{w\in L_2(0,T;\mathcal{V})}\bigg( || \partial_t w + \mathcal{L}w -F||^2_{L_2(0,T;\mathcal{H})}+ || \alpha w +\beta\mathbf{n}\cdot \nabla w- g ||^2_{L_2(0,T;\partial \mathcal{H})}+ || w(0,\cdot) - u_0 ||^2_{\mathcal{H}}\bigg),
\mathbf{x}^1 (\mathbf{R}- \nu\mathbf{1})= f_i^S - \nu g_i^S - \sum_{j \in S}(r_j^S - \nu w_j^S) x_j^0 + \sum_{j \in S^c}(r_j^S - \nu w_j^S) x_j^1.
\operatorname*{E}\left( \log_{10}\left( X\right) \right) =\frac{1}{\ln(2)}{\displaystyle\int\limits_{0}^{1}}\frac{\log_{10}(x)}{1+x}dx=-\frac{\pi^{2}}{12\ln(2)\ln(10)}=-0.5153204170... \ =\frac{1}{\ln(2)}{\displaystyle\int\limits_{0}^{1}}\frac{\log_{10}\{1/x\}}{1+x}dx=\operatorname*{E}\left( \log_{10}\left\{\frac{1}{X}\right\}\right) ,
{\mathcal D}^{( 1 / 2 , 1 / 2 )}\otimes{\mathcal D}^{( 1 / 2 , 1 / 2 )}={\mathcal D}^{( 1 , 1 )}\oplus{\mathcal D}^{( 1 , 0 )}\oplus{\mathcal D}^{( 0 , 1 )}\oplus{\mathcal D}^{( 0 , 0 )}.
\frac{1}{\sqrt{2\pi}}\frac{\int_{c_1+\sqrt{n_1}G_m^{-1}(RQL)}\left[ 1 - \Phi( c_2 - z + (\sqrt{n_1}+ \sqrt{n_2}) G_m^{-1}(RQL) ) \right] e^{-z^2/2}\, dz}{1- \Phi( c_1 + \sqrt{n_1}G_m^{-1}(RQL) )}= \beta_2,
\begin{aligned}\|z_1 - z_2\|_{C^{\frac{1+\beta_0}{2},1+\beta_0}(D)}\leq C_2(C_{02}\|g_1-g_2\|_{C^1([0,T_0])}+T_0^{\frac{\beta_0+1}{2}}\|w_1-w_2\|_{C^{\frac{1+\beta_0}{2},1+\beta_0}(D)}&+\|z_{01}-z_{02}\|_{C^{2}([0,1])}+ \|w_{01}-w_{02}\|_{C^{2}([0,1])}). \end{aligned}
\eta_{(j_1-1)2^{{r\over 2}-1}+k}= \theta_{j_1}+ \lambda_k^+, \eta_{[{m_1\over2}]2^{{r\over 2}-1}+l}= \lambda_l^+ , \eta_{[{m_1\over2}]2^{{r\over 2}-1}+2^{{r\over 2}-2}+(j_2-1)2^{{r\over 2}-1}+k}= \theta_{j_2}'+ \lambda_k^- ,
X_{L}( a , b , c ) = \sum_{a_{2}, \ldots , a_{L}}q^{\, \sum_{j = 1}^{L}j \, H ( a_{j}, a_{j + 1}, a_{j + 2})}\qquad a_{1}= a , \: a_{L + 1}= b , \: a_{L + 2}= c .
I(a_1) = \bigcup_{a_2 = a_1^2-a_1+1}^{\infty}J(a_1,a_2) = \bigcup_{a_2 = a_1^2-a_1+1}^{\infty}\left(\frac{1}{a_1}+ \frac{1}{a_2}, \frac{1}{a_1}+ \frac{1}{a_2-1}\right].
