harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\deg( F_{{\bf v}_k})= \sum_{j=1}^{n_q}{\bf v}_{k,j}\deg(\mathfrak{F}_j)= \sum_{j=1}^{n_q}{\bf v}_{k,j}\deg\left(\phi^{j-1}\left(f_1f_2^2\cdots f_{\ell-1}^{\ell-1}\right)\right) \equiv \sum_{j=1}^{n_q}q^{k+1-j}\sum_{h=1}^{\ell-1}h \deg(\phi^{j-1}( f_h))\equiv \sum_{h=1}^{\ell-1}h \deg(f_h)\sum_{j=1}^{n_q}q^{k+1-j}\bmod{\ell}.
t_1^{I_1}t^{\lambda,n}t_{p+1}^{-I_1}\,(1-t_1)\, g_2 = \mathop{\underbrace{\mathop{\left( \sum_{u=0}^{n-I_1-1}t_1^{I_1}t_2^u t^{\lambda,n}t_{p+1}^{-I_1-u}\right)}}}_{f_2}(1-t_1) (1-t_2)\, g_3 + t_1^{I_1}t_2^{n-I_1}t^{\lambda,n}t_{p+1}^{-n}(1-t_1)g_2
\begin{cases}\partial_\tau\phi = -(-\Delta)^s_y\phi + pU^{p-1}(y)\phi + H[\lambda,\xi,\dot{\lambda},\dot{\xi},\phi](y,t(\tau)), y\in \mathbb{R}^n, \tau\geq \tau_0, \phi(y,\tau_0) = e_{0}Z_0(y), y\in \mathbb{R}^n,\end{cases}
\int_{\lambda(t)^{1-\alpha}}^{\frac{1}{2}}|\frac{-1}{4(\lambda(t)^{1-\alpha}+w)}+\frac{1}{4(\lambda(t)^{1-\alpha}+w)(1+w)^{3}}| dw &\leq C \int_{\lambda(t)^{1-\alpha}}^{\frac{1}{2}}\frac{w dw}{(\lambda(t)^{1-\alpha}+w)}&\leq C
\lim_{R\to \infty}\,\frac{1}{R}\,\#\left\{\lambda\in \Lambda,\,\,t\le \lambda\le t+R,\,\,\lambda\,({\rm mod}\,a^{-1})\in I \right\}&= \lim_{R\to \infty}\,\frac{1}{R}\,\#\left\{\lambda_n,\,\,M(t,R)\le n\le M(t,R)+N(t,R)-1,\,\,\lambda_n\,({\rm mod}\,a^{-1})\in I \right\}=|I|\,a\,A,
\frac{1}{n}\ln \Omega_{n}\sim -\frac{n+1}{2n}\ln \frac{n}{2}+\frac{1}{2}\ln\left( \pi e\right) -\frac{\ln 2\pi}{2n}-\sum_{j=1}^{\infty}\frac{2^{2j-2}B_{2j}}{j\left( 2j-1\right) n^{2j}}.
{\bf f}( x ) = \left( \begin{array}{c}{u_{-}( x )}\\{u_{+}( x )}\\ \end{array}\right) , \hat{x}^{\mu}= x^{\mu}{\bf I}, \hat{\pi}_{\mu}= - \imath \partial_{\mu}{\bf I}, \hat{\psi}^{a}= \frac{\imath}{2}\gamma^{a},
\begin{array}{ll}\mbox{minimize}\max_{e \in E}\{z_e + L(e)\}\mbox{subject to}[z_{e},z_{e}+ L(e) - 1]\cap[z_{f},z_{f}+ L(f) - 1] = \emptyset, (e,f) \in U & z_e \in \{1,2,\ldots \}, e \in E,\end{array}
\frac{\det \mathbf{B}_{n+1}\left( x\right)}{\det \mathbf{B}_{n}\left(x\right)}=\frac{\Omega_{n}^{n}\left( x-1\right)}{\left[ \Omega_{n}^{n}\left( \frac{x-1}{2}\right) \right]^{2}},\ \ \ \ \n\geq 1,
R_{\mu \nu \, \, b}^{\quad a}= \partial_{\mu}\omega_{\nu \, \, b}^{\, a}- \partial_{\nu}\omega_{\mu \, \, b}^{\, a}+ \omega_{\mu \, \, c}^{\, a}\ast \omega_{\nu \, \, b}^{\, c}- \omega_{\nu \, \, c}^{\, a}\ast \omega_{\mu \, \, b}^{\, c}
y_{1}^{\prime}=\frac{x_{1}^{\prime}}{x_{3}^{\prime}}=\frac{r_{1}\cdot(\tilde{x}-t)}{r_{3}\cdot(\tilde{x}-t)}=\frac{r_{1}\cdot(y-t/x_{3})}{r_{3}\cdot(y-t/x_{3})}
\frac{d\langle\epsilon^{2}\rangle}{dt}=lim_{dt\rightarrow0}\frac{\langle\epsilon^{2}(t+dt)\rangle-\langle\epsilon^{2}(t)\rangle}{dt}=-\frac{\Gamma}{3}\langle\epsilon^{2}\rangle+\frac{2\Gamma}{3}\langle\epsilon\rangle^{2}
\lambda_{2p+1}>2\lambda_{2p-1}\left({\lambda_{2p-1}+\lambda_2-\lambda_1 \over \lambda_{2p-1}+\lambda_2}\right) =\lambda_{2p-1}\left({\lambda_{2p-1}+\lambda_2-\lambda_1 \over \lambda_{2p-1}+\lambda_2}\right)+\lambda_{2p}\left({\lambda_{2p}+\lambda_{2p-3}-\lambda_{2p-2}\over \lambda_{2p}+\lambda_{2p-3}}\right).
