harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\pi_{k-1}^k(t) =\frac{1}{k(k+1)}, \pi_{k-2}^k(t) =\frac{2k-1}{k-2}\,t\pi_{k-1}^k(t), \pi_j^k(t) =\frac{1}{j+1}\left[(2j+3)t\pi_{j+1}^k(t)-(j+2)\pi_{j+2}^k(t)\right], j=k-3,k-4,\dots1, \pi_0^k(t) =3t\pi_1^k(t)-2\pi_2^k(t),
e_1 = \frac{\sqrt{2}}{\sqrt{\rho (e^{-4\psi}\Omega_t^2 + 4 \psi_t^2)}}\, \partial_t, e_2 = \frac{e^{2\psi}}{\sqrt{2}\sqrt{\rho (e^{-4\psi}\Omega_t^2 + 4 \psi_t^2)}(\psi_t \Omega_\rho - \Omega_t \psi_\rho)}\bigg((e^{-4\psi}\Omega_t \Omega_\rho + 4 \psi_t \psi_\rho)\, \partial_t + (e^{-4\psi}\Omega_t^2 + 4 \psi_t^2)\ \partial_\rho \bigg).
\left\{j_{M}^{0}\left( t , \sigma \right) , \widehat{U_{N}^{0}}\left( t , \sigma^{\prime}\right) \right\}= - \delta_{M}U_{N}^{0}\cdot \delta \left( \sigma^{\prime}- \sigma \right) \; + \frac \partial{\partial \sigma^{r}}\left[ \delta_{M}\phi^{i}\frac{\partial \widehat{U_{N}^{0}}}{\partial \partial_{r}\phi^{i}}\cdot \delta \left( \sigma^{\prime}- \sigma \right) \right] \quad .
1+p_1\left(f^\ast\tau_N\right)+\cdots p_{\lfloor{\frac{m+k-1}{2}}\rfloor}\left(f^\ast\tau_N\right)=1+p_1\left( \tau_M\oplus \nu_f \right)+\cdots+p_{\lfloor\frac{m+k-1}{2}\rfloor}\left( \tau_M\oplus \nu_f \right) .
\left\Vert f\right\Vert_{\mathcal{B}^{p}}:=\sup_{\substack{R_{j}\geq1 j=1,...,n}}\left[ \frac{1}{R_{1}...R_{n}}{\displaystyle\int\nolimits_{\left[ -R_{1},R_{1}\right] \times...\times\left[ -R_{n},R_{n}\right]}}\left\vert f\left( x\right) \right\vert^{p}dx\right]^{1/p}
\mbox{where}t_\mu(x) = (x_1,\ldots,x_{\mu-1}, \frac{1}{2}(x_\mu + x_{\mu+1}), \frac{1}{2}(x_\mu + x_{\mu+1}), x_{\mu+2},\ldots,x_k), & r_\mu(x) = (0,\ldots,0,\frac{1}{2}(x_\mu-x_{\mu+1}),\frac{1}{2}(x_{\mu+1}-x_\mu),0,\ldots,0).
\hat{R}= \int \! d p_{0}\Lambda^{+ + +}{\frac{R}{1 - R}}\left[ G_{0 1}G_{0 2}K_{1 2}^{0}+ G_{0 2}G_{0 3}K_{2 3}^{0}+ G_{0 3}G_{0 1}K_{3 1}^{0}\right]
\Omega_{\nu}^{- 1}= \; \int d^{4}x \; \; \left[ \omega_{\nu}^{- 1}\right]_{5}\; \equiv \; \int d^{4}x \; \left( c^{a}\partial_{\nu}C^{* a}- A_{\mu}^{a}\partial_{\nu}A^{* a \mu}\; \right) \; ,
|\frac{\phi(r,\xi)}{r^{3/2}\langle \log(r)\rangle}| \leq C \frac{|a(\xi)|\sqrt{\xi}}{\langle \log(\xi)\rangle}+ C \begin{cases}\frac{1}{\langle \log(\xi)\rangle}, \xi \geq 1 1 , \xi \leq 1\end{cases}
\bar{\zeta}( s ) = \int_{- \infty}^{\infty}\frac{d k}{2 \pi}\int_{0}^{\infty}\frac{\lambda d \lambda}{( k^{2}+ \lambda^{2})^{s}}= \frac{1}{2 \sqrt{\pi}}\frac{\Gamma ( s - 1 / 2 )}{\Gamma ( s )}\int_{0}^{\infty}\lambda^{2 - 2 s}d \lambda{.}
g(u, v)= \tilde{g}(u, v)+\sum\limits_{i=i_0+1}^{\infty}\alpha_iu^i+\sum\limits_{j=j_0+1}^{\infty}\beta_jv^j = \tilde{g}(u, v)+\pi^{e_0+1}\Big(\sum\limits_{i=i_0+1}^{\infty}\pi^{-(e_0+1)}\alpha_iu^i+\sum\limits_{j=j_0+1}^{\infty}\pi^{-(e_0+1)}\beta_jv^j\Big),
\vert x_{n_k}- x([\ell]P_{n_k}) \vert_w = \Bigl \vert \frac{\Psi_{\ell-1}^\prime (x_n,y_n) \Psi_{\ell+1}^\prime (x_n,y_n)}{\left ( \Psi_\ell^\prime (x_n,y_n)\right)^2}\Bigr \vert_w \leq C^\prime \vert x_{n_k}- x_* \vert_w,
\sum_{i=1}^n \widehat{R}_{T,i}(\widehat{R}_{Y,i}- \theta_0\widehat{R}_{T,i}) = \sum_{i=1}^n R_{T,i}(R_{Y,i}- \theta_0R_{T,i}) + (\widehat{\beta}_{T,n}- \beta_T)^{\top}\sum_{i=1}^n X_i(R_{Y,i}- \theta_0 R_{T,i}).
