harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( 1 - \frac{w N}{6}e^{2 \phi}\right) \left( \delta^{2}\phi - 2 \delta \phi \, \delta \rho \right) - ( \delta \phi )^{2}= \frac{N}{1 2}e^{2 \phi}\left[ ( \delta Z )^{2}+ 2 \delta^{2}Z - 4 \delta Z \delta \rho \right] ;
\frac{dL}{dt}\left( t\right) =0,\ \ \ \frac{dA}{dt}\left( t\right)=-\int_{0}^{2\pi}k^{\alpha-1}\left( \theta,t\right) d\theta+\dfrac{{L\!}\left( t\right)}{2\pi}\int_{0}^{2\pi}k^{\alpha}\left( \theta,t\right) d\theta\geq0,\ \ \t\in\lbrack0,T),
[w_k]_{W^{s_k,p}(Q)}\le [v_k]_{W^{s_k,p}(Q)}+\left(\int_Q \int_{Q\setminus\tau\,Q}\frac{|\Psi_a(x)-\Psi_a(y)|^p}{|x-y|^{N+s_k\,p}}\,dx\,dy\right)^\frac{1}{p}&+\frac{C}{1-\tau}\,\left(\int_Q \left(\int_Q \frac{dx}{|x-y|^{N+s_k\,p-p}}\right)\,|v_k(y)-\Psi_a(y)|^p\,dy\right)^\frac{1}{p}.
M\left(\int_\Omega|(-\Delta)^{\frac{\alpha}{2}}u|^2dx\right)(-\Delta)^{\alpha}u = \lambda f(x)|u|^{q-2}u+|u|^{2^*_\alpha-2}u\;\; \; \Omega,\;u=0\;\textrm{in}\;\mathbb R^n\setminus \Omega,
\Gamma(\overline{\eta}^{\gamma \to i}) = \Gamma(\overline{\eta}) \;\mbox{and}\; \tilde{\Gamma}(\overline{\eta}^{\gamma \to i}) = \tilde{\Gamma}(\overline{\eta}) \cup \{\overline{\eta}^{\mbox{\tiny $J$}}_{\gamma,I}\}\setminus \{i\}\;.
\widetilde{\nabla}^{\alpha}\widetilde{Q}_{\alpha\beta}[\phi]=\Omega^{4}\Big(\Re\big(f\cdot\overline{\frac{1}{\Omega}D_{\beta}(\Omega\phi)}\big)+F_{\beta\gamma}\Im\big(\phi\overline{\frac{1}{\Omega}D^{\gamma}(\Omega\phi)}\big)\Big).
\mathop{m}\limits^{(n)}(0)\in (\rho+1)^{n}(f^{n}V)(x_{0})+u_{1}^{}\xi_{n}(\rho;x_{0}) &+span\left\{u_{1}^{k}\pi_{k}(\rho,\rho+1;x_{0}),k=2,...,n-2\right\}+\rho^{n-1}(\rho+1)u_{1}^{n-1}([\ldots [[f,\underbrace{g],g],\ldots ,g}_{n-1\, \,{\kern 1pt}{\kern 1pt}}]V)(x_{0})
2\hat{b}_+^i(\tilde\Sigma^{-1})_{ik}b_-^k+\nabla\cdot b_- = \beta^{-1}\partial_k\beta(\tilde\gamma^{-1})^{\eta k}H_{\eta\ell}(\tilde\gamma^{-1})^{\ell j}F_j+(\tilde\gamma^{-1})^{ki}H_{k\ell}(\tilde\gamma^{-1})^{\ell j}\partial_{q^i}F_j.
\| u \|_{L^2(t_1,t_2,W^{1,2}_\lambda ( \tilde\Omega ))}:=\biggl (\int\limits_{t_1}^{t_2}\int\limits_{\tilde\Omega}\biggl (|u(X,t)|^2+|\nabla u(X,t)|^2\biggr )\lambda(X)dXdt\biggr )^{1/2}<\infty.
\int \frac{d^{2}k}{( 2 \pi )^{2}}\frac{k^{2 n}}{( k^{2}+ M^{2})^{N}}= \frac{\Gamma ( n + 1 ) \Gamma ( N - n - 1 )}{4 \pi \Gamma ( N )}( M^{2})^{n - N + 1}\,
f(x)A_{13}A\sp\prime_{31}-g(y)A_{32}A\sp\prime_{23}+h(z)A_{32}A\sp\prime_{23}-f(x)A_{21}A\sp\prime_{12}+g(y)A_{21}A\sp\prime_{12}-h(z)A_{13}A\sp\prime_{31}=0.
{\hat{H}}\psi_{s}= \frac{1}{r^{2}}\frac{d}{d r}r^{2}{\rho}^{\prime}\left( \begin{array}{c}{{\begin{array}{c c}{{- 1 - E_{c}^{2}}}&{{E_{c}^{2}}}\\{{E_{c}^{2}}}&{{1 - E_{c}^{2}}}\\ \end{array}}}\\ \end{array}\right) \left( \begin{array}{c}{{1}}\\{{1}}\\ \end{array}\right) = \frac{1}{r^{2}}\frac{d}{d r}r^{2}{\rho}^{\prime}\left( \begin{array}{c}{{- 1}}\\{{1}}\\ \end{array}\right)
\Phi(\!A,B\!) = \sum_{j=1}^N \sum_{i=1}^{M-1}K_{ij}^2 +\sum_{j=1}^N L_j^2 = \sum_{j=1}^N \sum_{i=1}^{M-1}K_{ij}^2 +\sum_{j=1}^N \left( \sum_{i=1}^{M-1}K_{ij}\right)^2.