[ c_s (\vec{k}), c_{s^\prime}(\vec{k}^\prime) ]_+ =[ c^\dag_s (\vec{k}), c^\dag_{s^\prime}(\vec{k}^\prime) ]_+ =[ d_s (\vec{k}), d_{s^\prime}(\vec{k}^\prime) ]_+ =[ d^\dag_s (\vec{k}), d^\dag_{s^\prime}(\vec{k}^\prime) ]_+ =0 .
(A-N-Y'')|_{Y'}=-\lfloor B_{Y'}+\varepsilon f^*E\rfloor+(B_{Y'}+\varepsilon f^*E)^{=1}&=\lceil -(B_{Y'}+\varepsilon f^*E)^{<1}\rceil -\lfloor (B_{Y'}+\varepsilon f^*E)^{>1}\rfloor.
\Omega_{\tau,t-s}= \sum_{i=0}^{f-1}p^i\left( a_{\tau \circ \varphi^i}+ e- 2s_{\tau \circ \varphi^i}\right) = \sum_{i=0}^{f-1}p^i (a_{\tau \circ \varphi^i}+1)(-1)^{\tau \circ \varphi^i \in J}+ \sum_{i=0}^{f-1}p^i (e -1 -2 x_{\tau \circ \varphi^i}),
\mathrm{d e t}{\cal O}= \left[ \frac{b}{\alpha r}\tilde{\lambda}^{2}+ \frac{b^{2}}{4 Q^{2}}\right]^{2}- \left[ \frac{2 Q_{M}b \alpha r}{4 Q^{2}}- 4 Q_{M}( 1 - \tilde{\lambda}^{2}) \right]^{2}.
E_{x}=[j\frac{k_{x\epsilon}k_{z}}{\omega\epsilon_{o}\epsilon_{r}}(Ae^{-jk_{x\epsilon}x}-Be^{jk_{x\epsilon}x})+\frac{m\pi}{a}(Ce^{-jk_{x\epsilon}x}+De^{jk_{x\epsilon}x})]sin(\frac{m\pi}{a}y)e^{-jk_{z}z}(33)
I_{1,a}\ge \int_{0}^{1/4c_r}\frac{2\sqrt{2}c_r(1-\delta)v}{\pi v}(1-v)^{\ell^2}\,dv= \frac{2\sqrt{2}c_r(1-\delta)}{\pi(\ell^2+1)}\left(1- \left(1-\frac{1}{4c_r}\right)^{\ell^2+1}\right).
\begin{array}{l}<\chi_{\pm}\mid\{{\hat{\gamma}}^{(\lambda,\mu,\alpha)}(p,s),{\hat{\gamma}}_{(\tau,\nu,\beta)}(p',s')\}\mid\chi_{\pm}>= =<e^{(\lambda,\mu,\alpha)}(p,s),e_{(\tau,\nu,\beta)}(p',s')>=\delta_{ss'}\delta^{\lambda}_{\tau}\delta^{\nu}_{\mu}\delta^{\alpha}_{\beta}\delta^{(3)}(\vec{p}-\vec{p'}).\end{array}
\frac{\partial}{\bar{\partial}\xi_j^{\alpha}}\left( \sum_{a,b=1}^n h_{ab}^j(z,t)\frac{\partial \xi_j^{\alpha}}{\partial z_a}\frac{\partial \xi_j^{\beta}}{\partial z_b}\right)=\sum_{a,b=1}^n \frac{\partial}{\bar{\partial}\xi_j^{\alpha}}\left(h_{ab}^j(z,t)\frac{\partial \xi_j^{\alpha}}{\partial z_a}\frac{\partial \xi_j^{\beta}}{\partial z_b}\right)=0
\phi_{k,n}(y_1,\cdots,y_k)=(n/4)^k(2\ln n)^{\frac{k}{2}}\times & \mathbb{P}(\xi^{(n)}(I_{n,k})=0,\ \xi^{(n)}(J_{n,k,j})>0,\ \forall\ 1\leq j\leq k).
\left( jO_{n}^{(3)}\right)^{2}+3JO_{n+3}^{(3)}\cdot jO_{n+3}^{(3)}=4^{n+3}\underline{\alpha}^{2}+\frac{3\cdot 2^{n+1}}{49}\left(25\underline{\alpha}\cdot \widehat{\epsilon_{n}}-31\widehat{\epsilon_{n}}\cdot \underline{\alpha}\right),
\hat{g}_{\theta,F_n}(x_i) =\frac{w_{1n}(\partial_x \int \mathrm{d}F_{1n})(x_i)}{1 - w_{2n}\int{f(y|x_i;\theta)}/{f_Y(y;\theta,\hat{g}_{\theta,F_n})}\,\mathrm{d}F_{2n}}, i \in V,
\bigl|K_{\theta_\ell,\theta_{3-\ell}}\bigl((y,\omega),(z,\eta)\bigr)\bigr| = \sqrt{\!\frac{w(\Phi(\eta))}{w(\Phi(\omega))}}\cdot L^{(\ell)}_{\Phi(\eta)}\bigl(A^T (\Phi(\eta))\langle z-y \rangle,\Phi(\omega) - \Phi(\eta)\bigr).
\Delta_{n_1,\ldots,n_r}({\boldsymbol{X}}_{n_1,\ldots,n_r}) = \sum_{j=0}^{N_r}e_{j}^{({\boldsymbol{q}})}({\boldsymbol{X}}_{n_1,\ldots,n_r}) t^{N_r-j}, \!e_{0}^{({\boldsymbol{q}})}({\boldsymbol{x}})=1,\!e_{-p}^{({\boldsymbol{q}})}({\boldsymbol{x}})=0, \!p > 0.