\frac{e^{- 2 \rho}}{\sqrt{\kappa - 2}}+ \frac{\kappa}{2 \sqrt{\kappa - 2}}\rho = - \frac{\lambda^{2}x^{+}x^{-}}{\sqrt{\kappa - 2}}- \frac{\kappa}{4 \sqrt{\kappa - 2}}\ln ( - \lambda^{2}x^{+}x^{-}) - \frac{m ( x^{+}- x_{0}^{+})}{\lambda x_{0}^{+}\sqrt{\kappa - 2}}\theta ( x^{+}- x_{0}^{+})
\begin{aligned}\vec{\mathbf{\upsilon}}_{\omega, k}(t) \, = \, \frac{\alpha_{\omega}}{\jmath \, \omega}\bigg( \varphi_{\omega, k, t}- \frac{\sum_{j = 1}^{N_{Tot}}\varrho_{\omega, k, j}\, \lambda_{k,t}\, \varphi_{\omega, k, t}}{1 \,+\, \sum_{j = 1}^{N_{Tot}}\lambda_{k,j}\, \varrho_{\omega, k, j}}\bigg) \end{aligned}
c_k = b_k + \frac{1}{\mathcal D_k (1- \varepsilon_K) \vert \Lambda \vert}\sum_{p \in \mathcal P_L}\frac{\widehat g(k)}{\sqrt{1-\alpha_{k}^{2}\vphantom{\alpha_{p-k}^2}}\sqrt{1-\alpha_{p-k}^2}}\Big( z b_{p-k}^\dagger a_p + \alpha_k \alpha_{p-k}\bar z a_{-p}^\dagger b_{p-k}^\dagger \Big) ,
\prod_{\ell\in I}|\cos(\pi\theta_{\ell})|=\frac{\prod_{\ell\in \tilde{I}}|\cos(\pi\theta_{\ell})|}{\prod_{\ell\in I^c}|\cos(\pi\theta_{\ell})|}\geq e^{-\varepsilon(2q_n-|I|)}e^{(-\ln 2)|I|}.
\hat{T}_n(q)=\sup\limits_{0<F_0(t)<1}\dfrac{\sqrt{n}|\mathbb{F}_n(t)-F_0(t)|}{q({\mathbb{F}_n(t)})},\hat{T}_n^+(q)=\sup\limits_{0<F_0(t)<1}\dfrac{\sqrt{n}(\mathbb{F}_n(t)-F_0(t))}{q({\mathbb{F}_n(t)})},
p_{N,k}^{P_m}=\prod_{l=1}^k\int_{-1}^{+1}\mathrm{d}\lambda_l\prod_{j=1}^{(N-k)/2}\int_{D_{+}}\mathrm{d}x_j\mathrm{d}y_j Q_{N,k}\left(\{\lambda_l\}_{l=1}^k,\{x_j\pm\mathrm{i}y_j\}_{j=1}^{(N-k)/2}\right),
C_i = \bigg( \prod_{j=1}^i g_j \bigg) \cdot s, D_i = \bigg( \prod_{\substack{j=1 j\ne i}}^{n-1}g_j \bigg) \cdots,\intertext{and finally}K = C_n = \bigg( \prod_{j=1}^n g_j \bigg) \cdot s.
0 = \sum_{m \in \mathcal{I}}\overline{\alpha_m}G_{k,m,0}(z) = \sum_{m \in \mathcal{I}}\xi_{2-k}\biggl(\alpha_m \frac{(4\pi m)^{1-k}}{k-1}F_{2-k,-m,0}(z)\biggr) &= \frac{(4\pi)^{1-k}}{k-1}\xi_{2-k}\bigg( \sum_{m \in \mathcal{I}}\frac{\alpha_m}{m^{k-1}}F_{2-k,-m,0}(z)\bigg).
qm_nc_1c_2c_3\dotsm m_n =m_nc_1c_2c_3\dotsm T_1m_n &=m_nc_1T_1c_1c_3\dotsm m_n &=m_n(T_1c_2+(q-1)(c_1-c_2))c_1c_3\dotsm m_n &=a(q-1)m_nc_3\dotsm m_n-m_nc_1c_2c_3\dotsm m_n
\left(W_{n}^{M}\right)'(t) = \lim_{h \searrow 0}\left[ \frac{W_{n}^{M}(t) - W_{n}^{M}(t-h)}{h}\right] = \lim_{h \searrow 0}\left[ \frac{W_{n}(x_{t},t) - W_{n}(x_{t-h}, t-h)}{h}\right] begin{align*}1ex] \le \lim_{h \searrow 0}\left[ \frac{W_{n}(x_{t},t)-W_{n}(x_{t},t-h)}{h}\right] = \partial_{t}W_{n}(x_{t},t),
\psi_l(k,r)=2^{-1/2}\,(-1)^m\,i^{|l|}\, e^{i \delta_l}\,(J_{\scriptscriptstyle{W}}(kr)\,\cos \Delta_{\scriptscriptstyle{l}}- Y_{\scriptscriptstyle{W}}(kr) \,\sin \Delta_{\scriptscriptstyle{l}})\,,
B^{q-2}[B,A] = |D_0|^{\frac{2-q}{3}}[|D_0|^{-\frac13},a^4] = -|D_0|^{\frac{1-q}{3}}[|D_0|^{\frac13},a^4]|D_0|^{-\frac13}= -[|D_0|^{\frac13},|D_0|^{\frac{1-q}3}a^4|D_0|^{-\frac13}].