\frac{d^m}{d\epsilon^m}\vert_{\epsilon =0}\lim_{\theta\to 1}\Bigl(\Omega,T_{\theta ({\cal L}_0+\epsilon{\cal L}_1)}(f^{\otimes l})\Omega\Bigr)=i^m\lim_{\theta\to 1}\Bigl(\Omega,T_{\theta{\cal L}_0}((\theta{\cal L}_1)^{\otimes m}\otimes f^{\otimes l})\Omega\Bigr)
V ( r ) = ( 1 - H^{2}r^{2}- \frac{2 M}{r})^{2}[ \frac{1}{E^{2}}( \mu r + \frac{\Omega^{2}}{2 r})^{2}( 1 - H^{2}r^{2}- \frac{2 M}{r}) - 1 ] .
I+II= \dfrac{2\delta}{p-1}+\dfrac{(1-\delta)^p}{p(p-1)}-\dfrac{1}{p(p-1)}-\dfrac{\delta^p}{p(p-1)}&=\dfrac{2\delta}{p-1}+\dfrac{p\xi^{p-1}(-\delta)}{p(p-1)}-\dfrac{\delta^p}{p(p-1)}, &\geq\dfrac{2\delta}{p-1}-\dfrac{\delta}{p-1}-\dfrac{\delta^p}{p(p-1)}=\dfrac{\delta}{p-1}-\dfrac{\delta^p}{p(p-1)}
\begin{array}{l c l}{{F_{- 1 \mu}+ F_{n \mu}}}&{{=}}&{{( X_{- 1 \lambda}+ X_{n \lambda}){X_{\mu}}^{\lambda}+ ( X_{n , - 1}X_{- 1 , \mu}- X_{- 1 , n}X_{n , \mu}) - \frac{1}{2}n ( X_{- 1 , \mu}+ X_{n \mu})}}\\{{}}&{{}}&{{}}\\{{}}&{{=}}&{{P_{\lambda}({X_{\mu}}^{\lambda}+ ( D + 1 - \frac{n}{2}) \delta_{\mu}^{\lambda})}}\\ \end{array}
\delta \mathcal{P}_{a}^{(1)}=-\pi_{a}^{0},\;\delta \mathcal{P}_{a}=\partial_{i}B_{a}^{0i},\;\delta P_{a}^{(1)i}=p_{a}^{i},\;\delta P_{a}^{i}=-\partial^{i}\varphi_{a},
\begin{aligned}\|u^{n}_{h}\|^{2}_{2}\leq K_{4}, \sum_{m=1}^{N}\|u^{m}_{h}-u^{m-1}_{h}\|^{2}_{2}\leq 32 K_{4}, \Delta t\sum_{m=1}^{N}\|\nabla u^{m}_{h}\|^{2}_{2}\leq 4 K_{4}. \end{aligned}
|M^{(m)}| =\prod_{j=0}^{m-2}p_j+\prod_{j=0}^{m-3}p_j(1-p_{m-1}) +\prod_{j=0}^{m-4}p_j(1-p_{m-2})(1-p_{m-1}) +\cdots+ p_0\prod_{j=2}^{m-1}(1-p_{j})+\prod_{j=1}^{m-1}(1-p_j) &= \sum_{j=0}^{m-1}\prod_{i_1=0}^{j-1}p_{i_1}\prod_{i_2=j+1}^{m-1}(1-p_{i_2}),
\begin{cases}dx_i^o =( Ax_i^o +BR^{-1}B^Ty_i^o +G \bar m_i+\gamma \bar p_i) dt +D dW_i, dy_i^o=\left\{-A^T y_i^o +Q[x_i^o-(\Gamma \bar m_i+\eta)]\right\}dt +\zeta_i^o dW_i, \end{cases}
\int\left(\sum_{n=1}^\infty a_n(x) \right)dx = \sum_{n=1}^\infty \left(\int a_n(x)\,dx \right),\;\;\;\int \lim_{n \rightarrow \infty}s_n(x)\,dx = \lim_{n \rightarrow \infty}\int s_n(x) \,dx,
\sum_{\sigma \in S_n}x_{\sigma(1)}^{d_1}\ldots x_{\sigma(j)}^{d_j}= \sum_{\pi \in S_n}\big(x_{\pi(1)}\ldots x_{\pi(j)}\big)^{d_j}\cdot \sum_{\rho \in S_n}x_{\rho(1)}^{d_1 - d_j}\ldots x_{\rho(j-1)}^{d_{j-1}- d_j}.