Z = \int{\cal D}{^+}G \,{\cal D}{^-}G \, \, d e t{\bf G}^{- 1 / 2}\, \, d e t ({^+}M )^{- 1 / 2}d e t ({^-}M )^{- 1 / 2}\, \, e^{- \int d^{4}x \tilde{L}}
\lim_{u\to-\infty}\frac{\partial}{\partial u}\left(e^{z\psi(s,u)}-1\right)=\exp\left\lbrace \frac{-2bz}{\sigma^{2}\left(e^{bs}-1\right)}\right\rbrace \lim_{u\to-\infty}\frac{ze^{-bs}}{\left(1-\frac{\sigma^{2}u}{2b}\left(1-e^{-bs}\right)\right)^{2}}=0.
\frac{K_{\nu/2}(s|\tilde{\omega}|)}{(s|\tilde{\omega}|)^{\frac{\nu}{2}}}= \frac{2^{\frac{\nu-2}{2}}\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\pi}|\tilde{\omega}|^{\nu}}\int_{\mathbb{R}}\left(s^2+\tilde{n}^2\right)^{-\tfrac{\nu+1}{2}}e^{i\tilde{\omega}\tilde{n}}~\mathrm{d}\tilde{n}
\varphi = \sum_{n \in \mathbb{Z}}\frac{1}{\sqrt{k_p}}A_n H^{(1)}_n(k_p r)^{in\theta_x}, \psi = \sum_{n \in \mathbb{Z}}\frac{1}{\sqrt{k_s}}B_n H^{(1)}_n(k_s r)^{in\theta_x}.
(y)_d = \prod_{i \in Q_0}(y)_{d_i}, \;\;\;\;\;\;\;\;\;\;\;(y)^{\sigma}_e = \prod_{i \in Q_0^{\sigma}}(y^2)_{\lfloor \frac{e_i}{2}\rfloor}\times \prod_{i \in Q_0^+}(y)_{e_i}
\sigma_{\mathrm{p}}^{(\omega )}\left( \mathbf{x},\mathbf{y},t\right):=\int\nolimits_{0}^{t}\varrho^{(\beta ,\omega ,\vartheta ,\lambda )}\left(i[I_{\mathbf{y}}^{(\omega ,\vartheta )},\tau_{s}^{(\omega ,\vartheta,\lambda )}(I_{\mathbf{x}}^{(\omega ,\vartheta )})]\right) \mathrm{d}s\ .
\begin{array}{l l}{D_{\mu}\phi = ( \nabla_{\mu}- i \frac{q}{2 \sqrt{\beta}}H_{\mu}) \phi ,}\\{D_{\mu}\sigma = ( \nabla_{\mu}+ i \frac{e}{2}A_{\mu}- i \frac{\alpha e}{2 \sqrt{\beta}}H_{\mu}) \sigma ,}\\ \end{array}
B^{\varepsilon ,\varphi^{\varepsilon}}(t)[\phi ]=\int_{\mathbb{X}\times\lbrack 0,t]}D_{x}\left( G(X^{0}(s),y)[\phi ]\right) \bar{Y}^{\varepsilon,\varphi}(s)\nu (dy)ds+\mathcal{E}_{2}^{\varepsilon ,\varphi^{\varepsilon},\phi}(t),
r ( \rho ) = \frac{( g a^{3})^{1 / 2}}{2}\sqrt{\frac{3}{5}}\Big [ \frac{2}{3}\rho_{\: \quad 2}^{2 / 3}F_{1}\big [ - \frac{9}{2 0}, \frac{1}{4}, \frac{1 1}{2 0}; \frac{l^{1 0 / 3}}{\rho^{1 0 / 3}}\big ] + 3 l^{3 / 2}\frac{\Gamma \left( - \frac{9}{2 0}\right) \Gamma \left( \frac{3}{4}\right)}{1 0 \Gamma \left( \frac{3}{1 0}\right)}\Big ] ,
\varphi ({\overline{{{\alpha V}}}},{\overline{{{\beta V}}}}) = ({\overline{{{\alpha V}}}}- W ({\overline{{{\alpha V}}}}) ) \cdot ({\overline{{{\beta V}}}}- W ({\overline{{{\beta V}}}}) ) + f ({\overline{{{\alpha V}}}},{\overline{{{\beta V}}}}) + \chi ({\overline{{{\alpha V}}}},{\overline{{{\beta V}}}}) ~ ,
8\rho^3 cos\theta -6s\rho^2-2l\rho^2cos(2\theta - \omega)+4m\rho cos(\theta-\alpha)-4d=0 8i\rho^3 sin\theta -6it\rho^2-2ih\rho^2sin(2\theta - \varphi)+4ik\rho sin(\theta+\mu)-4if=0
\int_0^{\infty}t^{-1-\alpha/2}\|tL_{\mu_2}(I+tL_{\mu_2})^{-1}f\|^2_{L^2(\mathbb{R}^n, \mu_2)}dt \leq C\int_{2B}\int_{2B}\frac{|f(x)-f(y)|^2}{d_g(x,y)^{n+\alpha}}\omega(x)\omega(y)dxdy.
|\eta^{k+1}G_k(s)|=\eta\Big|\sum_{p\in[N_0,N_1]}\frac{\Lambda_{\pi}(p)+\Lambda_{\pi\otimes\chi}(p)p^{\beta_{\chi}-1}}{p^{1+\eta+i\tau}}\frac{(\eta\log p)^k}{k!}\Big|+O\Big(\frac{m^2\eta\log(qQT)}{(110)^k}\Big).
T^{-1}(1 - x) = 1 - x - \sum_{\overline \pi_0(\beta)< \overline \pi_0(\alpha)}\lambda_\beta + \sum_{\overline \pi_1(\beta) < \overline \pi_1(\alpha)}\lambda_\beta & = 1 - x - \sum_{\pi_1(\beta)<\pi_1(\alpha)}\lambda_\beta + \sum_{\pi_0(\beta) < \pi_0(\alpha)}\lambda_\beta = 1 - T(x).