p_{k-1}(z)=z^{k-1}+ \sum_{j=0}^{k-2}\beta_j z^j, &\beta_j=(-1)^{j+k-1}(-z_0)^{-\frac{3}{2}-j}\frac{(2j+1)!!}{2^j j!}, &z_0=- \left(\frac{2^{k-1}(k-1)!}{(2k-1)!!}\right)^{-\frac{2}{2k+1}},
L=-\rho\{\int_{-h(x,y)}^{\zeta(x,y,t)}[\frac{\partial\Phi}{\partial t}+\frac{1}{2}((\frac{\partial\Phi}{\partial x})^{2}+(\frac{\partial\Phi}{\partial y})^{2}+(\frac{\partial\Phi}{\partial z})^{2})]dz+\frac{1}{2}g(\zeta^{2}-h^{2})\},
H_3^{(3)}=c_1\int_{\mathbb{Z}}(z_x^3+3\,z_x^2\,v_x)\,dx+c_2\int_{\mathbb{Z}}(z_x^2\,z+z_x^2\,v+2\,v_x\,z_x\,z)\,dx+c_3\int_{\mathbb{T}}(z^3+v\,z^2)\,dx.
\underline{{{A}}}^{\mu}= \frac 1 \ell \underline{{{\Theta}}}^{\mu}+{\frac{1}{2}}\epsilon^{\mu}\_{\nu \rho}\underline{{{\Omega}}}^{\nu \rho}\qquad , \qquad \underline{{{\tilde{A}}}}^{\mu}= - \frac 1 \ell \underline{{{\Theta}}}^{\mu}+{\frac{1}{2}}\epsilon^{\mu}\_{\nu \rho}\underline{{{\Omega}}}^{\nu \rho}\qquad .
\beta_{1}\beta_{2}=\frac{(\sum_{i=1}^{n}(x_{i}-\overline{x})(y_{i}-\overline{y}))^{2}}{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}=r^{2}
\left\| \delta{X}\right\|_{\tilde{g}}^{2}= \int{d^{2}}\xi \sqrt{\tilde{g}}\delta{X}\cdot \delta{X}, \quad \int{\mathcal{D}_{\tilde{g}}}\delta{X}{e^{- \left\| \delta{X}\right\|_{\tilde{g}}^{2}}}= 1 .
S = \int_{M}d^{3}x \epsilon^{\sigma \mu \nu}\left[{B_{\sigma}}\!^{a}({F_{\mu \nu}}\!^{a}+ \epsilon^{a b c}{\phi_{\nu}}\!^{b}{\phi_{\mu}}\!^{c}) +{\eta_{\sigma}}\!^{a}D_{[ \mu}{\phi_{\nu ]}}^{a}\right] .
\Psi(\sigma_1,\sigma_2)=\sum_{s,t\in{\bf Z}}~ e^{is\sigma_1/(n-m\theta)}e^{-it\sigma_2/n}\sum_{i=sb}^{n+sb}~ \sum_{j=mbt}^{n+mbt}c_{s,t,i,j}~V'^{i}U'^{j}.
U_k = \frac{y}{10}+\alpha (k-n_0) - 10 \mathcal{S}_{\mathcal{L}}(x \mapsto \log(x))(k), L_k = - 10 y +\alpha (k-n_0) - \mathcal{S}_{\mathcal{L}}(x \mapsto x^{3/4})(k).
\mathrm{Mant}_{G,b,\mu}(\mathrm{Red}_b(I^G_{M_S}(\rho))) =[I^G_{M_S}(\rho)] \left[ \bigoplus_{(M_S, \mu_S) \in \mathrm{Rel}^{G, \mu}_{M_S, b}}r_{-\mu_S}\circ LL(\rho)|_{W_{\{\mu_S\}_{M_S}}}| \cdot|^{- \langle \rho_G, \mu \rangle}\right].
\chi^{a b}= \sum_{n = 1}^{\infty}( - 1 )^{n}\, \left( \Lambda^{n}\right)^{a b}- \frac{g \bar{\theta}^{b a}}{2 !}+ \sum_{n = 1}^{\infty}d_{n}\, \left[ ( g \bar{\theta})^{2 n}\right]^{b a}\, .
\lim_{T\rightarrow\infty}\overline{\Psi}_{f_{1}}\left(u,T\right)=\left(\prod_{i=1}^{n}\left|u_{2i-1}\right|\right)^{p+q-3}\left|u_{2n}\right|\int_{\left(S_{p}\times S_{q}\right)^{n}}f_{1}\left(\left\Vert g_{0}^{-1}\left(u_{1}\eta_{1},u_{1}\eta_{2}\dots,u_{2n}\eta_{2n}\right)\right\Vert \right)d\eta.
p_1=-\frac{v_1 (v_2-1)}{(v_1-1) (v_1 - v_2)},\p_2=\frac{v_1^2 - v_2}{(v_1-1) (v_1 - v_2)},\p_3=\frac{4 v_1^2 (v_1 + v_2^2) v_3}{(v_1 - v_2)^3 (v_1 v_2+1)}.
F_{1 2}= - \frac{M_{1}M_{2}}{r_{1 2}^{2}}+ \frac{Q_{1}Q_{2}}{r_{1 2}^{2}}+ \frac{P_{1}P_{2}}{r_{1 2}^{2}}- \frac{\S_{1}\S_{2}}{r_{1 2}^{2}}\ .