\widehat{L}_{\beta}( \lambda ) = \frac{1}{\sin \gamma}\left( \begin{array}{c c}{\sinh [ \gamma ( \lambda + \mathrm{i}S^{3}) ]}&{\mathrm{i}\sin \gamma e^{( \gamma - \beta ) \lambda}S^{-}}\\{\mathrm{i}\sin \gamma e^{( \beta - \gamma ) \lambda}S^{+}}&{\sinh [ \gamma ( \lambda - \mathrm{i}S^{3}) ]}\\ \end{array}\right)
\begin{cases}\widetilde{c}_{\jmath\imath}(-2+r^{(\jmath)}+4k)=\delta_{0, k}\ 1\leq \jmath\leq n-1, \widetilde{c}_{2n-\jmath, \imath}(-2+r^{(\jmath)}+4k)=\delta_{0, k}\n+1\leq \jmath\leq 2n-1, \widetilde{c}_{n, \imath}(-1+r^{(n)}+2k)=\delta_{0, k}\ \jmath=n, \end{cases}
\frac{\partial\mu_{s,\tau}}{\partial s}=\sum_{j=1}^{\dim\mathfrak{k}}\tilde{X}_{j}^{2}\mu_{s,\tau};\quad\frac{\partial\mu_{s,\tau}}{\partial u}=\sum_{j=1}^{\dim\mathfrak{k}}\tilde{X}_{j}\tilde{Y}_{j}\mu_{s,\tau}.
q = \left( \begin{matrix}{{P_{1}+ P_{2}\cos^{2}\alpha + P_{3}\cos^{2}\beta}}&{{- P_{2}\sin \alpha \cos \alpha}}&{{- P_{3}\sin \beta \cos \beta}}\\{{- P_{2}\sin \alpha \cos \alpha}}&{{P_{2}\sin^{2}\alpha}}&{{0}}\\{{- P_{3}\sin \beta \cos \beta}}&{{0}}&{{P_{3}\sin^{2}\beta}}\\ \end{matrix}\right)
\sum_{k=0}^\infty\mu(k)(1-{E_2(i,m)})^k \leq\sum_{k=0}^\infty\mu(k)\left(1-\left(\frac{1}{2A(m+1)}\right)^{i+2}\right)^k &\leq\sum_{k=0}^\infty\mu(k)\left(1-\left(\frac{1}{2A(m+2)}\right)^{i+2}\right)^k.
\langle x,y\rangle=\bigg\langle x,\frac{P_Y x}{\|P_Y x\|_2}\bigg\rangle=\|P_Y x\|_2=\sqrt{\|x\|_2^2-\|P_{Y^\perp}x\|_2^2}>\sqrt{1-\alpha^2}\|x\|_2,
Z_{n,0}-uZ_n+\frac{u^2}{2}\Gamma = \sum_{i=1}^nY_{i,n}(u) -uZ_n +\frac{1}{2}[u^2\Gamma -\sum_{i=1}^nY_{i,n}(u)^2] &+ Z_{n,0}-\sum_{i=1}^nY_{i,n}(u)+\frac12 \sum_{i=1}^nY_{i,n}(u)^2
\tau=\begin{pmatrix}1 1_{n-2}& 1\end{pmatrix}\begin{pmatrix}\xi_{n}^{-1}&\ddots & \xi_{3}^{-1}& -y\zeta_{v}^{-1}\xi_{2}^{-1}& -\zeta_{v}\xi_{1}^{-1}\end{pmatrix}.