I = \int_{-{\frac{\pi}{2}}}^{\frac{\pi}{2}}d \mu \, f^{- \tilde{\gamma}}\left[ ( K^{\prime})^{2}+ ( \tilde{\gamma}^{2}- 9 / 4 ) K^{2}\right] >{-{\frac{9}{4}}}\int_{-{\frac{\pi}{2}}}^{\frac{\pi}{2}}d \mu \, f^{- \tilde{\gamma}}K^{2}\ ,
\lesssim\|(\tilde{d}_{N})^{2n}\|_{\ell_N^2}\|(\tilde{d}_{N}')^{m-2n}\|_{\ell_N^{2}}=\|\tilde{d}_{N}\|_{\ell_N^{4n}}^{2n}\|\tilde{d}_{N}'\|_{\ell_N^{2(m-2n)}}^{m-2n}&\leq \|\tilde{d}_{N}\|_{\ell_N^2}^{2n}\|\tilde{d}_{N}'\|_{\ell_N^2}^{m-2n}\lesssim \|d_{N}\|_{\ell_N^2}^{2n}\|d_{N}'\|_{\ell_N^2}^{m-2n}=\|d_{N}\|_{\ell_N^2}^{2n}\|d_{N}'\|_{\ell_N^2}^{(p-3)n}.
\Xi(x) = \zeta(1)^{-1}\biggl(\sum_{i=1}^\infty \bigl(-2i-v(\Delta'/\varpi)\bigr)(-i)t^i\biggr)\cdot (\log q)^2 = |u|^{-1}\frac{t \bigl(2 (1+ t) + v(\Delta'/\varpi)(1- t)\bigr)}{(1 - t)^2}\cdot (\log q)^2.
\left( R^{k}|_{S^{m}}f \right)_{p_1 q_1 \dots p_{m-k}q_{m-k}}^{i_{1}\dots i_{k}}= \left( R^{0}|_{S^{m-k}}f^{i_{1}\cdots i_{k}}\right)_{p_{1}q_{1}\cdots p_{m-k}q_{m-k}},
dd^c\max(u,v)\wedge T\wedge S_{|\{\max(u,v)>a\}}=dd^c\max(\max(u,v),a)\wedge T \wedge S_{|\{\max(u,v)>a\}}&=dd^c\max(u,v,a)\wedge T\wedge S_{|\{\max(u,v)>a\}}dd^cu\wedge T\wedge S_{|\{u>a\}}=dd^c\max(u,a)\wedge T \wedge S_{|\{v>a\}}.
e^{(1+\frac{\alpha}{2\beta})\sup_{X}\varphi_t(x)}\ = \e^{(1+\frac{\alpha}{2\beta})\varphi_t(x_{1})}\ \leq \ \frac{(t\omega_{0}-_{\omega_{0}}))^{n}}{\omega_{0}^{n}}\ \leq \C,
(\sum_{i=1}^{M}G_a(y_i\phi_i x)^p)' = \sum_{i=1}^{M}p\cdot G_a(y_i\phi_i x)^{p-1}\cdot(G_a(y_i\phi_i x))' = \sum_{i=1}^{M}p\cdot G_a(y_i\phi_i x)^{p-1}\cdot(-\frac{a}{2})G_a(y_i\phi_i x)(2-G_a(y_i\phi_i x))\cdot y_i\phi_i^T = -\frac{1}{2}ap\sum_{i=1}^{M}\phi_i^T\cdot [y_iG_a(y_i\phi_i x)^p(2-G_a(y_i\phi_i x))].
\langle \overline{{n_{m}, \ldots , n_{1}}}| = ( - 1 )^{\sum_{a}\bar{n}_{a}( a - 1 )}\langle \bar{n}_{m}, \ldots , \bar{n}_{1}| = \langle \bar{0}| ( \hat{\eta}^{m})^{n_{m}}\cdots ( \hat{\eta}^{1})^{n_{1}}
u_{2}(x)= K \left(\frac{1}{H^{o}(x)^{\frac{p-q}{q-(p-1)}}}- 1\right), K=\frac{q-(p-1)}{p-q}\left(\frac{(N-1)q-(p-1)N}{q-(p-1)}\right)^{\frac{1}{q-(p-1)}}.
\pi\|f\|_{S_n^p}=\pi\left(|f(0)|+|f'(0)|+\cdots+|f^{(n-2)}(0)|+\|f^{(n-1)}\|_{S_1^p}\right) &\geq |f(0)|+|f'(0)|+\cdots+|f^{(n-2)}(0)|+\|f^{(n-1)}\|_{\infty}&\geq |f(0)|+|f'(0)|+\cdots+|f^{(n-2)}(0)|+\|f^{(n-1)}\|_{H^p}&=\|f\|_{S_{n-1}^p}\,.
\begin{aligned}\int_{\mathbb{R}^N}g(x)|u_{n_k}(x)|^p dx \to \int_{\mathbb{R}^N}g(x)|u(x)|^p dx, \int_{\mathbb{R}^N}g(x)|u_{n_k}(x)|^{p-2}u_{n_k}(x)u(x) dx \to \int_{\mathbb{R}^N}g(x)|u(x)|^p dx. \end{aligned}
\| u^{*}(\varphi_n(f\otimes a) \oplus \overbrace{\bigoplus_{k=1}^n f(z_k)\sigma_n (a) \oplus \cdots \oplus\bigoplus_{k=1}^n f(z_k)\sigma_n (a)}^n) u & - \psi_n(f\otimes a)\oplus \overbrace{\bigoplus_{k=1}^n f(z_k)\sigma_n (a) \oplus \cdots \oplus \bigoplus_{k=1}^n f(z_k)\sigma_n (a)}^n\| < \varepsilon
U_1(x, a,b,\theta) = \frac{W_q(x)}{W_q(b)}U_1^0(a,b,\theta) - U_1^0(a,x,\theta), \theta \geq 0, U_i(x, a,b) = \frac{W_q(x)}{W_q(b)}U_i^0(a,b) - U_i^0(a,x), i = 2,3, 4.