|F_p(u(t))| \le \|F(u(\cdot,t))\|_{L^2(\Omega)}\le \|F(0)\|_{L^2(\Omega)}+ K\|u\|_{L^2(\Omega)}&\le \underbrace{\|F(0)\|_{L^2(\Omega)}+ \max(K,1) \sup_{0 \le t \le T}\|u(.,t)\|_{L^2(\Omega)}}_{:=E},
- ( 2 - k z ) \left( 1 2 z^{2}h_{0}^{\prime}( z ) + 6 z h_{0}\right) - 6 q ( z ) = ( 2 - k z ) \left( - \frac{1}{6}z A^{\prime}( z ) + \frac{1}{1 2}A ( z ) \right) + \frac{1}{4}B ( z ) ,
g_i(x) = (\textup{row $i$ of $B$}) \cdot (g\circ S^{1-i})(x)+d' &= \sum_{j=1}^n b_j(g\circ S^{1-j-(1-(1+(i-1)k))})(x) +d'=\sum_{j=1}^n b_j(g\circ S^{1-j+(1-i)k})(x) +d',
\mathbb{C}[NCP(n)_0] \cong \mathbb{C}[\Pi_0(n)]/ \mathrm{ker}(p_\Pi) = \mathbb{C}[\Pi(n)_0]/\mathrm{ker}(h\circ \iota_\Pi) \cong \mathrm{im}(h\circ \iota_\Pi) \subset M_n \cong \mathbb{C}[NCM(n)]
E = \frac{\rho F}{2}\int_{0}^{l}d x \left( \dot{y}^{2}+ r^{2}\dot{\psi}^{2}\right) + \frac{E I}{2}\int_{0}^{l}d x \psi^{2}+ \frac{k F G}{2}\int_{0}^{l}d x \left( y^{\prime}- \psi \right)^{2}.
\displaystyle\langle u'(t),\varphi\rangle_{H^1(\Omega\setminus\overline{D})}+\int_{\Omega\setminus\overline{D}}\nabla u(x,t)\cdot\nabla\varphi(x)dx &- \langle{\rho}u(t),\,\varphi\vert_{\partial{D}}\rangle_{H^{1/2}(\partial D)}=\langle f(t),\,\varphi\vert_{\partial\Omega}\rangle_{H^{1/2}(\partial\Omega)}\,\,\,]0,\,T[
\delta S = \frac{i}{2}\int \{D_{\bar{\eta}}({I \! \! L}D_{\eta}f ({I \! \! N},{I \! \! R}, Z , \bar{Z}) ) + D_{\eta}({I \! \! L}D_{\bar{\eta}}f ({I \! \! N},{I \! \! R}, Z , \bar{Z}) ) \}d \eta d \bar{\eta}d t ,
\left\langle \prod_{j}T^{\mathrm{\scriptscriptstyle L}}( w_{j}) \prod_{k}\widetilde T^{\mathrm{\scriptscriptstyle L}}( \bar{w}_{k}) \prod_{r = 1}^{N}P ( z_{r}, \bar{z}_{r}) \right\rangle^{\! \! \! \mathrm{\scriptscriptstyle L}}\ .
Z_1^2 =\big(x\frac{\partial}{\partial x}-y\frac{\partial}{\partial y}\big)^2=x^2 \frac{\partial^2}{\partial x^2}+ x\frac{\partial}{\partial x}-2xy\frac{\partial^2}{\partial x\partial y}+y^2\frac{\partial^2}{\partial y^2}+y\frac{\partial}{\partial y},
W_{5}( p , q , r ; \psi , \phi , \eta ) = \left( \begin{array}{c c c c c}{{1 + \frac{p^{2}+ q^{2}+ r^{2}}{2}}}&{{p}}&{{q}}&{{r}}&{{- \frac{p^{2}+ q^{2}+ r^{2}}{2}}}\\{{p}}&{{}}&{{}}&{{}}&{{- p}}\\{{q}}&{{}}&{{R ( \psi , \phi , \eta )}}&{{}}&{{- q}}\\{{r}}&{{}}&{{}}&{{}}&{{- r}}\\{{\frac{p^{2}+ q^{2}+ r^{2}}{2}}}&{{p}}&{{q}}&{{r}}&{{1 - \frac{p^{2}+ q^{2}+ r^{2}}{2}}}\\ \end{array}\right)
|M_1\cap M_2|=|M_1\cap(M_2\cup\Psi)|=|(M_1\cup\Psi)\cap M_2| =|f(M_1)\cap f(M_2)|, |(M_1\cup\Psi)\cap(M_2\cup\Psi)| =|f(M_1)\cap f(M_2)|+2,
\lambda^{\#,(n,k)}_h = i \, R_1 - h^{\frac 23}a'_n e^{\pi i/3}+ h\, (2k-1)\, \frac{e^{-\pi i/4}}{\sqrt{2R_1}}& - h^{\frac 43}\frac{e^{\pi i/6}}{a'_n}\left(\hat \kappa + \frac{1}{2R_1}\right) + O(h^{\frac 53})\,.
\tilde \nabla_a \beta_b = \tilde \nabla_a \bar W_{Lb L \underline L}= \bar W_{a b L \underline L}+\bar W_{L\underline{L}L \underline L}\tilde \sigma_{ab}+ \bar W_{Lb a \underline L}- \frac{1}{2}\bar W_{Lb L a}= -\frac{3}{4}\sigma \epsilon_{ab}+ \frac{3}{2}\rho \tilde \sigma_{ab}- \frac{1}{2}\alpha_{ab}.
\zeta_{2d}^{h^W}= 2{2d \choose d}{2k - 2d \choose k -d}^2{2k \choose k}^{-2}(\zeta_1^h)^2 = 4{2k - 2 \choose k -1}^2{2k\choose k}^{-2}(\zeta_1^h)^2.
