harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sqrt{1-\mathrm{LHS}^2}=\max_{\substack{x\in X \|x\|_2=1}}\|P_{Y^\perp}x\|_2=\max_{\substack{z\in\mathbb{R}^N \|z\|_2=1}}\|P_{Y^\perp}P_Xz\|_2=\|P_{Y^\perp}P_X\|_2.
\begin{aligned}\sum_{2\le |a|\le N}(f,e^{itH}\left(\frac{x}{t}-D\right)^2\tilde P^ae^{-itH}f) &\sim \sum_{2\le |a|\le N}(f,e^{itH}\left(\frac{x_a}{t}-D_a\right)^2\tilde P^ae^{-itH}f)+2\sum_{2\le |a|\le N}(f,e^{itH}H_0^a\tilde P^ae^{-itH}f).\end{aligned}
\frac{\mathrm{d}}{\mathrm{d}t}_{|t=0}\int_{\Gamma_t}(c+tw) \circ \theta_t^{-1}\, \mathrm{d}\mathcal{H}^d = \int_\Sigma \partial^\square \big((c+tw) \circ \theta_t^{-1}\big) - \big((c+tw)\circ \theta_t^{-1}\big)_{|t=0}HV \, \mathrm{d}\mathcal{H}^d &= \int_\Sigma w - cHV \, \mathrm{d}\mathcal{H}^d= 0
H_Q(p)=4\pi d_Q(R)\int \frac{d^4l}{{(2\pi)}^4}\left\{\frac{(l+p)^2-m_Q^2}{{[(l+p)^2-m_Q^2]}^2+\varepsilon^2}+(p\to -p) \right\}\delta(l^2-m_Q^2)\sin^2\theta(l^0,\mu_Q),
\exp \left[ \frac{i}{\hbar}S_{c l}( b , a ) \right] = \prod_{k = 1}^{n - 1}\exp \left[ \frac{i m}{2 \hbar \epsilon}( r_{k}^{2}+ r_{k - 1}^{2}- 2 r_{k}r_{k - 1}\cos ( \theta_{k}- \theta_{k - 1}) \right] .
\frac{\ell+1}{\sin^4 \phi}\big[ (1+2 \cos^2 \phi ) \, p_{0,\ell}( \phi)-5 \cos \phi \, p_{0,\ell+1}( \phi)+2 p_{0,\ell+2}( \phi) \big] &=\frac{\ell}{\sin^4 \phi}\big[ \sin \phi \, \sin \psi_{0,\ell-1}h_0(0) s_{0,0}(\ell,\phi) \big] + \phi^{-4-1/2}O(\ell^{-1/2})+O(\phi^{-2}).
\int_{V}[\varphi(x^{\prime})\delta(x-x^{\prime})-G(x,x^{\prime}){\nabla^{\prime}}^{2}\varphi(x^{\prime})]d^{3}x^{\prime}=\int_{S}[\varphi(x^{\prime})\nabla^{\prime}G(x,x^{\prime})-G(x,x^{\prime})\nabla^{\prime}\varphi(x^{\prime})]\cdot d\hat{\sigma}^{\prime}.
r^{1-\epsilon}\leq \tilde{r}' \leq r^{1+\epsilon}, (1-\epsilon) r^{-\epsilon}g_C \leq \tilde{g}_C' \leq (1+\epsilon) r^\epsilon g_C, \sum_{k=1}^K r^k|\nabla^k_{g_C}\tilde{g}_C'|_{g_C}\leq \epsilon r^{\epsilon}.
0=[P_{k_1},G_1]|_a^b=0+\dots+0+ a_{i,m+1}[P_{k_1},f_1]|_a^b+\dots+a_{i,2m}[P_{k_1},f_m]|_a^b, 0=[P_{k_2},G_1]|_a^b=0+\dots+0+ a_{i,m+1}[P_{k_2},f_1]|_a^b+\dots+a_{i,2m}[P_{k_2},f_m]|_a^b, &\vdots 0=[P_{k_m},G_1]|_a^b=0+\dots+0+ a_{i,m+1}[P_{k_m},f_1]|_a^b+\dots+a_{i,2m}[P_{k_m},f_m]|_a^b.
\begin{aligned}K: = \{\rho=(\rho_1, \cdots, \rho_s), u=(u_1, \cdots, u_s): \partial_t \rho_i +\nabla\cdot(\rho_i u_i)=0, (\rho_iu_i)\cdot \mathbf{n}= 0 \partial \Omega \times [0, 1], \rho_i\in \mathcal{P}(\Omega), \rho_i(x,0)=\rho_i^0(x), \rho_i( x,1)=\rho_i^1(x) \}. \end{aligned}
\left[ \arraycolsep=3pt \begin{array}{rrrrrrrrrr}0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 -1 -1 0 0 0 0 0 0 0 2 0 0 -1 -1 -1 -1 -1 0 0 2 0 0 -1 -1 -1 0 0 -1 -1 4 -2 -2 -1 -1 -1 -2 0 -1 -1 4 -2 -2 -1 -1 -1 -1 -1 -2 0 \end{array}\right]
A(z_{1},\cdot\cdot\cdot,z_{n})=\sum_{i_{1}\in\mathbb{Z}}\cdot\cdot\cdot\sum_{i_{n}\in\mathbb{Z}}A_{i_{1},\cdot\cdot\cdot,i_{n}}z_{1}^{i_{1}}\cdot\cdot\cdot z_{n}^{i_{n}},
f_{\mathfrak{m}}^{\sharp}=\underset{^{\ast}\text{-uppersemicont.}}{\underbrace{\Delta_{\mathfrak{a}_{+}}}}+\underset{^{\ast}}{\underbrace{\left( -\Delta_{\mathfrak{a}_{-}}+e_{\Phi}-\beta^{-1}s\right)}}\ .