H_{1n}(t,x,y^1,y^2,z^1,z^2)=\psi(\frac{y^1}{n},\frac{y^2}{n},\frac{z^1}{n},\frac{z^2}{n})\tilde{H}_{1n}(t,x,y^1,y^2,z^1,z^2)
|I_{21}| \leq \sum_{i=b+1}^n \sum_{k,j=1}^p \int_{0}^1 \mathbb{E}|\partial^2_{jk}F(\mathcal{Z}^{(i)}(t))| \left| \mathbb{E}[\dot{Z}_{ij}(t)V_k^{(i)}(t)] \right|dt & \leq C \phi(M_x, M_y) \psi^2 p^4.
\frac{\partial P_{p}}{\partial t}= \frac{1}{2}\frac{\partial}{\partial \phi}\left( \frac{H^{3 / 2}( \phi )}{2 \pi}\frac{\partial}{\partial \phi}\Bigl ( \frac{H^{3 / 2}( \phi )}{2 \pi}P_{p}\Bigr ) + \frac{V^{\prime}( \phi )}{3 H ( \phi )}P_{p}\right) + 3 H ( \phi ) P_{p}\ .
(K_H^{\ast}\varphi)(s) = \, c_H \Gamma\left( H+\frac{1}{2}\right)\left(D_{T^-}^{\frac{1}{2}-H}\varphi(s)\right) (s) &+ c_H \left( \frac{1}{2}-H\right)\int_s^T \varphi(t) (t-s)^{H-\frac{3}{2}}\left(1- \left(\frac{t}{s}\right)^{H-\frac{1}{2}}\right)dt.
\sqrt{\frac{K}{{\rm tr}\left(\hat{\bf H}^H{\bf F}{\bf F}^H{\bf F}{\bf F}^H\hat{\bf H}\right)}},\gamma_{ZF}=\sqrt{\frac{K}{{\rm tr}\left(\left(\hat{\bf H}^H{\bf F}{\bf F}^H \hat{\bf H}\right)^{-1}\right)}}.
\tilde q^{(ab)}(r)=\frac{1}{8\pi^2b^2r^2}\sum_{l=1}^{\infty}(2l+1)\sum_{\lambda =0,1}\int_{0}^{\infty}{z\tilde \Omega_{\lambda l}(z,\sigma )F_{\lambda l}^{(q)}(z,zy)dz},
J_{10}= \mathcal{O}\left( a^{2}\; \left\Vert V \right\Vert_{\mathbb{L}^{2}(D_{j_{0}})}\; d^{-4}\; \max_{j}\left\Vert \overset{3}{\mathbb{P}}\left(E^{T}_j\right) \right\Vert_{\mathbb{L}^{2}(D_j)}\right) \overset{(\ref{max-P1P3})}{=}\mathcal{O}\left( \left\Vert V \right\Vert_{\mathbb{L}^{2}(D_{j_{0}})}\; d^{-4}\; a^{\frac{10+h}{2}}\right).
\|v_n\|^2_{L^2}=\|u_n(t_n)\|^2_{L^2}=\|u_{n,0}\|^2_{L^2}\to \|\phi_{\omega}\|^2_{L^2}E(v_n)=E(u_n(t_n))=E(u_{n,0})\to E(\phi_{\omega}),
I_{m + 2, \alpha}(a, b) = \frac{m + 1}{\alpha}\frac{C_m}{\alpha^{m/2}}- \frac{m + 1}{\alpha}\sum_{i = 0}^{\lfloor \frac{m - 1}{2}\rfloor}\frac{\tilde C_{i, m}}{\alpha^{i + 1}}J_{m - 1 - 2i, \alpha}(a, b) - \frac{1}{\alpha}J_{m + 1, \alpha}(a, b),
P_{B}(\eta) : = \int_{B}\frac{F_r(\eta)}{1-r}\,dr &= \sum_{0 \le j<N(\eta)}\int_{\{2^{-(j+1)}<|1-r|<2^{-j}\}}\chi_{\{|1-r|<\delta_\eta \}}(r) \frac{F_r(\eta)}{1-r}\,dr .
Z = \int{\cal D}g_{i j}\,{\cal D}\pi^{i j}\,{\cal D}\theta^{a}\,{\cal D}\chi^{a}\delta ( ( M^{- 1})_{b}^{a}[ \theta ] \chi^{b}) e x p \int ( -{\cal H}^{'}+ i \pi^{i j}\partial_{0}g_{i j}+ i \chi^{a}\partial_{0}\theta^{a}) .
f_7f_{12}f_{15}t + f_1f_{10}f_{15}t = t f_7f_{26}\widehat{c}+ f_1f_{19}\overline{c}= t -f_Af_{25}\widehat{c}- f_Af_{20}\overline{c}= t f_Af_{25}\widehat{c}+ f_Af_{20}\overline{c}= -t.
\begin{cases}\partial_t \theta + v \cdot \nabla \theta =0, (t,x) \in (0,\infty) \times \mathbb R^2; v = D^{-1+\gamma}\nabla^{\perp}\theta=(-D^{-1+\gamma}\partial_2 \theta,D^{-1+\gamma}\partial_1 \theta), 0<\gamma<1; \theta(0,x)= \theta_0(x).\end{cases}
\{\mathrm{tr}(\rho(\alpha)),\mathrm{tr}(\rho(\beta))\}=\sum_{p\in \alpha\cap\beta}\epsilon(p,\alpha,\beta)\big((\mathrm{tr}(\rho(\alpha_p\beta_p))-(1/3)\mathrm{tr}(\rho(\alpha))\mathrm{tr}(\rho(\beta))\big)
V_1 = \mathrm{Span}_F\{e_{2m-n+1}, \ldots, e_{2m}, e_{-2m}, \ldots, e_{-2m+n-1}\}, V_2 = \mathrm{Span}_F\{e_1, \ldots, e_{2m-n}, e_{-2m+n}, \ldots, e_1\}.