F^{\Psi}_{(-1)q(n-1)}\bigl( \bigl( e_{1}\wedge\ldots \wedge e_{n}\bigr) \otimes e_{b_1}\otimes \ldots \otimes e_{b_{q-1}}\bigr) \qquad=\Bar{z}_{b_1}\ldots \Bar{z}_{b_{q-1}}\sum_{i=1}^{n}(-1)^{i+n}\Bar{z}_{i}{\zeta}_{1}\wedge \ldots \wedge{\zeta}_{i-1}\wedge{\zeta}_{i+1}\wedge \ldots \wedge{\zeta}_{n}\,.
\sum_{j=1}^m\frac{(-m-i+1)_{j-1}(-m+i)_{j-1}}{(\tfrac{\beta}{2}-m+1)_{j-1}(-\tfrac{\beta}{2}-m)_{j-1}}=-\frac{1}{(\tfrac{\beta}{2}+i)(\tfrac{\beta}{2}-i+1)}\frac{1}{(\tfrac{\beta}{2}-m+2)_{-1}(-\tfrac{\beta}{2}-m+1)_{-1}}=-\frac{(\tfrac{\beta}{2}-m+1)(-\tfrac{\beta}{2}-m)}{(\tfrac{\beta}{2}+i)(\tfrac{\beta}{2}-i+1)},
\begin{array}{rl}p_B \geq{\rm E}\Big[1-\sum_{i \in B^c}\max \Big\{0,\Big(\frac{\sum_{j\in B}\alpha_{ji}c_j}{\eta_i}\Big)^{\gamma_i}\Big\}\Big], = 1-\sum_{i\in B^c}{\rm E}\Big[ \max \Big\{0,\frac{\sum_{j\in B}\alpha_{ji}c_j}{\eta_i}\Big\}^{\gamma_i}\Big].\end{array}
{\rm ess}\inf L' - \frac{E[L'^2]}{2E[L']}= \lceil-\log_2{\delta M}\rceil-\frac{E[L'^2]}{2E[L']}\ge -\log_2{\delta M}-1-\frac{E[L^2]}{2E[L]}-\frac{3}{2}\ge -\log_2{\delta M}-1-\frac{1}{2}h(X)+\frac{1}{2}\log_2{\delta}-\frac{3}{2}-\epsilon = -\frac{1}{2}\log_2{\delta}-\log_2{M}-\frac{1}{2}h(X)-\frac{5}{2}-\epsilon > 0.
\sum_{k = 0}^{\infty}\frac{q^{2 k^{2}+ ( 2 \mu_{a}- 1 ) k}}{( q )_{2 k + \mu_{a}}}= \frac{Q ( q^{2})}{Q ( q )}= q^{- 1 / 2 4}\chi_{1 , 2}^{( 4 )}( q ) \equiv q^{- 1 / 2 4}\chi_{1 / 1 6}( q ) .
P_{-}= \int_{-{\infty}}^{\infty}d x^{-}{\theta}_{-}^{+}( x ) + \frac{1}{2}\int_{-{\infty}}^{\infty}d x^{+}B ( x ) \frac{n_{-}}{m^{2}- n_{-}{\partial}_{+}^{2}}B ( x ) .
\int_a^1\max(t-w,0)^mdw = \int_{t-a}^{t-1}\max(W,0)^m(-dW) = -\int_{-\infty}^{t-1}\max(W,0)^mdW+\int_{-\infty}^{t-a}\max(W,0)^mdW = -\frac{1}{m+1}M(t-1)^{m+1}+\frac{1}{m+1}M(t-a)^{m+1}.
\left( \begin{array}{c}{\dot{\Phi}}\\{\Phi^{\prime}}\\ \end{array}\right) \rightarrow \left( \begin{array}{c c}{c o s h \theta}&{s i n h \theta}\\{s i n h \theta}&{c o s h \theta}\\ \end{array}\right) \left( \begin{array}{c}{\tilde{\dot{\Phi}}}\\{\tilde{\Phi}^{\prime}}\\ \end{array}\right) .
\|\eta\overline{u}^{\alpha+1}\|_{m^\star}^m\le C\int\overline{u}^{m(\alpha+1)}|\nabla \eta|^m+\frac{C}{m\alpha+1-Cm\epsilon^{m'}}\int mg_1(x,\epsilon)\overline{u}^{m(\alpha+1)}+C\left(\frac{\|f\|_q}{(m\alpha+1-Cm\epsilon^{m'})k^{m-1}}\right)^{N/(mq-N)}\int \eta^m\overline{u}^{m(\alpha+1)}.