\left[ i \gamma^{\mu}\partial_{\mu}- m{\frac{1 \pm b_{1}\pm b_{2}}{a}}\right] \times \left[ i \gamma^{\nu}\partial_{\nu}+{\frac{a}{2 m}}\partial^{\nu}\partial_{\nu}+ m{\frac{b_{1}^{2}- 1}{2 a}}\right] \Psi_{1 , 2}= 0 \, .
T_k(t)= \frac{\varphi_{k}}{\Gamma(\alpha)}t^{\alpha-1}E_{\alpha,1+\frac{\beta}{\alpha},1+\frac{\beta-1}{\alpha}}(-\lambda_kt^{\alpha+\beta})+\frac{\Gamma(\mu+1){f_{k}}}{\Gamma(\mu+\alpha+1)}t^{\alpha+\mu}E_{\alpha,1+\frac{\beta}{\alpha},1+\frac{\beta+\mu}{\alpha}}(-\lambda_{k}t^{\alpha+\beta}).
\begin{aligned}(\mu_1, A_1) \le (\mu_2, A_2) \Leftrightarrow \mu_1 \le \mu_2 A_1 \le A_2 & (\mu_1, A_1) \ll (\mu_2, A_2) \Leftrightarrow \mu_1 < \mu_2 A_1 < A_2.\end{aligned}
\varphi_4 = \frac{1}{3}\left(x^{4}+ 8 x y^{3}\right), \varphi_{12}= 243 \left( 61 x^{12}+ 440 x^{9}y^{3}+ 14784 x^{6}y^{6}+ 28160 x^{3}y^{9}+ 1024 y^{12}\right).
\sigma = \int{\frac{d^{2}k}{( 2 \pi )^{2}}}[ \not \! k + \lambda^{2}\gamma^{5}\tilde{\sigma}+ \lambda^{2}\tilde{\sigma}]_{\nu \nu}^{- 1}= \int{\frac{d^{2}k}{( 2 \pi )^{2}}}[{\frac{4 i \lambda^{2}\sigma}{k^{2}- 4 \lambda^{4}( \sigma^{2}+ \tilde{\sigma}^{2})}}]
\frac{d}{ds}\int_0^1 H(u(s,t), t) dt = \int_0^1 \frac{d}{ds}H(u(s,t),t)dt & = \int_0^1 dH \left( \frac{\partial u}{\partial s}\right) dt = -\int_0^1 \omega_s \left( X^{\omega_s}_H, \frac{\partial u}{\partial s}\right) dt.
S [ \{\nu^j \}, \{\mu^j \}] =\sum_{i \neq k =1}^N \frac{1}{2}\left[ \frac{\mu^3_i \mu^3_k +\mu^4_i \mu^4_k}{g}+ g ( \epsilon_i^3 \epsilon_k^3 + \epsilon_i^4\epsilon_k^4 ) \right] \ln \left| \frac{r_i - r_k}{a}\right|
0 \leq 1 - \left\langle e_0 , \widetilde{\rho}_t \, e_0 \right\rangle= \sum_{n \geq 1}\varphi_n \left( 0 \right) \left( 1 - \left( 1- \hbox{\rm e}^{-2 \kappa t}\right)^{n}\right) \leq \hbox{\rm e}^{-2 \kappa t}\sum_{n \geq 1}n \, \varphi_n \left( 0 \right) ,
\begin{cases}\mathcal{B}_{q}e_{0}=\sum_{k=0}^{p}a_{-k-q}^{(k)}\, e_{k}, \mathcal{B}_{q}e_{n}=e_{n-1}+\sum_{k=0}^{p}a_{-k-q-n}^{(k)}\, e_{n+k},n\geq 1.\end{cases}
Q_n(t+1) = Q_n^r(t+1) + Q_n^f(t+1) = Q_n^r(t) - b_n^r(t) + a_n^{r'}(t) + Q_n^f(t) + a_n^{f'}(t) = Q_n(t) - b_n^r(t) + a_n^{r'}(t) + a_n^{f'}(t).
{\cal M}= i g_{YM}^2 C_2(R) \int{d^4 p \over (2\pi)^4}{1 \over p^2 (p-k)^2}\sum_{j=0}^{n-1}(\Delta \cdot p)^j (\Delta \cdot k)^{n-j}= c_\phi (\Delta \cdot k)^n \ln\Lambda^2 + \ldots \,,
B^{\mathrm{T}}( l_{3}, l_{2}, l_{1}) = e^{\pi i ( \Delta_{l_{3}}- \Delta_{l_{2}}- \Delta_{l_{1}})}, \qquad \Delta_{l}= \frac{Q^{2}}{4}+ \Bigl ( \frac{l}{4 \pi b}\Bigr )^{2}.
Z [ \lambda , T ] = \frac{\int d A_{\mu}e^{- \int_{0}^{1 / T}F_{\mu \nu}^{2}/ 4 + \lambda \int d x e^{i e \int_{0}^{1 / T}d \tau A_{0}( \tau , x ) + \mathrm{c . c .}}}}{\int d A_{\mu}e^{- \int_{0}^{1 / T}F_{\mu \nu}^{2}/ 4}}
{\cal A}={\frac{8 \pi J}{a}}\left[ r_{+}- M +{\frac{\left( M^{2}+ \Sigma^{2}- P^{2}- Q^{2}\right) \left( M + \Sigma / \sqrt{3}\right)}{\left( M + \Sigma / \sqrt{3}\right)^{2}- Q^{2}}}\right] .