\phi_{\omega,k}(x)={1 \over 4\pi^3 \kappa^{1/2}}\left(\sinh (\pi \omega/\kappa)\right)^{1/2}~K_{i\omega/\kappa}(\rho(m^2+k_j^2)^{1/2})e^{-i\omega t}e^{-ik_jz^j}~~~,
E_{\varepsilon,\delta}(t,x,w,l,p,X)=\min\{E_*(t',x',w,l,p,X) \mid |t-t'|\le M\delta^{1/2},x'\in B_{x,\varepsilon}\}, &E^{\varepsilon,\delta}(t,x,w,l,p,X)=\max\{E^*(t',x',w,l,p,X) \mid |t-t'|\le M\delta^{1/2},x'\in B_{x,\varepsilon}\},
T_\psi^{A_0,A_1,A_2}(X,Y)=\sum_{i,j,k=1}^n \psi(\lambda_i^{(0)},\lambda_k^{(1)},\lambda_j^{(2)})P_{\xi_i^{(0)}}XP_{\xi_k^{(1)}}YP_{\xi_j^{(2)}}
\textbf{1}= \sum_{n = 0}^{\infty}a_{m_{1}}^{\dagger}\ldots a_{m_{n}}^{\dagger}\mid 0 \rangle \langle 0 \mid a_{m_{n}}\ldots a_{m_{1}}= \sum_{n = 0}^{\infty}\tilde{a}_{m_{1}}^{\dagger}\ldots \tilde{a}_{m_{n}}^{\dagger}\mid 0 \rangle \langle 0 \mid \tilde{a}_{m_{n}}\ldots \tilde{a}_{m_{1}}
\sigma(\tilde{t}+s, X (\tilde{t}, \tilde{x_1},\tilde{x_2}, \tilde{x_3}, \tilde{m}; s)) = 0\ \mbox{if}\ \M(\tilde{t}, \tilde{x_1},\tilde{x_2}, \tilde{x_3}, \tilde{m};\overline s_1 ) > \overline{m}_B .
+ \frac 5 8 F^{\lambda \nu}F^{\mu \rho}F_{\lambda \mu}F_{\nu \rho}+ 2 F_{\mu \rho}^{; \rho}F^{\mu \lambda}_{; \lambda}- 2 ( F_{\rho}^{\nu}F^{\rho \mu})_{; \nu \mu}+ ( F_{\nu \rho}F^{\nu \rho})^{; \mu}_{\mu}.
\frac 1{2 \pi i}\int_{c - i \infty}^{c + i \infty}d s \frac{( - s )}{\mu}\left( \frac{t}{\mu}\right)^{- s - 1}\frac{2^{s - 1}}{\sqrt{\pi}}\Gamma \left( \frac{s + 1}2 \right) \left[ \mu^{- s}\Gamma \left( \frac s 2 \right) \zeta \left( \frac s 2 , \frac{D_{B}}{\mu^{2}}\right) \right] ,
|\hat{Q}\cap U_Q| = |\hat{Q}\cap \tilde{U}| \ge |\hat{Q}\cap \tilde{U}\cap N_2(\Gamma)| \ge | (N_2(\Gamma) \setminus X) \cap \hat{Q}| - C(B\lambda m)^2 & \ge -CB^2\lambda^2 m^2 + C\bar{c}^2Bm \lambda^2 \ge cm \lambda^2
P_e^{(n)}\leq P\left\{\mathcal{F}_1^c \cup \cdots \mathcal{F}_{k_n}^c \right\}+ \frac{1}{M^{k_n}}\sum\limits_{A^{\ast}}\sum\limits_{k=1}^{k_n}P\{\mathcal{E}_{k}| A^{\ast}\}.
\mathrm{T r}( L^{2}) = 2 ( p_{1}^{2}+ p_{2}^{2}) + 4 m^{2}e^{q_{1}- q_{2}}+ 2 m^{2}\left( e^{2 q_{2}}+ e^{- 2 q_{1}}\right) = 4{\cal H}_{C_{2}}^{( 1 )},
L_S d_W(\mathbb{\hat{Q}}_{K_S}, \mathbb{P}) \geq L_S \left ( \mathbb{E}_{\mathbb{\hat{Q}}_{K_S}}\left[\frac{1}{L_S}u_S(\xi)\right] - \mathbb{E}_{\mathbb{P}}\left[\frac{1}{L_S}u_S(\xi)\right] \right ),
\left(\frac{e^{\theta_n u}-1}{e^{\theta_n}-1}\right)^{\alpha+\beta}=u^{\alpha+\beta}\left( \frac{e^{\theta_n u}-1}{\theta_n u}\right)^{\alpha+\beta}\left( \frac{\theta_n}{e^{\theta_n}-1}\right)^{\alpha+\beta},
\|Ux\|^p=\left\|\sum_{j\in \mathbb{J}}\widehat{F}_jA_jx\right\|^p =\left\|\sum_{j\in \mathbb{J}}\widehat{F}_j\widehat{L}_j\left(\sum_{k\in \mathbb{J}}L_kA_kx\right)\right\|^p=\sum_{j\in \mathbb{J}}\left\| \widehat{L}_j\left(\sum_{k\in \mathbb{J}}L_kA_kx\right)\right\|^p &=\sum_{j\in \mathbb{J}}\|A_jx\|^p=\|x\|^p,
\begin{array}{c}\varGamma_{2}=\left\{\left(\omega_{1},\omega_{2}\right)\left|\omega_{1}T_{0}=2k_{1}\pi,\omega_{2}\left(T-T_{0}\right)=2k_{2}\pi,k_{1}\in\mathbb{Z},k_{2}\in\mathbb{Z}\right.\right\}.\end{array}
\begin{aligned}|\mathcal{K}^2(u_h - v,f,\theta_f)| \leq C\left(h^k\|u_h - u\|_{\infty,I_h}+ h^{k+1}\|u\|_{W^{1,\infty}(I)}\right)\|f\|_{k+1,\Omega}\|\theta_f\|_{0,\mathcal{T}_h}.\end{aligned}
\nabla \Phi_{\gamma_{1}\ldots \gamma_{k}\dot{\gamma}_{1}\ldots \dot{\gamma}_{k}}= i e^{\alpha \dot{\alpha}}\Phi_{\gamma_{1}\ldots \gamma_{k}\alpha \dot{\gamma}_{1}\ldots \dot{\gamma}_{k}\dot{\alpha}}- i k^{2}e_{( \gamma_{k}| ( \dot{\gamma}_{k}|}\Phi_{| \gamma_{1}\ldots \gamma_{k - 1}) | \dot{\gamma}_{1}\ldots \dot{\gamma}_{k - 1})}.