T(z) = \left(\mathbb{I}+\mathcal{O}\left(\frac{1}{z}\right)\right)\begin{pmatrix}1 0 0 0 z^{\frac{1}{4}}0 0 0 z^{-\frac{1}{4}}\end{pmatrix}\begin{pmatrix}1 0 0 0 \frac{1}{\sqrt{2}}\frac{i}{\sqrt{2}}0 \frac{i}{\sqrt{2}}\frac{1}{\sqrt{2}}\end{pmatrix}.
\check{R}(u_0) = \sum_{i}^{n}\left[ \prod_{j=1}^{i-1}\left( \frac{\lambda_j}{\lambda_{j+1}}f_j(u_0) +1 \right) \prod_{k=i}^{n-1}\left( f_k(u_0) + \frac{\lambda_k}{\lambda_{k+1}}\right) \lambda_n \right] P_i
\{z_{j}\}_{j = 1}^{M}: = \{y_{l}^{( b )}+ \hat{\tau},{y^{\prime}}_{k}^{( b )}\; | \; \; l = 1 , . . . m_{b}; \; k = 1 , . . . m_{b}^{\prime}; \; b = 1 , . . . N \}\; .
=q\langle(\sqrt{2}-1)^{5}\{(\sqrt{2}+1)tan[arctan(\frac{1}{3}\sqrt{5}-\frac{1}{3}\sqrt[3]{6\sqrt{30}+4\sqrt{5}}+\frac{1}{3}\sqrt[3]{6\sqrt{30}-4\sqrt{5}})-\frac{1}{8}\pi]\}^{4}\rangle=
w = s_{\alpha_1}s_{\alpha_2 + \alpha_3}s_{\alpha_4}s_{\alpha_2}= s_{\alpha_2 + \alpha_3 + \alpha_1}s_{\alpha_1}s_{\alpha_4}s_{\alpha_2}= s_{\widetilde\alpha_3}s_{\alpha_1}s_{\alpha_4}s_{\alpha_2},
\beta_{i}(S/I_{n,d,{\bf t}})\ =\ (-1)^{i+1}\prod_{j \neq i}\frac{d_{j}}{d_{j}-d_{i}}= (-1)^{i+1}\prod_{j=1}^{i-1}\frac{d+j-1}{j-i}\prod_{j=i+1}^{p}\frac{d+j-1}{j-i}&=\ \binom{d+i-2}{d-1}\binom{n-\sum_{j=1}^{d-1}t_j+d-1}{d+i-1}.
\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j,x_k\rangle \langle \tau_k,x_j\rangle =\frac{1}{\dim\mathcal{H}}\left(\sum_{j=1}^n\langle x_j,\tau_j \rangle\right)^2 &=\frac{1}{\dim\mathcal{H}}\left(\sum_{j=1}^n\langle \tau_j,x_j \rangle\right)^2 =\sum_{j=1}^n\sum_{k=1}^n\langle \tau_j,\tau_k\rangle \langle x_k,x_j\rangle .