\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\frac{\partial G}{\partial r})+\frac{1}{\alpha}\delta(t-\tau)\frac{\delta(r-r^{\prime})}{4\pi r^{2}}=\frac{1}{\alpha}\frac{\partial G}{\partial t};0<r<b;t>0
\int_{B_r}y^{1-2s}a_{jk}\frac{\partial\beta_l}{x_k}\frac{\partial}{\partial x_l}U\frac{\partial}{\partial x_j}Udz =\int_{B_r}y^{1-2s}a_{jk}(\delta_{lk}+O(z))\frac{\partial}{\partial x_l}U\frac{\partial}{\partial x_j}Udz &=\int_{B_r}y^{1-2s}\langle A\nabla U,\nabla U\rangle dz+O(r)\int_{B_r}y^{1-2s}|\nabla U|^2dz &=D(r)+O(r)D(r).
h_{\alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2}}( r_{1}, r_{2}, j_{1}, j_{2}) = \sum_{\gamma_{1}, \gamma_{2}}c_{\alpha_{1}, \alpha_{2}, \beta_{1}, \beta_{2}}^{\gamma_{1}, \gamma_{2}}\varphi_{\gamma_{1}, \gamma_{2}}( r_{1}, r_{2}, j_{1}, j_{2}) .
0\geq\nabla_{ii}Q =\frac{-h_{tii}}{h-\varepsilon_{0}}+\frac{2h_{ti}h_{i}+h_{t}h_{ii}}{(h-\varepsilon_{0})^{2}}-\frac{2h_{t}h^{2}_{i}}{(h-\varepsilon_{0})^{3}}&=\frac{-h_{tii}}{h-\varepsilon_{0}}+\frac{h_{t}h_{ii}}{(h-\varepsilon_{0})^{2}}.
G_k=\Phi_0(G_k)=\langle x,y\rangle \Phi_1(G_k) G_k/\Phi_1(G_k) \cong C_2 \times C_2, \Phi_1(G_k) = \langle x^2,y^2, [y,x], [y,x,x] \rangle\Phi_2(G_k) \Phi_1(G_k)/\Phi_2(G_k) \cong C_2^{\, 4},
K = i \int_{0}^{\infty}d \lambda^{\prime}s e^{i \lambda^{\prime}( \pi^{\prime \prime}- \pi^{\prime})}e^{- i [ s \lambda^{\prime}( \gamma^{\mu}P_{\mu}^{\prime}- m ) +{\frac{i}{s}}({\bar{\eta}}^{\prime \prime}-{\bar{\eta}}^{\prime}) ( \eta^{\prime \prime}- \eta^{\prime}) ]}\delta^{4}( P^{\prime \prime}- P^{\prime}) .
=\sum_{(j,h),(i,k)\in I\times \{1, \dots,n\}}\sum_{m',m\in F}\bar{z}_{j,h}E_{\mathcal{B}; j,i}\left(b^*_{h,m'}b_{k,m}\right)E_{\mathcal{C}; j,i}\left(c^*_{h,m'}c_{k,m}\right)z_{i,k}.
i \frac{\partial}{\partial t}\langle q^{\prime}, t \mid q , 0 \rangle_{l}= - \frac{1}{2}\frac{\partial^{2}}{\partial q^{2}}\langle q^{\prime}, t \mid q , 0 \rangle_{l}+ \left( \frac{l ( l + 1 )}{2}\frac{1}{q^{2}}+ V ( q^{\prime}) \right) \langle q^{\prime}, t \mid q , 0 \rangle_{l},
\left.\frac{d}{d\varepsilon}V(\tilde{x}(T))\right|_{\varepsilon = 0}= \left.2\tilde{x}'(T) P\frac{d}{d\varepsilon}\tilde{x}(T) \right|_{\varepsilon=0}= 2\sum_{i=1}^r (a_i-u_i^*(\tau))\tilde{x}'(T)P \times C^*(T,\tau)A_i x^*(\tau),
\lim_{T\rightarrow \infty}\underset{\kappa \in (0,1)}{sup}\mathbb{E}\left|\langle :X_T \otimes \ldots \otimes X_T:,\Phi_{\epsilon,\psi_\kappa}^f \rangle - \rho_{f,\epsilon,\psi_\kappa}^T\right|^2=0,\epsilon >0,
\frac{r_{p}^{3}}{(2-\sigma r_{p}^{2}(1-\cos{(\alpha_{1p}-\alpha_{jp})}))^{\frac{3}{2}}}=\frac{r_{p}^{3}}{(2+r_{p}^{2}(1-\cos{(\alpha_{1p}-\alpha_{jp})}))^{\frac{3}{2}}}
[ \tilde{\cal T}_{\mu}, \tilde{\cal T}_{\nu}] = 2 ( p^{2}\eta_{\mu \nu}- p_{\mu}p_{\nu}) , \quad [ \tilde{\cal T}_{\mu}, \tilde{\cal T}_{\nu}]_{{}_{+}}= 4 i \epsilon_{\mu \nu \lambda}p^{\lambda}( p \widehat{J}) ,
\sum_{i=1}^M\sum\limits_{\substack{j=1 j\neq l}}^M\sum_{l=1}^Ma_l^2\phi^{\left(i,l\right)^*}\phi^{\left(j,l\right)}=2\sum_{i=1}^{M-1}\sum_{q=i+1}^{M}\sum_{r=1}^M a_r^2\Re\left[\phi^{\left(i,r\right)^*}\phi^{\left(q,r\right)}\right].
f_j = \frac{1}{(L_0-c_{j,1})!}\cdot \frac{1}{(L_0-c_{j,2})!}\cdots \frac{1}{(L_0-c_{j,l_j})!}\cdot \prod_{1 \le h < k \le l_j}((L_0-c_{j,k}) - (L_0-c_{j,h})) &= \frac{1}{(L_0-c_{j,1})!}\cdot \frac{1}{(L_0-c_{j,2})!}\cdots \frac{1}{(L_0-c_{j,l_j})!}\cdot \prod_{1 \le h < k \le l_j}(h-k) \neq 0
\left( \cdots \left( \varphi_{k_1l_1}^{r_2'\cdots r_s'}\right) \cdots \right)_{k_sl_s}(X_1,\cdots, X_j+2\pi ,\cdots, X_n,Y_1,\cdots, Y_n) &=\left( \cdots \left( \varphi_{k_1l_1}^{r_2'\cdots r_s'}\right) \cdots \right)_{k_sl_s}(X_1,\cdots, X_n,Y_1,\cdots, Y_n).