\dot{\bar{x}}(t) = \bar{A}\bar{x}(t) + \bar{B}u(t) + \displaystyle \sum_{j=1}^{n_{\rm in}}u_j(t) \bar{N}_j \bar{x}(t) + \bar{H}( \bar{x}(t) \otimes \bar{x}(t) ) \bar{y}(t) = \bar{c}^\top \bar{x}(t)
f = \begin{pmatrix}0 1 \end{pmatrix}, f'_1 = \begin{pmatrix}2 1 \end{pmatrix}, f'_2 = \begin{pmatrix}6 1 \end{pmatrix}, g = \begin{pmatrix}0 0 1 0 0 1 \end{pmatrix}, g'_1 = \begin{pmatrix}2 1 0 0 0 1 \end{pmatrix}, g'_2 = \begin{pmatrix}6 1 0 0 0 1 \end{pmatrix}.
\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{(\frac{b}{a})_{n}(q)_{n}(a)_{N-n}a^{n}}{(1-cq^{n})(b)_n(a)_N}=\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]\frac{(\frac{b}{c})_{n-1}(q)_n (cq)_{N-n}c^{n-1}}{(b)_{n-1}(cq)_N}\left(\frac{aq^{n-1}}{1-aq^{n-1}}-\frac{bq^{n-1}}{1-bq^{n-1}}\right),
(\lambda-A)^{-1}C (\lambda-B)^{-1}&= \begin{pmatrix}(\lambda-A_m)^{-1}C_{m,n}(\lambda-B_m)^{-1}(\lambda-A_m)^{-1}C_{m,\infty}(\lambda-B_\infty)^{-1}(\lambda-A_\infty)^{-1}C_{\infty,n}(\lambda-B_n)^{-1}(\lambda-A_\infty)^{-1}C_{\infty,\infty}(\lambda-B_\infty)^{-1}\end{pmatrix}.
\begin{cases}\partial_{t}z_{k}-\Delta z_{k}=-\mathrm{div}(u^{c,\gamma}\otimes z_{k-1}+z_{k-1}\otimes u^{c,\gamma}+w_{1}\otimes w_{2})-\nabla\pi_{k-1}, \mathrm{div}z_{k}=0, \pi_{k-1}=(-\Delta)^{-1}\partial_{i}\partial_{j}(u^{c,\gamma}\otimes z_{k-1}+z_{k-1}\otimes u^{c,\gamma}+w_{1}\otimes w_{2}), z_{k}(x,0)=0.\end{cases}
\widetilde{\omega}^{-, \iota}_n := \prod_{1 \leq m \leq n, m: \textrm{odd}}\Phi_m \left(\frac{1}{1+X}\right) = \prod_{1 \leq m \leq n, m: \textrm{odd}}\left( \Phi_m (1+X) \cdot (1+X)^{-p^{m-1}(p-1)}\right) = \left( \prod_{1 \leq m \leq n, m: \textrm{odd}}(1+X)^{-p^{m-1}(p-1)}\right) \cdot \widetilde{\omega}^{-}_n .
i=i_1+i_2\cdot n_1+i_3\cdot n_1n_2, &j = j_1+j_2\cdot (N_1-n_1+1)+j_3\cdot (N_1-n_1+1)(N_2-n_2+1), &l_k=i_k+j_k, ~1\leq k \leq 3.
(A_{n,4}V_{n,4}+ \epsilon V_{n,4})(\zeta, \eta,y,x) \leq K_4- K_{4,\zeta}\; \zeta^2- K_{4,\eta}\; \eta^2- K_{4,y}\; y^2 - K_{4,x}\; x^2 + \delta |y| + |x|.
C_{2}^{+}(\beta)=\frac{\alpha}{2}log(1+c_{31}^{2}P_{1}^{(1)})+\frac{1-\alpha}{2}log(1+c_{31}^{2}P_{1}^{(2)}+c_{32}^{2}P_{2}+2\sqrt{\beta C_{31}^{2}P_{1}^{(2)}C_{32}^{2}P_{2}})
\omega^{i_{p}}(g,\lambda) = \sum_{\alpha=0}^{\infty}\lambda^\alpha \omega^{i_{p},\alpha}(g) = \omega^{i_{p},0}(g) +\lambda \omega^{i_{p},1}(g) + \lambda^2 \omega^{i_{p},2}(g) +\ldots \; ; \quad p=0,1 \; ,
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\frac{1}{l^4}\big [40+\frac{48M^2z^4}{\Sigma^6}(r^2-a^2 cos^2{\theta})(r^4-14a^2r^2cos{\theta}+a^4cos^4{\theta})\big ]
\tilde{\psi}_i(u_i)=\sum_{n=0}^{\infty}\frac{\lambda^n}{2^n n!}\oint_{C_1}\frac{dx_1}{2\pi}\dots \oint_{C_n}\frac{dx_n}{2\pi}\psi_i(_i)\psi(x_n)\dots \psi(x_1) \ .