R_{1 p+1}(z) R_{2 p+1}(z\tau^2 )\cdots R_{p p+1}(z\tau^{2p-2})v^{(i_1 \cdots i_p )}\otimes v_i=\prod_{j=1}^{p-1}b(z\tau^{2j}) v^{(i_1 \cdots i_p )}\otimes v_i .
\min_{\{\theta_{k,m}\}, t\in \mathbb{R}_{+}}t\left({\frac{1}{\omega}\sum\limits_{k=1}^K P_k+P_c+\kappa\phi\left(B\sum\limits_{k=1}^K\sum\limits_{m=1}^M\theta_{k,m}\right)}\right) s.t. t(B\sum\limits_{k=1}^K \sum\limits_{m=1}^M\theta_{k,m})=1
\Gamma(U_{\sigma\tau}, \mathcal{E}xt^i(I,J\otimes M)) = \Gamma(U_\sigma \cap U_\tau, \mathcal{E}xt^i(I|_{U_{\sigma\tau}},(J\otimes M)|_{U_{\sigma\tau}})) & = \Gamma(U_{\sigma\tau}, \mathcal{E}xt^i(\O,M))
\overline{(x_1, x_2,\ldots, x_t)}^{v, p}= (x_1, x_2,\ldots, x_{a-1}, \overline{x_{\overline{v}^{\{t\}}}+ p}^{\{r\}},x_{a+1},\ldots, x_t) &= (x_1, x_2,\ldots, x_{a-1}, \overline{x_{a}+ p}^{\{r\}},x_{a+1},\ldots, x_t).
\sum_{n=0}^\infty \frac{|x|^{n/s}|D^\beta f(x)|^{1/s}}{n!}\left(\frac{A}{s}\right)^n = \exp\left( \frac{A}{s}|x|^{1/s}\right) |D^\beta f(x)|^{1/s}\leqslant | f |_A^{1/s}\beta! A^{-|\beta|/s},
\dfrac{d}{ds}Z(Q^{-1}, s)\Big|_{s=0}= C + \Big[ \dfrac{\pi^s}{\Gamma(s)}F'_Q(s)+ \dfrac{\pi^s \log \pi}{\Gamma(s)}F_Q(s) +\pi^s \, \dfrac{d}{ds}\Big(\dfrac{1}{\Gamma(s)}\Big) F_Q(s) \Big]_{s=0}.
|H_0| \sqrt{V^2+ V^4 |\nabla \tau|^2 +\frac{(div V^2 \nabla \tau)^2}{|H_0|^2}}= A \Big [ \frac{2}{r}+ \frac{2AB_i \tilde{X}^i}{A^2}+ r(h_0^{(1)}+ \frac{A^2+(C_i \tilde X^i)^2 +g_1+\frac{g_2}{4}- h_0^{(1)}\sum_{ij}C_iC_j \tilde{X}^i \tilde{X}^j}{A^2}) \Big ] + O(r^2).
{\rm d}\big(|u^\varepsilon|^2\big)= 2u^\varepsilon{\rm d}u^\varepsilon+({\rm d}u^\varepsilon)^2 = 2u^\varepsilon \left(u^\varepsilon_{xx}{\rm d}t+\left(x+\varepsilon\right)^{2\gamma}u^\varepsilon_{yy}{\rm d}t+\frac{\sigma}{(x+\varepsilon)^2}u^\varepsilon{\rm d}t+f{\rm d}t+F{\rm d}B(t)\right)+|F|^2{\rm d}t.
\mathbb{P}( \Pi_1 = 1, \Pi_2 = 2 ) = \mathbb{P}(Y_1 = 1, \Pi_1 = 1, \Pi_2 = 2 ) + \mathbb{P}( Y_1 \ge 2 , \Pi_1 = 1, \Pi_2 = 2 ) &= (1-q) \cdot \frac{1-q}{q}\lambda_1(q) + \sum_{y=2}^\infty q^{y-1}( 1-q) \frac{1}{y ( y - 1)}&= \frac{(1-q)^2}{q}\lambda_1(q) + \frac{1 - q}{q}\lambda_2(q),
f(x;\frac{1}{3},1,1,0)=\frac{1}{\pi}\frac{2\sqrt{2}}{3^{\frac{7}{4}}}\frac{1}{\sqrt{x^{3}}}K_{\frac{1}{3}}(\frac{4\sqrt{2}}{3^{\frac{9}{4}}}\frac{1}{\sqrt{x}})
S(x) \ll e^{\varepsilon x}{\sum_p}^* \left|\sum_{j=1}^m a_{L_j}(p)\Delta_j(\log p)\right|\sum_{j=1}^m |\Delta_j(\log p)| &\ll e^{\varepsilon x}{\sum_p}^* \left|\sum_{j=1}^m a_{L_j}(p)\Delta_j(\log p)\right|\sum_{j=1}^m |\Delta_j(x^{(j)}_0)|+O(e^{(-A+1+\varepsilon-\sigma_1)x}).