\| R \|_{\varrho ; [s,t]^2}:= \sup_{\mathbb{D}, \mathbb{D}'}\left( \sum_{t_i \in \mathbb{D}, t'_j \in \mathbb{D}'}\big|R(t_{i+1},t'_{j+1}) - R(t_{i},t'_{j+1}) - R(t_{i+1},t'_{j}) + R(t_{i},t'_{j})\big|^{\varrho}\right)^{\frac{1}{\varrho}}
A_{M}= \left( \begin{array}{c}{{a}}\\{{\psi}}\\ \end{array}\right) , \; \; A_{M}^{\dagger}= \left( \begin{array}{c}{{a^{\dagger}}}\\{{\psi^{\dagger}}}\\ \end{array}\right) , \; \; \tilde{A}_{M}= \left( \begin{array}{c}{{\tilde{a}}}\\{{\tilde{\psi}}}\\ \end{array}\right) , \; \; \tilde{A}_{M}^{\dagger}= \left( \begin{array}{c}{{\tilde{a}^{\dagger}}}\\{{\tilde{\psi}^{\dagger}}}\\ \end{array}\right)
\int d^3 p \sum_{\lambda}\left[U_{\vec{p},\lambda}(t,\vec{x})U^{+}_{\vec{p},\lambda}(t,\vec{x}^{\,\prime})+V_{\vec{p},\lambda}(t,\vec{x})V^{+}_{\vec{p},\lambda}(t,\vec{x}^{\,\prime})\right]=e^{-3\omega t}\delta^3 (\vec{x}-\vec{x}^{\,\prime})\,.
\times \Bigg \{\sum_{\sigma \in Z_{N}}\frac{\omega^{\sigma k_{2}}w ( x_{3}, x_{1 3}, x_{1}| \sigma ) w ( x_{4}, x_{2 4}, x_{2}| k_{4}+ \sigma )}{w ( x_{4}, x_{1 4}, x_{1}| k_{3}+ \sigma ) w ( x_{3}/ \omega , x_{2 3}, x_{2}| k_{1}+ \sigma )}\Bigg \}_{0}
\begin{cases}\mathcal{E}(t)+d_P(t)^{-1}+d_I(t)^{-1}\leq 2(\mathcal{E}(0)+d_P(0)^{-1}+d_I(0)^{-1}), \|z_{tt}(\cdot,t)\|_{H^s}\leq 2\|w_0\|_{H^s}, \frac{C_1}{2}|\alpha-\beta|\leq |z(\alpha,t)-z(\beta,t)|\leq 2C_2|\alpha-\beta| \inf_{\alpha\in \mathbb{R}}a(\alpha,t)|z_{\alpha}|\geq \frac{1}{2}\alpha_0.\end{cases}
\cdot \sqrt{\frac{( n - l - 1 ) !}{( n - l^{\prime}- 1 ) !}\frac{\Gamma ( n + l + \delta_{1}+ \delta_{2}+ 1 )}{\Gamma ( n + l^{\prime}+ \delta_{1}+ \delta_{2}+ 1 )}}\frac{1}{\Gamma ( l - l^{\prime}+ 1 ) \Gamma ( l^{\prime}- l + 1 )}.
\int d^{4}x \psi_{[ \omega , \frac{l + 1}{2}, \frac{l}{2}, m m^{\prime}]}^{\dagger}( x ) \psi_{k_{x}, +}( x ) = ( 2 \pi ) \delta ( \omega - k^{0}) \cdot f_{\frac{l + 1}{2}, \frac{l}{2}, m m^{\prime}}.
J^{a}t_{a}= \left( \begin{array}{c c c c}{{\frac{J^{7}}{2}+ J^{8}+ J^{9}}}&{{J^{5}+ J^{6}}}&{{J^{2}+ J^{3}}}&{{J^{1}}}\\{{J^{1 0}+ J^{1 1}}}&{{\frac{J^{7}}{2}- J^{8}- J^{9}}}&{{J^{4}}}&{{J^{2}- J^{3}}}\\{{J^{1 2}+ J^{1 3}}}&{{J^{1 4}}}&{{- \frac{J^{7}}{2}+ J^{8}- J^{9}}}&{{J^{5}- J^{6}}}\\{{J^{1 5}}}&{{J^{1 2}- J^{1 3}}}&{{J^{1 0}- J^{1 1}}}&{{- \frac{J^{7}}{2}- J^{8}+ J^{9}}}\\ \end{array}\right)
\sigma_{ac|Bz}= \sigma_{ac|B}-\frac{(-\Sigma_{aB}\Sigma^{-1}_{BB}\Sigma_{Bz})(-\Sigma_{cB}\Sigma^{-1}_{BB}\Sigma_{Bz})}{\sigma_{zz|B}}=\sigma_{ac|B}-\sigma_{ax}\sigma_{cx}(\Sigma_{xB}\Sigma^{-1}_{BB}\Sigma_{Bz})^2\sigma^{-1}_{zz|B}&=\sigma_{ac|B}\left\{1+(\Sigma_{xB}\Sigma^{-1}_{BB}\Sigma_{Bz})^2/(Q_1\sigma^{-1}_{zz|B})\right\}.
II_1^p:=\sum_{|k|\le N}\int_{\eta_{k-1}}^{\eta_k}v_1^{p}(x)\bigl[V_1^-(x)\bigr]^{-p}\bigl[v_1^{-p'}(a(x))\,a'(x)\bigr]^p \biggl(\int_{\eta_{k-1}}^x v_1^{-p'}(t) \bigl|G^{(0)}_{1,k}(t)\bigr|^{p'-1}\,dt\biggr)^p\,dx.