K_{\nu ,m}(z,w):=\frac{(2m+2\nu +n)\Gamma (m+n+2\nu )}{\pi^{n}\Gamma(m+2\nu +1)}\left( \frac{\left( 1+\langle w,z\rangle \right)^{2}}{(1+|z|^{2})(1+|w|^{2})}\right)^{\nu}P_{m}^{(n-1,2\nu )}(\cos 2d_{FS}(z,w)),
\left. \int_{K}n \varphi \,dx \right|_{t_0}^{t_1}+\int_{t_0}^{t_1}\int_{K}\left\{-n\varphi_{t}+ \nabla n^{1 +\alpha}\nabla \varphi + B n \nabla \varphi \right\}\,dx\,dt = 0
d s^{2}= \left( \frac{d r^{2}}{2 M r \pm ( 2 M r )^{2 \beta / ( 2 \beta - 1 )}}- \left[ 2 M r \pm ( 2 M r )^{2 \beta / ( 2 \beta - 1 )}\right] d t^{2}) \right) \left( 1 \mp \beta m ( 2 M r )^{1 / ( 2 \beta - 1 )}) \right)
|\partial_{t}^{2}\left(-\lambda(t) \langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}}(r))-1}{r^{2}}\right)v_{5}\left(1-\chi_{\geq 1}(\frac{4r}{t})\right)\vert_{r=R\lambda(t)},\phi_{0}\rangle\right)| \leq \frac{C}{t^{4}\log^{3N-2}(t)}
\left[ \widetilde{I}_\mu ,I_\nu \right] =2\widetilde{I}\delta_{\mu \nu},\hspace{0.3in}\left[ I_\mu ,I_\nu \right] =4I_{\mu\nu},\hspace{0.3in}\left[ \widetilde{I}_\mu ,\widetilde{I}_\nu\right] =-4I_{\mu \nu},
E(z) = \frac{1}{\sqrt{1-z}}[(1-\mu) - \mu \sqrt{1-z}] \frac{1}{1- 2\mu + \mu^2 z}&= \frac{\frac{1-\mu}{\sqrt{1-z}}-\mu}{1- 2\mu}\sum_{k=0}^{\infty}\left( \frac{\mu^2}{1-2 \mu}\right)^k(- z)^k.
S_n'(y_1'- S_k'(y_1') ) = S_n' \Big[\frac{1}{k+1}\sum_{l=0}^{k}\Big( y_1' - \sigma_l'(y_1') \Big) \Big] &= \frac{1}{k+1}\sum_{l=0}^{k}\Big[ S_n' \Big( y_1' - \sigma_l'(y_1') \Big) \Big].
\begin{array}{l}\mathbb{E}[<\mu_t^e,\hat{\psi}_0> f_2(e_t)]= \mathbb{E}[\mathbb{E}[<\mu_t^e,\hat{\psi}_0>|e]f_2(e_t)]= \mathbb{E}[\mathbb{E}[<m_0,\psi_t^{t,e}>|e]f_2(e_t)]\end{array}
c_{n}=\sum_{i=0}^{n}\frac{x^{i}}{i!}\frac{y^{n-i}}{(n-i)!}=\frac{1}{n!}\sum_{i=0}^{n}(\begin{matrix}n\\i\end{matrix})x^{i}y^{n-i}=\frac{(x+y)^{n}}{n!}
T^{{\underline{a}}}={\cal D}E^{{\underline{a}}}= -{\frac{i}{2}}\hat{E}^{\hat{\underline{\alpha}}}\wedge \hat{E}^{\hat{\underline{\beta}}}\Gamma_{{\underline{{\hat{\alpha}\hat{\beta}}}}}^{{\underline{a}}}= -{\frac{i}{2}}( \hat{E}\Gamma^{{\underline{a}}}\hat{E})
{\cal L}= \frac{1}{2 g l_{s}}\mathrm{T r}\left[ \dot{X}^{a}\dot{X}^{a}+ \frac{1}{2}[ X^{a}, X^{b}]^{2}+ \theta^{T}( i \dot{\theta}- \Gamma_{a}[ X^{a}, \theta ] ) \right]
\begin{array}{cccccccccl}\ \ \ \ g \ \ \ \{\rm Level}\ 0 \ \ \ \swarrow \ \searrow \ \ \ CS \sim KM_k \ \ \ g_q \ \{\rm Level}\ 1 \ \swarrow \ \searrow \ \swarrow \ \searrow \ ? \ \ \ KM_{k,q}\ \ \ g_{q,k}{\rm Level}\ 2\end{array}
W_{1 3 2 4}\equiv \langle F_{a b}( x_{1}) F_{c d}( x_{3}) \rangle \langle F_{c d}( x_{3}) F_{e f}( x_{2}) \rangle \langle F_{e f}( x_{2}) F_{g h}( x_{4}) \rangle \langle F_{g h}( x_{4}) F_{a b}( x_{1}) \rangle
\delta U=-\int_{0}^{T}\{\int_{\Omega^{0}}[N_{\alpha\beta,\alpha}\delta u_{\beta}^{0}+M_{\alpha\beta,\beta\alpha}\delta w^{0}]dA-\int_{\Gamma^{0}}[n_{\alpha}N_{\alpha\beta}\delta u_{\beta}^{0}+n_{\alpha}M_{\alpha\beta,\beta}\delta w^{0}+n_{\beta}M_{\alpha\beta,\alpha}\delta w^{0}]ds\}dt
\theta^{*}=argmax_{\theta\in\Theta_{H}}\{\frac{\pi_{n}(\theta)}{r(\theta)}\},s_{n}^{*}=max_{\theta\in\Theta_{H}}\{\frac{\pi_{n}(\theta)}{r(\theta)}\}=\{\frac{\pi_{n}(\theta^{*})}{r(\theta^{*})}\}.
|u_n(\lambda_n)| \geq r(u_n+\theta H^\infty)\geq \frac{1}{\|\Phi^{-1}\|}\|u_n+\theta H^\infty\| &= \frac{1}{\|\Phi^{-1}\|}\|u_n(S_{\theta})\|= \frac{1}{\|\Phi^{-1}\|}
{\frac{2}{\alpha}}k_{\rho}\tilde{\Pi}_{\mu \nu , \rho \sigma}= -{\frac{k_{\sigma}}{2}}\left( \eta_{\mu \nu}-{\frac{k_{\mu}k_{\nu}}{k^{2}}}\right){\frac{1}{1 6 \pi^{2}}}\mathrm{l n}{\frac{\Lambda^{2}}{\mu^{2}}}\neq 0 .