R_{itit}=\frac{-aa_{tt}}{(1+\frac{K}{4}\vec{x}\vec{x})^2},\;\;\;\;\;\;R_{ijij}=\frac{a^2(K+a_{t}^2)}{(1+\frac{K}{4}\vec{x}\vec{x})^4};\;\;\;\;i\neq j
f_{9}^{A}= 2 \zeta ( 3 ) r_{A}e^{- 2 \phi}+ \frac{2 \pi^{2}}{3 r_{A}}+ 0 + ~ 4 \pi e^{- \phi}\sum_{m \ne 0}\sum_{n \ne 0}\left| \frac{m}{n}\right| K_{1}\left( 2 \pi r_{A}e^{- \phi}| m n | \right) e^{2 \pi i m n{\cal A}}
c=\vec{w}\cdot\frac{1}{2}(\vec{\mu}_{0}+\vec{\mu}_{1})=\frac{1}{2}\vec{\mu}_{1}^{T}\Sigma_{1}^{-1}\vec{\mu}_{1}-\frac{1}{2}\vec{\mu}_{0}^{T}\Sigma_{0}^{-1}\vec{\mu}_{0}
L_{2}^{\ast\ast}=\frac{c_{1}\left[\xi_{p}\overline{G}_{(2)}\left(\frac{1}{2}\xi_{p}^{2}\right)-\xi_{q}\overline{G}_{(2)}\left(\frac{1}{2}\xi_{q}^{2}\right)\right]}{F_{Y}(\xi_{p},\xi_{q})}+\frac{c_{1}}{c_{(2)}^{\ast}}\frac{F_{Y_{(2)}}(\xi_{p},\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})}.
\begin{aligned}\int_{V_{\min}}^{V_{F}}\left(\partial_{t}p+\partial_{v}(hp)-a\partial_{v v}p\right)\phi dv-h(V_R)\phi(V_R)\left(p(V_R^-)-p(V_R^+)\right) + a\phi(V_R)\left(\partial_vp(V_R^-)-\partial_vp(V_R^+)+\partial_vp(V_F)\right)-a\partial_vp(V_{\min})\phi(V_{\min})=0, \end{aligned}
{}^{(n-1)}R_{ABCD}={}^{(n)}R_{A'B'C'D'}^\pm \; \;q^{A'}{}_A \; q^{B'}{}_B \; q^{C'}{}_C \; q^{D'}{}_D \;+ K_{AC}^\pm\; K_{BD}^\pm - K_{AD}^\pm \; K_{BC}^\pm.
H = - \frac{1}{2}\sum_{j = 1}^{L}\{( \sigma_{x}^{j}\sigma_{x}^{j + 1}+ \sigma_{y}^{j}\sigma_{y}^{j + 1}) ( 1 - U \tau_{z}^{j + 1}) + ( \tau_{x}^{j}\tau_{x}^{j + 1}+ \tau_{y}^{j}\tau_{y}^{j + 1}) ( 1 - U \sigma_{z}^{j}) \}
{\frac{\kappa}{4 \pi}}\{\xi_{\alpha L}( x ) , \xi_{\beta R}( z ) \}\; = \; \Bigl ( L^{T \; - 1}( x ) H ( z ) R^{- 1}( z ) \Bigr )_{\alpha \beta}\; + \; \Bigl ( \tilde{h}( x ) R^{- 1}( z ) \Bigr )_{\alpha \beta}\quad ,
\Phi(x,t) = \sum_{n=1}^\infty \left[ \alpha_n \Phi^{\rm odd}(n,x,t) + \alpha_n^*\overline{\Phi^{\rm odd}(n,x,t)}\right]+ \sum_{j=1}^\infty\left[ \beta_j\Phi^{\rm even}(j,x,t) + \beta_j^* \overline{\Phi^{\rm even}(j,x,t)}\right]
V_n(f;x) =\sum^{n}_{k=0}{\binom{n}{k}(r_n(x))^k(1-r_n(x))^{n-k}f\left(\frac{k}{n}\right)}&=\sum^{n}_{k=0}{v_{n,k}(x)\cdot f\left(\frac{k}{n}\right)},
[\hat{B}^{0i},\hat{Q}^a]=i\varepsilon^{ab}\hat{\pi}^i_{(C)b},\; [\hat{\pi}_{(B)0i},\hat{Q}^a]=-i\varepsilon^{ab}(\partial_i \hat\pi_{(C)0b}+\hat{A}_i\wedge\pi_{(C)0b}),
\sum_{n=0}^\infty B(n) \frac{x^n}{n!}\frac{d}{da}(a)_n =\sum_{n=0}^\infty B(n) \frac{x^n}{n!}(a)_{n}\frac{1}{a}\sum_{k=0}^{n-1}\frac{(a)_{k}}{(a+1)_{k}}
\varphi_{0}(\theta,\phi)={}\sum_{l=1}^\infty\sum_{m=-l}^{l}a_{l,m}\thinspace{}_1 Y_{l,m}(\theta,\phi), \varphi_{2}(\theta,\phi)={}\sum_{l=1}^\infty\sum_{m=-l}^{l}b_{l,m}\thinspace{}_{-1}Y_{l,m}(\theta,\phi).