u_{t}= \mathrm{div}\,\big(g(v) \nabla u\big) (0, \infty) \times G, v_{t}+ v = F\big(|\nabla u|\big) (0, \infty) \times G, \frac{\partial u}{\partial \mathbf{n}}= 0 (0, \infty) \times \Gamma, u(0, \cdot) = u^{0}, v(0, \cdot) = v^{0}\Omega
s(x,y)=\begin{cases}-(1,0) \ \ \mbox{if}\x\geq 1, -(x,y) \ \ \mbox{if}\ (x-\frac{1}{2})^2 + y^2 \leq \frac{1}{4}, \frac{-1}{(x-1)^2+y^2}(y^2,y(1-x)) \ \ \ \mbox{else}.\end{cases}
Q =[e_1 \e_3 \e_4 \ \mbox{-}e_2], Q^2 =[e_1 \e_4 \ \mbox{-}e_2 \ \mbox{-}e_3], Q^3 =[e_1 \ \mbox{-}e_2 \ \mbox{-}e_3 \ \mbox{-}e_4], Q^4 = [e_1 \ \mbox{-}e_3 \ \mbox{-}e_4 \e_2], Q^5 =[e_1 \ \mbox{-}e_4 \e_2 \e_3], Q^6 =[e_1 \e_2 \e_3 \e_4].
\widetilde{\Delta}(\widehat{x}_{i})=\widehat{x}_{i}\otimes e^{\frac{\widehat{p}_{0}}{\kappa}},\widetilde{\Delta}(\widehat{x}_{0})=\widehat{x}_{0}\otimes1+\frac{1}{\kappa}\widehat{x}_{i}\otimes e^{\frac{\widehat{p}_{0}}{\kappa}}\widehat{p}_{i}.
(\mu+1-s)\int_0^{\infty}e^{-(\mu+1)t}(\sinh t)^{-s}\,dt=\frac{2^{s}\Gamma(1-s)\Gamma\big(\frac{\mu}{2}+\frac{1+s}{2}\big)}{\Gamma\big(\frac{\mu}{2}+\frac{1-s}{2}\big)}.
\mathbf{a}_{\mathrm{h}}\left(\phi,\theta\right) = \frac{1}{\sqrt{M_{\mathrm{h}}}}\left[1, \,\, e^{j2\pi \frac{d_{\mathrm{h}}}{\lambda}\cos\phi\sin\theta}, \,\, \right. & \left. \cdots, \,\, e^{j2\pi \left(M_{\mathrm{h}}-1\right)\frac{d_{\mathrm{h}}}{\lambda}\cos\phi\sin\theta}\right]^{\mathrm{T}},
Q_1 \approx \frac{4(\lambda_c\pi)^{t}}{(t-m-1)!(m-1)!}\sum^{t-m-1}_{p=0}(-1)^p{t-m-1 \choose p}\times \sum^{N}_{l=1}\frac{\pi\left(\tau_2-\tau_1\right)}{2 N}f_m\left(\frac{\tau_2-\tau_1}{2}w_l+\frac{\tau_2+\tau_1}{2}\right)\sqrt{1-w_l^2}.
\sum_{\lambda}\epsilon^\mu(\lambda) \epsilon^{*\nu}(\lambda) =-g_\perp^{\mu \nu}+{1 \over q^+}\bigl(\eta^\mu q_\perp^\nu +\eta^\nu q_\perp^\mu \bigr) +{{\bf q}_\perp^2 \over (q^+)^2}\eta^\mu \eta^\nu \;,
(D^n_1)_{n-i,n-j}=(-1)^{n+j}\sqrt{2}\sqrt{2(n-i)+1}\sqrt{2(n-j)+1}\frac{(-n+j)_{n-i}(n-j+1)_{n-i}}{(2n-i+1)!}&\times\sum_{l=0}^{i-j}\frac{(j-i)_l(2n-i-j+1)_l}{(2n-i+2)_l l!}\prod_{m=0}^{\frac{i-1}{2}}(i-l-2m)\prod_{m=0}^{\frac{i-3}{2}}(2n-i+l-2m).
\lim_{k\rightarrow+\infty}(\mathbf{P}_{\mathcal{B}|\mathcal{A}}\mathbf{P}_{\mathcal{A}|\mathcal{B}})^{k}\cdot V_{1}(f)=\mathbf{0}_{m_{1},1},\;\lim_{k\rightarrow+\infty}(\mathbf{P}_{\mathcal{A}|\mathcal{B}}\mathbf{P}_{\mathcal{B}|\mathcal{A}})^{k}\cdot V_{2}(f)=\mathbf{0}_{m_{2},1}.
F (2R_2) = e^{2\xi_1}\left( e^{-2\xi_1}-e^{- 2\frac{(2^{2R_2}-1)}{\rho\alpha_2^2}}\right) \approx \left( 2\frac{(2^{2R_2}-1)}{\rho\alpha_2^2}-2\xi_1\right)=\frac{\gamma}{\rho},
\delta S = \frac{i}{2}\int \{D_{\bar \eta}({I\!\!L}D_{\eta}f({I\!\!N},{I\!\!R}, Z, \bar Z)) + D_{\eta}({I\!\!L}D_{\bar \eta}f({I\!\!N},{I\!\!R}, Z, \bar Z)) \}d\eta d\bar\eta dt,
eT_{\mu\nu}= e\Theta_{\mu\nu}+ \frac{1}{2}g_{\mu\nu}\left({\tilde{\overline\Psi}}_{L_0}\frac{\delta{\tilde S}_F}{\delta{\tilde{\overline\Psi}}_{L_0}}+{\tilde\Psi}_{L_0}\frac{\delta{\tilde S}_F}{\delta{\tilde\Psi}_{L_0}}\right).