\left( \sum_{j=1}^{n}|a_{j}|^{2}\right)^{\frac{1}{2}}\leq\mathrm{A}_{p}\left( \int_{0}^{1}\left\vert \sum_{j=1}^{n}a_{j}r_{j}(t)\right\vert^{p}dt\right)^{\frac{1}{p}},
sn[\frac{3}{5}K(k);k]=\{\frac{\sqrt{5}\vartheta_{01}[q(k)^{5}]}{\vartheta_{01}[q(k)]}+1\}\{\frac{5\vartheta_{01}[q(k)^{10}]^{2}}{\vartheta_{01}[q(k)^{2}]^{2}}-1\}^{-1}
s_{\mu}s_{\nu^*}=s_{\mu}p_{s(\mu)}s_{\nu^*}=\sum_{\beta \in B_{s(\mu)}}s_{\mu}s_{\beta}s_{\beta^*}s_{\nu^*}=\sum_{\beta \in B_{s(\mu)}}s_{\mu \beta}s_{s(\beta)^*}s_{s(\beta)}s_{(\nu \beta)^*}\in M^*M.
\begin{aligned}K(z)=q^{\nu+1/2}\,T_{q},P(z)=e R_{z}, A_{+}(z)=z, A_{-}(z)=\frac{q^{\nu}}{q-q^{-1}}\left(\frac{T_{q}-R_{z}}{z}\right)-\frac{q^{-\nu}}{q-q^{-1}}\left(\frac{T_{q}^{-1}-R_{z}}{z}\right), \end{aligned}
S_{D}= - \int d^{4}\sigma ( \sqrt{- \mathrm{d e t}\, ( G_{\mu \nu}+ \tilde{\cal F}_{\mu \nu}^{\prime})}+ \int ( C_{4}^{\prime}+ C_{2}^{\prime}\wedge \tilde{\cal F}^{\prime}- \frac{1}{2}C_{0}\tilde{F}^{\prime}\wedge \tilde{F}^{\prime}) .
\vec{\theta}-\vec{\beta}=\nabla_{\vec{\theta}}\psi(\vec{\theta})=\sum_{i}\frac{\theta_{Ei}^{2}}{|\vec{\theta}-\vec{\theta}_{i}|},\pi\theta_{Ei}^{2}\equiv\frac{4\pi GM_{i}D_{is}}{c^{2}D_{s}D_{i}}
\int_{0}^{\infty}\frac{\cosh \alpha t - 1}{\sinh t}d t = \psi \left( \frac{1}{2}\right) - \frac{1}{2}\left( \psi \left( \frac{1 + \alpha}{2}\right) + \psi \left( \frac{1 - \alpha}{2}\right) \right) ,
P_{\ell+u}(\cos \phi)= \Big( \frac{\phi}{\sin \phi}\Big)^{1/2}\Big[\sum_{n=0}^{m-1}A_n(\phi) \frac{J_{n}((\ell+u+1/2) \phi)}{(\ell+u+1/2)^{n}}+\phi^m O((\ell+u+1/2)^{-m}) \Big],
\rho_{k,\mathcal{A}}=\frac{1}{2^l}\sum_{a_1,\dots,a_l=0}^3\langle\psi_k|\sigma_{1}^{( a_1 )}\dots\sigma_{l}^{( a_l )}|\psi_k\rangle\sigma^{( a_1 )}\otimes\dots\otimes\sigma^{( a_l )}
D_r=(\gamma^\mu \otimes \mathbf{1})(\mathbf{f}_\mu - \frac{1}{4}\Gamma_{\mu\nu\rho}\gamma^\nu\gamma^\rho + \frac{1}{4}A_\mu^{\;m}t_{mkl}\mathbf{1}\otimes \gamma^k \gamma^l) -\frac{1}{8}F_{\mu\nu k}\omega \gamma^\mu \gamma^\nu \otimes \gamma^k + \omega \otimes \epsilon =0 \, ,
\begin{cases}x \frac{\partial Y}{\partial x}= A(x, y) Y = x^{-3}\begin{bmatrix}x^3 +y y^2 -1 -y + x^3 \end{bmatrix}Y y \frac{\partial Y}{\partial y}= B(x, y) Y = y^{-1}\begin{bmatrix}y y^2 -2 -3\end{bmatrix}Y.\end{cases}.
\varphi(1) =1, \varphi(3) =2, \varphi(5) =4, \varphi(7) =6, \varphi(16) =8, \varphi(31) =30, \varphi(71) =70, \varphi(625) =500, \varphi(1057) =900, \varphi(991) =990, \varphi(5591) =5590, \varphi(9551) =9550,
s^1(z) = - \frac{1}{2 z}\left(z^2-c+1 - \sqrt{(z^2-c+1)^2-4z^2}\right) s^2(z) = - \frac{1}{2 c z}\left(z^2+c-1 - \sqrt{(z^2-1+c)^2-4cz^2}\right)
\int_{\mathcal{H}}\int_{\mathcal{H}}\| x-y\|^{a}d\mu(x)d\nu(y) = \frac{a(a -2)}{4\Gamma(2-a/2)}\int_{(0, \infty)}\left ( \int_{\mathcal{H}}\int_{\mathcal{H}}e^{-r\| x-y\|^{2}}d\mu(x)d\nu(y) \right ) \frac{1}{r^{a+1}}dr \geq 0.
\sum_{j=1}^3\sum_k\int_{\mathbb{R}}(\mathbf{1}_{|k-k_0|\geq D k_0}+\mathbf{1}_{|\xi-\eta_0|\geq Dk_0})e^{(|k-k_0|+\sqrt{\sigma}|\xi-\eta_0|)/2}|\widehat{g_j'}(t,k,\xi)|^2\,\mathrm{d}\xi\lesssim \varepsilon_1^2e^{-Dk_0/2}.