\bigg(\frac{1}{r}\frac{\mathrm{d}}{\mathrm{d}r}\bigg)^{k-1}\left(r^{2k-1}\phi(r)\right) = \sum_{j=0}^{k-1}\beta^{(k)}_j r^{j+1}\frac{\mathrm{d}^j \phi}{\mathrm{d}r^j}(r),
g(x_{1},x_{2},x_{3})=\sum_{m_{1},m_{2},m_{3}\in\mathbb{Z}}h^{three}(m_{1},m_{2},m_{3})\cdot e^{i2\pi(\frac{m_{1}}{a_{1}}x_{1}+\frac{m_{2}}{a_{2}}x_{2}+\frac{m_{3}}{a_{3}}x_{3})}.
\|x\widetilde f_{R,\delta}\|\leq\Lambda\left\|\nabla u_{R,\delta}^-\right\|+\epsilon^2(R) \,, \|x\widetilde f_{R,\delta}\|\leq\Lambda\left\|\nabla u_{R,\delta}\right\| +\epsilon^2(R) \,, \lim_{R\to\infty}\epsilon^2(R)=0 \,,
{\delta^\dagger}^{(l)}_j(\boldsymbol{u}^{(k)})^{(l)}_i -{\delta^\dagger}^{(k)}_i(\boldsymbol{u}^{(l)})^{(l)}_j = 0 \quad\forall (i,j)\in B_e(l,k), \;\forall l\in{\mathcal{I}}_{\mathcal{F}}^{(k)},
\mathbb E\left[\|\Delta_j\mathcal SB^{\mathfrak H}\|_H^{8}\right]\leq \int_0^2 \mathbb E \left[\left(B^{\mathfrak H}_{x+i\Delta_n}-B^{\mathfrak H}_{x+(i-1)\Delta_n}\right)^8 \right]dx= 210 \Delta_n^{8H}=210 \Delta_n^2,
\begin{aligned}\left(\frac{\mathcal{Z}}{\sqrt{m}}\right)\ll\frac{1}{N^2}R^\epsilon N\left(\left(1+R\cdot \rho^{1/3}\right)+\frac{N}{\rho^2 R^2}\right) \ll\frac{1}{N^2}R^\epsilon N\left(R\cdot \rho^{1/3}+\frac{1}{\rho^2 R}\right).\end{aligned}
\lim_{m \rightarrow 0^+}\; \lim_{\beta \rightarrow \infty}\langle{\bar{\psi}}\psi{\rangle}_{\beta}= 0 \;\;\; \lim_{m \rightarrow 0^-}\; \lim_{\beta \rightarrow \infty}\langle{\bar{\psi}}\psi{\rangle}_{\beta}= \frac{eB}{2 \pi}\; ,
- i \bar{\kappa}_{( \alpha )}\gamma^{a}\kappa_{( \beta )}e_{a}= - i \Gamma_{s}( u^{- 1})^{\dagger}\mathcal{D}\mathcal{S}\Gamma_{s}( P_{b}) \Gamma_{s}( u^{- 1}) \eta^{b a}e_{a}\, ,
\beta_{k,q}=c_{n,k,q}\beta\wedge\theta_0^{n-k+q}\wedge\theta_1^{k-2q-1}\wedge\theta_2^q,q<\frac{k}{2}, \gamma_{k,q}=\frac{c_{n,k,q}}{2}\gamma\wedge\theta_0^{n-k+q-1}\wedge\theta_1^{k-2q}\wedge\theta_2^q, k-n< q,
(f(\mathcal{\partial},\beta )\varphi)(x):=\mathcal{F}_{\xi\rightarrow x}^{-1}(|f(\xi )|_{p}^{\beta}\mathcal{F}_{x\rightarrow \xi}\varphi ),\varphi \in \mathcal{D}(\mathcal{\mathbf{\mathbb{Q}}}_{p}^{n})\beta >0,