-f(ae_x)g(axe_y)+f(ae_y)(g(ae_x)-g(ae_y)+g(axe_y)) &\qquad-g(ae_x)(f(ae_xy)-f(ae_x)+f(ae_y))+g(ae_y)f(ae_{x}y) &=(f(ae_y)-f(ae_x))g(axe_y)-h(ae_y)+h(ae_x)+(g(ae_y)-g(ae_x))f(ae_{x}y) &=(f(axe_y)-f(ae_xy))g(axe_y)-h(ae_y)+h(ae_x)+(g(axe_y)-g(ae_xy))f(ae_{x}y) &=h(axe_y)-h(ae_y)+h(ae_x)-h(ae_{x}y)=(d^*h)(ae_xe_y),
\begin{aligned}a_j = - \overline{A}_{z_j}- \overline{B}_{z_j}- \bar{r}_{z_j}, & b = \overline{A}_{z_1}+ \overline{B}_{z_1}+ \bar{r}_{z_1}, & c = a_j (A_{z_1}+ B_{z_1}+ r_{z_1}) + b (A_{z_j}+ B_{z_j}+ r_{z_j}) .\end{aligned}
(P_k)_j=\begin{bmatrix}\overbrace{0 \ldots 0}^{(k-1)}U_0(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\mu^{\frac{1}{2}}U_1(\alpha_r) \ldots \overbrace{0 \ldots 0}^{(k-1)}\mu^{\frac{n-k}{2k}}U_{\frac{n-k}{k}}(\alpha_r)\end{bmatrix}^{T}
\int_{0}^{\infty}s^n e^{-s^2+rs}ds = \frac{1}{2}\Gamma\left(\frac{n+1}{2}\right){}_1F_1\left(\frac{n+1}{2};\frac{1}{2};\frac{r^2}{4}\right) +\frac{r}{2}\Gamma\left(\frac{n}{2}+1\right){}_1F_1\left(\frac{n}{2}+1;\frac{3}{2};\frac{r^2}{4}\right)
\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}
f_X(x)=\frac{\theta}{\lambda}x^{\theta-1}e^{-\frac{x^{\theta}}{\lambda}}=\frac{1}{\lambda^{\frac{1}{\theta}}}I^{1,0}_{0,1}\Bigg[\frac{x}{\lambda^{\frac{1}{\theta}}}\left|\begin{matrix}\left(1-\frac{1}{\theta},\frac{1}{\theta},1\right) \end{matrix}\right.\Bigg],\ \x>0.
E=-\sum_{l=1}^\infty(2l+1)\int_{-\infty}^\infty{d\omega\over2\pi}e^{-i\omega\tau}\left\{\int_0^a r^2dr\,k^{\prime 3}(A_F+A_G)j_l^2(k'r)+\int_a^\infty r^2dr\,k^3(B_F+B_G)h_l^2(kr)\right\}.
|K(x,y)| \lesssim \int_0^\infty |S_t(x,y)|\frac{dt}{t}= \int_0^{d(x,y)}\,{1\over V(x,y,t+d(x,y))}\frac{t}{t+\|x-y\|}\frac{dt}{t}+\int_{d(x,y)}^{\|x-y\|}\,{1\over V(x,y,t+d(x,y))}\frac{t}{t+\|x-y\|}\frac{dt}{t}+\int_{\|x-y\|}^\infty{1\over V(x,y,t+d(x,y))}\frac{t}{t+\|x-y\|}\frac{dt}{t}=:I_1+I_2+I_3.
V(\log (f_{h_1}\circ f_{h_2}^{-1})') = V(\log((f_{h_1}'\circ f_{h_2}^{-1})\cdot ((f_{h_2}')^{-1}\cdot f_{h_2}^{-1}))) &= V((\log f_{h_1}'-\log f_{h_2}')\circ f_{h_2}^{-1}) &= V((F_{h_1}-F_{h_2})\circ f_{h_2}^{-1}+f_{h_1}'(0)+f_{h_2}'(0)) &= V(F_{h_1}-F_{h_2}) &= 4n_0+1>4n_0,
\theta_{\Lambda', P}= (a_{N+1}q^{N+1}+ a_{N+2}q^{N+2}+ \cdots)(1 + 2q + 2q^4 + \cdots) &= a_{N+1}q^{N+1}+ (2a_{N+1}+ a_{N+2})q^{N+2}+ \cdots
\begin{array}{c}\sum \limits_{i=1}^l \frac{\log(1+ \theta_i/\tau)}{\log(1+1/\tau)}\thickapprox \sum \limits_{i=1}^l \left\{\frac{\log(1+ \theta^{(k)}_i/\tau)}{\log(1+1/\tau)}+ \right. \left.\frac{\tau^{-1}}{(\log(1+\tau^{{-1}}))(1+\theta^{(k)}_i\tau^{{-1}})}(\theta_i-\theta^{[k]}_i) \right\}\end{array},
\frac{\partial^2h_{kk}^{(2)}}{\partial{(x^{i})}^2}-\frac{\partial^2h_{ki}^{(2)}}{\partial x^k\partial x^i}= \frac{1}{2(n-2)(n-1)}\left([nR_{,ii}+2(1-n)Ric_{kk,ii}-2(1-n)Ric_{ki,ki}-R_{,ii}\right].
W^{[ 2 , 3 ]}( \Phi^{1}, \Phi^{2}) ={\frac{( \Phi^{1})^{6}}{6}}- ( \Phi^{1})^{4}\Phi^{2}+{\frac{2}{3}}( \Phi^{1})^{2}( \Phi^{2})^{2}- ( \Phi^{2})^{3}- \mu \Phi^{1}~ .
Y=\left[\begin{smallmatrix}y_1^{1/2}&\ddots & y_n^{1/2}\end{smallmatrix}\right], C=\left[\begin{smallmatrix}\cos(\theta_1) &\ddots & \cos(\theta_n)\end{smallmatrix}\right], S=\left[\begin{smallmatrix}\sin(\theta_1) &\ddots & \sin(\theta_n)\end{smallmatrix}\right].