\frac{16 I_n}{\sinh(n\pi)}= \frac{32\sinh(n\pi/2)}{\pi^4 \sinh(n\pi)}\left( \frac{6\sinh(n\pi/2)}{n^4}- \frac{3\pi\cosh(n\pi/2)}{2n^3}\right) = \frac{24}{\pi^4}a_n,
P = (a - (2 +\frac{p + q}{2p.q}).b)(a - (\frac{1}{2}+ \frac{p.u+q.v+2p.q}{2p.q.(u+v)}).b)(a - (\frac{p.u+q.v+2p.q}{2p.q.(u+v)}).b) \ + & -4\lambda^{2}.(a - (1 + \frac{p.u+q.v+2p.q+p.(u+v)}{2p.q.(u+v)}).b)(a - (\frac{p.u+q.v+2p.q+ q.(u+v)}{2p.q.(u+v)}).b).
\liminf_{n\to\infty}\frac{1}{n}\sum_{l=1}^{n}\sum_{k=1}^{N_{p}-1}X^{kl}\geq\sum_{k=1}^{N_{p}-1}\liminf_{n\to\infty}\frac{1}{n}\sum_{l=1}^{n}X^{kl}=pN_p-p.
\left(\tau(D^\star{\boldsymbol u}),B D\varphi\right)+\delta\left({\tau}_{33}(D^\star{\boldsymbol u}),\varphi_2\right)+{\cal R}e\, \left(Bu\cdot \nabla u,\varphi\right)={\cal R}e\,\delta\left(u_3^2,\phi_2\right),
M^{\prime}:= \displaystyle\sup_{x_0 \in \mathbb{R}^2}\displaystyle\sup_{R >0}R^{-2 \alpha}\displaystyle\inf_{l \in \textrm{Span}\{1, x_1 \}}\| u - \sigma_i(x_0) v_i(\cdot, a(x_0)) - l \|_{B_R(x_0)}.
G^{(c,a,\gamma)}_{(\alpha,\beta,\epsilon)}(\partial)= \left(\begin{array}{cc}a(u'+2u\partial)+\alpha \partial + c\partial^3 a\, v \partial+\beta \partial + \gamma \partial^2 a \partial\circ v + \beta \partial -\gamma \partial^2 \epsilon \partial\end{array}\right)\,.
\begin{aligned}\lim_{n\to\infty}P_{n,k}(\mathcal{S}(n,k;M_n))=k\sum_{l=1}^k\binom kl(\frac c{1-c})^l\sum_{s=1}^\infty(1-c)^{k(s+1)}\frac 1{k(s+1)-l}+ &(1-c)^k\sum_{l=1}^{k-1}\binom kl(\frac c{1-c})^l,\ \M_n\sim cn,\c\in(0,1).\end{aligned}
{\cal V}_{N;g}\, =\,{\cal C}_{g}< \Omega| \int [{\mbox d}m]^{g}_{N}\exp \left\{\frac{1}{2}\sum_{i=1}^{N}\sqrt{\frac{\alpha'}{2}}p^{(i)}\cdot\left.\! \left[ \sqrt{\frac{\alpha'}{2}}p^{(i)}+{\alpha}_{1}^{(i)}\partial_{z}+ \bar{\alpha}_{1}^{(i)}\partial_{\bar{z}}\right] \log |V'_{i}(z)|^2\right|_{z=0}\right\}
\widetilde{\cal J}^{\nu}= (\eta^{\mu\nu}\bot_{\sigma\rho}+2 \eta^{\nu}{_{[\sigma}}\eta_{\rho ]}{^{\mu}})[\acute{\xi}^{\sigma}\widetilde{\nabla}_{\mu}\xi^{\rho}-(\widetilde{\nabla}_{\mu}\acute{\xi}^{\rho}) \xi^{\sigma}].
\big(|\cdot|^m D^\gamma O_{\alpha,q}\big)\ast f= \left(|\cdot|^m 2^{q|\gamma|}2^{3q}g\left(2^{2\alpha q}t,\,2^{q}\cdot\right)\right)\ast f = 2^{q(|\gamma|-m)}2^{3q}\left(|\cdot|^m g\right)\left(2^{2\alpha q}t,\,2^{q}\cdot\right)\ast f.
\mathcal{F}^{\omega,\{U_i\}_{i=0,\dots,K}}:=\left\{\psi:=(\psi_0,\psi_1,\dots,\psi_K)|\;\psi_i:\Phi_i(U_i)\rightarrow\psi_i(\Phi_i(U_i)) \right. &\qquad\qquad\qquad\qquad\left.i=0,\dots,KDF^{\omega}(\psi,T\psi)\right.\}.