\mathcal{C}_{1,j,k}^{J}=\frac{1+2x_{1}}{2+2x_{1}}\left[ \mathcal{A}_{1,j}^{J\setminus k}-\frac{1}{2+2x_{1}}\tilde{\mathcal{D}}_{1,j}^{J\setminus k}\right]\tilde{\mathcal{D}}_{1,k}^{J}+\frac{1+2x_{1}}{2+2x_{1}}\left[\mathcal{A}_{1,k}^{J\setminus j}-\frac{1}{2+2x_{1}}\tilde{\mathcal{D}}_{1,k}^{J\setminus j}\right]\tilde{\mathcal{D}}_{1,j}^{J}\,,
{\cal G}( i )^{- 1}{\cal Z}( z ){\cal K}( z ; i ) = \left( \begin{array}{c c}{1}&{0}\\{i x_{-}^{\prime}{\cdot \tilde{\sigma}}^{t}}&{- 2 \theta^{\prime a}{}^{t}}\\{2 \bar{\theta}_{b}^{\prime t}}&{\delta_{b}^{a}}\\ \end{array}\right) =
|Z(s)|_{min}=R_{1}\sqrt{1-\frac{(2Q^{2}+k^{2}-1)^{2}}{2Q^{4}+k^{2}(2Q^{2}+k^{2}-1)^{2}+2Q^{2}\sqrt{Q^{4}+k^{2}(2Q^{2}+k^{2}-1)}}}.
\sum_{j=1}^n \theta^j \frac{\partial}{\partial{\theta^j}}\left[\frac{\partial{U(\hat{x},\theta)}}{\partial{\theta^i}}\right]=\frac{\lambda(\hat{x},\theta)}{\theta^i}\left[\sum_{j=1}^n \theta^j \frac{\partial{s^i(\hat{x},\theta)}}{\partial{\theta^j}}\right]=0.
c_{ir}c_{is}=c_{is}c_{ir}\phantom{pha}\hbox{for all $1\leq i,r,s\leq n$}c_{jr}c_{is}=qc_{is}c_{jr}\phantom{pha}\hbox{for all $1\leq i<j\leq n, 1\leq r\leq s\leq n$}c_{js}c_{ir}=c_{ir}c_{js}+(q-1)c_{is}c_{jr}\phantom{pha}\hbox{for all $1\leq i<j\leq n, 1\leq r< s\leq n$}.
\mathcal{X}_{j , u , \theta}( \Omega ) = \left| \int_{0}^{\tau}d t \left\langle \mathbf{\Phi}_{j , u , \theta}^{A}( x , t )^{\dagger}x \mathbf{\Phi}_{j , u , \theta}^{A}( x , t ) \right\rangle \exp i \Omega t \right| .
\hat{f}_{\mathbf{t}}(x,y)=y^2(3-3 xy+x^2y^2)^2+ (1-xy)^2+ t_1 x^2 (1-xy)+t_2 x, \tilde{f}_{\mathbf{t}}(x,y)=(3-3 y+y^2)^2+x^{2}(1-y)^2+ t_1 x^{4}y^2 (1-y)+t_2 x^{3}y.
\begin{array}{l}{{\vert 0 , z \rangle_{0}= \displaystyle \exp_{q^{- 1}}( - \frac{1}{2}| q z | ) \sum_{m = 0}^{\infty}\frac{( - q z )^{m}}{[ m ]_{q^{- 1}}!}({\cal D}_{0})^{m}\vert 0 \rangle_{0}~ , ~ ~{\cal X}^{0}\vert 0 \rangle_{0}= 0 ~ ,}}\\{{{\cal X}^{0}\vert 0 , z \rangle_{0}= z \vert 0 , z \rangle_{0}~ .}}\\ \end{array}