I\!I_1 =\bigg( \int_{\|y-x_0\|\leqslant \frac{R}{4}}+\int_{d(y,x_0)\leqslant \frac{R}{4}\leqslant \|y-x_0\|}\bigg)[K(x,y)-K(x,x_0)]f_2(y)d\omega(y) &=: I\!I_{11}+I\!I_{12}.
|\!|\!| f |\!|\!|^{2}:= \iiint B(v-v_*,\sigma) \mu_* (f'-f)^2 dvdv_*d\sigma &+ \iiint B(v-v_*,\sigma) f^2_* (\sqrt{\mu'}-\sqrt{\mu})^2 dvdv_* d\sigma = J_1^{\Phi_\gamma}(g) + J_2^{\Phi_\gamma}(g)\,,
\left\|\sum_{k=1}^{\infty}A_{\overline{\i}}^{k-1}x\right\|\geq \|x\|-\left\|\sum_{k=1}^{\infty}A_{\overline{\i}}^{k}x\right\|\geq \|x\|-\sum_{k=1}^{\infty}\lambda^k\|x\|\geq \|x\|\left(1-\frac{\lambda}{1-\lambda}\right)
X(k_{1}+\frac{N}{2},k_{2}+\frac{N}{2})=S_{00}(k_{1},k_{2})-S_{01}(k_{1},k_{2})W_{N}^{k_{2}}-S_{10}(k_{1},k_{2})W_{N}^{k_{1}}+S_{11}(k_{1},k_{2})W_{N}^{k_{1}+k_{2}}
\Delta_{C_{12}}^{}=2^{-12}\gamma_{C_{12}}\!\left( 6,11\right)=2^{18}\cdot3^{7}\cdot5^{12}\cdot11^{12}\cdot61\cdot67^{4}\cdot73^{3}c_{4}=2^{-4}\alpha_{C_{12}}\!\left( 6,11\right) =3^{2}\cdot23\cdot107\cdot227\cdot27361\cdot320687 c_{6}=2^{-6}\beta_{C_{12}}\!\left( 6,11\right) =-3^{3}\cdot503\cdot769\cdot47221\cdot18748939480561.
\mathrm{~ H_{n m}^{\{{\bf ~ R}\}}( 0 , 0 ) ~}= \left( \frac{R_{1 1}^{n}R_{2 2}^{m}}{2^{n + m}}\right)^{\frac 1 2}\sum_{l = 0}^{\left[ \frac{\mu_{n m}}2 \right]}\frac{( - 1 )^{\frac{n + m}2}n ! m !}{l ! \left( \mu_{n m}- 2 l \right) ! \left( \frac{| n - m |}2 + l \right) !}( 2 r )^{\mu_{n m}- 2 l},
\bar{\psi}(\gamma,\alpha_{0},\theta_{0}) =\frac{d[\tau\bar{\psi}(\gamma,\alpha_{0},\theta_{0})]}{d\tau}=\frac{d[(1-\tau)\bar{\psi}(\gamma_{0},\alpha_{0},\theta_{0})+\tau\bar{\psi}(\gamma,\alpha_{0},\theta_{0})]}{d\tau}& =\frac{\bar{\psi}((1-\tau)\gamma_{0}+\tau\gamma,\alpha_{0})}{d\tau}=0.Q.E.D.
V(\overline{\rho})=\frac{m_{R}^{2}}{2}\overline{\rho}^{2}+\frac{\lambda_{R}}{24}\overline{\rho}^{4}+\frac{3e^{4}}{64\pi^{2}}\overline{\rho}^{4}\ln\frac{e^{2}\overline{\rho}^{2}}{\mu^{2}}-\frac{3e^{4}}{128\pi^{2}}\overline{\rho}^{4}.
\Gamma (s_1) \Gamma (s_2) \zeta_2 (s_1,s_2 \,; a) = \int_0^\infty \frac{y^{s_2-1}}{e^y-1}\int_0^\infty \frac{x^{s_1-1}e^{(1-a)(x+y)}}{e^{x+y}-1}dx dy.
\dim C_k^r(\Delta) \leq \binom{k+3}{3}+f_2^0\binom{k+2-r}{3}&-\sum_{i=1}^{f_1^0}\biggl[\tilde s_i\binom{k+2-r}{3}-\tilde b_i\binom{k+3-\tilde \Omega_i}{3}-\tilde a_i\binom{k+2-\tilde \Omega_i}{3}\biggr]
L_\xi(r,h_\beta) = \begin{cases}\left( 4\pi f/c \right)^2 \left( r^2+h_\beta^2 \right)\,\eta_{0},{\xi=0}\left( 4\pi f/c \right)^2 \left( r^2+h_\beta^2 \right) \,\eta_{1},{\xi=1},\end{cases}
\sum_{{s_1,s_2}={0,1}}(-1)^{s_1 + s_2}\theta(^{s_1}_{s_2})' \theta^3(^{s_1}_{s_2})\frac{d}{d \tau}[\sum_{s_1,s_2}(-1)^{s_1+s_2}\theta^4(^{s_1}_{s_2})] = 0,
\sum_{k=0}^{\infty}\frac{(a)_k(b)_k}{(1)_{k}(a+b)_{k}}x^kH_k(a-1) begin{align*}1mm] \:\:=-\{\log(1-x)\}\sum_{k=0}^{\infty}\frac{(a)_k(b)_k}{(1)_{k}(a+b)_{k}}x^k-\sum_{k=0}^{\infty}\frac{(a)_k(b)_k}{(1)_{k}(a+b)_{k}}x^kH_k(b-1).