\frac{d c_{n}( t )}{d t}= \sum_{n^{\prime}\ne n}~ \exp \Bigl [ \frac{i}{\hbar}\int_{0}^{t}\Bigl ( ( \theta_{n^{\prime}}( t^{\prime}) - \theta_{n}( t^{\prime}) \Bigr ) d t^{\prime}\Bigr ] \times \frac{\Bigl < n ; t | \frac{{\cal D}\Lambda ( t )}{{\cal D}t}| n^{\prime}; t \Bigr >}{\bigl ( \lambda_{n}( t ) - \lambda_{n^{\prime}}( t ) \bigr )}~ c_{n^{\prime}}( t ) ,
\|u\|_{1,\infty,Q_T}+b_\#\|u\|_{\ell-1,\Sigma_T}^{\ell-1}\leq \|u_0\|_{1,\Omega}+\|f\|_{1,Q_T}+\|h \|_{1,\Sigma_T}:= \mathcal{Z}; \|\nabla u\|_{p,Q_T}^p \leq \mathcal{B}+{rn^2 \left((Z_1)^{2-p}\mathcal{Z}^{(2-p)/ n}+1\right)\over a_\# (n+2-p(n+1)) (n-1)}\left({b^\#\over b_\#}\mathcal{Z}\right),
a_j' = \cos^\ell(\theta_j) \frac{16 \sin^4(\theta_j)}{n^2}\left( 1 + O\left(\frac{1}{n}\right)\right) a_j = \cos^\ell(\theta_j) \frac{16 \sin^4(\theta_j)}{n^2}\left(\frac{\ell}{\cos(\theta_j)}+ O\left(n^3\sin^3(\theta_j)\right)\right) \left(1+O\left(\frac{1}{n}\right)\right)
f'(\lambda) = \lambda \|\nabla u\|^2_{L^2}-\frac{3(p-1)}{2(p+1)}\lambda^{\frac{3(p-1)}{2}-1}\|u\|^{p+1}_{L^{p+1}}+ \frac{3(q-1)}{2(q+1)}\lambda^{\frac{3(q-1)}{2}-1}\|u\|^{q+1}_{L^{q+1}}= \frac{G(u_\lambda)}{\lambda}
\tilde{D}_{1}= \kappa_{5}^{2}e^{- 2 A_{+}- \alpha_{+}^{( 0 )}}\bar{\tau}_{+ ( T )}^{[ 0 ]}= - \kappa_{5}^{2}e^{- 2 A_{--}\alpha_{-}^{( 0 )}}\bar{\tau}_{- ( T )}^{[ 0 ]},
[(x_1\phi_1)_\lambda(x_2\phi_2)]=[(x_1\phi_1)_{\lambda}x_2]\phi_2+(-1)^{(\bar{x}_1+\bar{\phi}_1)\bar{x}_2}x_2[(x_1\phi_1)_{\lambda}\phi_2]+\int^\lambda_0[[(x_1\phi_1)_{\lambda}x_2]_\mu \phi_2]d \mu,
\partial_{\underline{i}}\epsilon_{I}+{\textstyle \frac{1}{4}}\biggl ({\frac{\partial_{\underline{i}}{\overline{{\cal H}}_{2}}}{\overline{{\cal H}}_{2}}}-{\frac{\partial_{\underline{i}}{\cal H}_{2}}{{\cal H}_{2}}}\biggr ) \epsilon_{I}-{\textstyle \frac{1}{2}}\left( \partial_{\underline{i}}U \right) \epsilon_{I}= 0 \, .
{\frac{1}{\Theta}}\left({\frac{\partial^{2}\Theta}{\partial \theta^{2}}}+{\frac{\cos \theta}{\sin \theta}}{\frac{\partial \Theta}{\partial \theta}}\right) +{\frac{1}{\sin^{2}\theta}}{\frac{1}{\Phi}}{\frac{\partial^{2}\Phi}{\partial \varphi^{2}}}\equiv - l ( l + 1 ) \, ,
{\bf ( 3 , 1 )_{- x}\otimes ( 1 , 2 )_{- y}\rightarrow ( 3 , 2 )_{- ( x + y )}}\qquad \quad{\bf ( \bar{3}, 1 )_{x}\otimes ( 1 , 2 )_{y}\rightarrow ( \bar{3}, 2 )_{x + y}}
\xi_{q}(z)=\frac{q^{1/2}}{1-q^{1/2}}{}_{1}\psi_{1}\left(q^{1/2};q^{3/2};q,-q^{1/2}z\right)=q^{1/2}\frac{\left(q,q,-qz,-z^{-1};q\right)_{\infty}}{\left(q^{1/2},q^{1/2},-q^{1/2}z,-q^{1/2}z^{-1};q\right)_{\infty}}.
m ={\frac{2^{\frac{1}{3}}\pi^{4 / 3}\Gamma ( 5 / 9 )}{\Gamma ( 2 / 3 ) \Gamma ( 8 / 9 )}}\, \left({\frac{2 \Gamma ( 1 1 / 1 6 )}{\Gamma ( 5 / 1 6 ) \Gamma ( 3 / 4 )}}\right)^{8 / 9}\, g^{8 / 9}\approx 4 . 9 2 7 7 9 1 2 2 4 4 \ldots g^{8 / 9}.
{\cal L}= \left( \begin{matrix}{- 2 c_{1}\displaystyle \frac{d^{2}}{d x^{2}}+ u ( x )}&{2 c_{2}\displaystyle \frac{d}{d x}+ v ( x ) \hfill}\\{\noalign{\smallskip}- 2 c_{2}\displaystyle \frac{d}{d x}+ v ( x )}&{4 c_{3}\hfill}\\{\noalign{\smallskip}}\\ \end{matrix}\right)
\psi [ (h_1, \ldots, h_{m+1}), (k_1, \ldots, k_{n+1})]_c =~ \sum_{i=1}^{m+n+1}\sum_{q+r = i+1}[h_q, k_r] =~ [h_1 + \cdots + h_{m+1}, ~k_1 + \cdots + k_{n+1}] =~ [\psi (h_1, \ldots, h_{m+1}), \psi (k_1, \ldots, k_{n+1}) ],
\tilde{y}^{[i,j]}= \sqrt{\gamma^{[i,j]}}x^{[i,j]}+\underbrace{\sum_{k=1, k\neq i}^{K}{\mathbf{f}_{k}^{[i,j]}}^{\mathsf{H}}\mathbf{V}_k\mathbf{x}_k}_{\textsf{inter-cell interference}}+{\mathbf{u}^{[i,j]}}^{\mathsf{H}}\mathbf{z}^{[i,j]},
\sum_{\tau \in T}D_{R}( \tau ) = \frac{1}{d_{R}}\frac{n ( n - 1 )}{2}\chi_{R}( \tau ) \mathrm{{\bf ~ 1}}_{R}\qquad \mathrm{\c i t e{T u n g}l e m m a ~ t o ~ t h . 3 . 7}
(-c^2\partial_r^2-\frac{c^2}{r}\partial_r+\frac{c^2n^2}{r}-\lambda^2)u_{1,n}(r) =0 (-\partial_r^2-\frac{1}{r}\partial_r+\frac{n^2}{r}-\lambda^2)u_{2,n}(r) =0
\beta^2\xi'(q) =\frac{1}{m}\Bigl(\frac{1}{1-q}-\frac{1}{1-q+mq}\Bigr), \beta^2\xi(q) =\frac{1}{m^2}\log\Bigl(\frac{1-q+q m}{1-q}\Bigr)-\frac{q}{m}\frac{1}{1-q+m q}.
\nu^{\pi^*}_n(1|1)\equiv{{c}_1(n)=\frac{2^{\mu_1(n)}}{1+2^{\mu_1(n)}}},~\pi^*_n(1|1)\equiv{d_1(n)}=\frac{\beta_n(1+2^{\mu_1(n)})-1}{(\beta_n-\delta_n)(1+2^{\mu_1(n)})},~\mu_1(n)\triangleq\frac{H(\beta_n)-H(\delta_n)}{\beta_n-\delta_n}.
\begin{array}{rlclr}\min \displaystyle{\sum_{j=1}^{\sigma}\frac{1}{j}x_{j}}\mbox{s.t.}\displaystyle{\sum_{j=1}^{\sigma}x_{j}}\geq \left\lceil\frac{c}{\gamma}\right\rceil \displaystyle{\sum_{j=1}^{\sigma}(j-1)x_{j}}\leq s-1 x_{j}\in \mathbb{R}_+ j\in\{1,\ldots,\sigma\}\, .\end{array}
EV\bigl(\Lambda_{\psi}(p)\otimes\Lambda(a)\bigr) =E(\Lambda_{\psi}\otimes\Lambda)\bigl((\Delta p)(1\otimes a)\bigr)=(\Lambda_{\psi}\otimes\Lambda)\bigl(E(\Delta p)(1\otimes a)\bigr) &=(\Lambda_{\psi}\otimes\Lambda)\bigl((\Delta p)(1\otimes a)\bigr)=V\bigl(\Lambda_{\psi}(p)\otimes\Lambda(a)\bigr).
\int_{e x t}\left< T_{0 0}( \vec{x}) \right>_{B}= - \frac 1{1 6 \pi^{2}}\left( a \int_{e x t}\zeta_{B}( 2 | 0 , x_{2}) + b \int_{e x t}\zeta_{B}( 2 | x_{1}, 0 ) \right) + \frac 1{3 2 \pi^{2}}\int_{e x t}\zeta_{B}( 2 | x_{1}, x_{2})
\eta_{M,N}(q\,|\,b) = \exp\Bigl(-\int\limits_0^\infty(1-e^{-tq})dK^{(f)}_{M,N}(t\,|\,b)/t\Bigr), dK^{(f)}_{M,N}(t\,|\,b) \triangleq e^{-b_0t}\prod\limits_{j=1}^N(1-e^{-b_j t}) f(t)\,dt/t^M.
\left|{{dx}\over{d\lambda}}\right|=H^{-{{2(p+1)}\over{(D-3)\Delta_p}}}_p\sqrt{E^2H^{{4(D-p-4)}\over{(D-3)\Delta_p}}_p-{{{\cal J}^2}\over{x^2H^{{4(p+1)}\over{(D-3)\Delta_p}}_p}}-\epsilon},
\frac{d x_1}{\mathcal{Q}^{(2)}_1}= \frac{d x_2}{\mathcal{Q}^{(2)}_2}= \frac{d \bar{\mathcal{F}}_2}{(\Delta_{L-1}- \mathcal{V}^{(2)}) \bar{\mathcal{F}}_2}\; .
|^3\mathbb{X}_N^{(f)}| = \sum_{\pi(N): N=\{N_1^{c_1};N_2^{c_2};\ldots N_f^{c_f}\}}\prod_{j=1}^f\binom{|^3\mathbb{X}_{N_j}^{(1)}|+c_j-1}{c_j}.
p(\boldsymbol{s}^{c}|\boldsymbol{s})=\prod_{m}\mathop{p(s_{m}^{c}|s_{m})}=\prod_{m}(1-s_{m})\delta(s_{m}^{c})+s_{m}\rho_{c}^{s_{m}^{c}}(1-\rho_{c})^{1-s_{m}^{c}}
\mathrm{P}(\mathcal{O}_{2}) = \sum^{N}_{l=1}\mathrm{P}\left(x_{n^*}<2R_2 ,|\mathcal{S}_r|=l\right) = \sum^{N}_{l=1}\mathrm{P}\left(x_{n^*}<2R_2 ||\mathcal{S}_r|=l\right)\mathrm{P}\left( |\mathcal{S}_r|=l\right).
u_i(s,t)-u_i(s,0)-u_i(0,t)+u_i(0,0) &=\int_0^t\int_0^s\{b_i(\xi,\zeta,W_{\xi,\zeta}(\omega)+u(\xi,\zeta))-b_i(\xi,\zeta,W_{\xi,\zeta}(\omega))\}\mathrm{d}\xi\mathrm{d}\zeta, \forall\,(s,t)\in[0,1]^2,\,\forall\,i,
0 = D(0)=[D((1\otimes E_{ii})(1\otimes E_{jj}))]_{ij}&=[D((1\otimes E_{ii}))(1\otimes E_{jj})]_{ij}+[(1\otimes E_{ii})D((1\otimes E_{jj}))]_{ij}&=[D(1\otimes E_{ii})]_{ij}+[D(1\otimes E_{jj})]_{ij}&=D^{ij}_{ii}(1)+D^{ij}_{jj}(1),