\int |f_{g,\delta,j}(x)|^2dx=\int |\hat{f}_{g,\delta,j}(\omega)|^2d\omega=\int |\hat{\nu}_{g,j}(\omega)|^2 |\hat{\phi}^D_\delta(\omega)|^2d\omega\leq (c'''')^2\int |\hat{\nu}_{g,j}(\omega)|^2d\omega.
R_Bf_\ell(y):=\int_{\frac{1}{2}B}\theta(a)(y-a)\lrcorner\,\Bigl(\int_0^1t^{\ell-1}f_\ell(a+t(y-a))\,{\rm d}t\Bigr)\,{\rm d}a &\mbox{for an $\ell$-form}f_\ell\ (\ell=1,\dots,n)
\hat{\rho}_{\mathrm{G}}= \left( \operatorname{s i n}\vartheta_{\mathrm{G}}\operatorname{c o s}\varphi_{\mathrm{G}}, \operatorname{s i n}\vartheta_{\mathrm{G}}\operatorname{s i n}\varphi_{\mathrm{G}}, \operatorname{c o s}\vartheta_{\mathrm{G}}\right) \, .
z_1^+ = B_1^+ z_2^+ = B_2^+(h_1-h_2) -E_{21}B_1^+ z_3^+ = B_3^+(h_1-h_3)(h_2-h_3) -E_{32}B_2^+(h_1-h_3) -E_{31}B_1^+(h_2-h_3-1) + E_{21}E_{32}B_1^+ z_1^- = B_1^-(h_1-h_2)(h_1-h_3) + E_{21}B_2^-(h_1-h_3) +E_{31}B_3^-(h_1-h_2) + E_{21}E_{32}B_3^- z_2^- = B_2^-(h_2-h_3) +E_{32}B_3^- z_3^- = B_3^-
\gamma_{l}\geq Z_{nl}- \hat{\varDelta}= \frac{\zeta_{nl}-\varDelta_{n}}{b_n}= O\bigl(b_n^{-1}\bigr), \gamma_{l}\leq W_{nl}+ \hat{\varDelta}= \frac{-\zeta'_{nl}+\varDelta_{n}}{b_n}=O\bigl(b_n^{-1}\bigr).
\Delta\left(\phi,\theta\right) = \left\lbrace \begin{array}{cc}1, \phi \in \left[\phi_{\mathrm{min}},\phi_{\mathrm{max}}\right] \textrm{and}\theta \in \left[\theta_{\mathrm{min}},\theta_{\mathrm{max}}\right], 0, \textrm{otherwise},\end{array}\right.
F(x^+,x^-,0)=\sum_{n,m,s,t}\,\,\left(\frac{\pi}{2L^2 l}\right)^2\langle 0|\frac{L}{\pi}T(n,m) e^{-iP^-_{op}x^+-iP^+x^-}\frac{L}{\pi}T(s,t)| 0 \rangle\,,
w^k:=\!\left(\begin{matrix}\nabla\!f(x^{k})\!-\!\nabla\!f(y^{k-1})-\frac{1}{\tau_{k-1}}(x^{k}\!-\!y^{k-1})+\delta(x^{k}\!-\!x^{k-1}) \delta(x^{k-1}-x^{k}) \end{matrix}\right)\in\partial H_{\delta}(z^{k}).
\theta(\ell) = \max_{1\leq i<j \leq 3}\{E(k_{\ell,i}) + E(k_{\ell,j})\}, \nu(\ell) = \min_{\substack{1\leq i,j \leq 3 i\neq j}}\{V(k_{\ell,i}) - E(k_{\ell,j})\},
y_{1}^{\alpha_{1}}y_{2}^{\alpha_{2}}\dots y_{n}^{\alpha_{n}}=q^{+\frac{1}{2}\sum_{i<j}B_{ij}\alpha_{i}\alpha_{j}}y^{\alpha}, y^{\alpha}=q^{-\frac{1}{2}\sum_{i<j}B_{ij}\alpha_{i}\alpha_{j}}y_{1}^{\alpha_{1}}y_{2}^{\alpha_{2}}\dots y_{n}^{\alpha_{n}}.
\langle x, u_k-e_{a_k}\rangle = z^* (x) - x^{a_k}\leq z^*(x) + \|x\|_\infty<0 \quad \langle \tilde{x}, u_k- e_{a_k}\rangle = \tilde{x}(z^*)-\tilde{x}^{a_k}\geq \tilde{x}(z^*)-\|\tilde{x}\|_\infty>0.
\begin{aligned}\mu_{\{b_k\},\{D_k\}}=\delta_{b_{1}^{-1}D_1}\ast\delta_{b_{1}^{-1}b_{2}^{-1}D_2}\ast\delta_{b_{1}^{-1}b_{2}^{-1}b_{3}^{-1}D_3}\ast\cdots=\mu_{k}\ast\nu_{>k}(b_{1}b_{2}\cdots b_{k}\cdot)\end{aligned}
a_{n}^{\alpha}= \int_{0}^{\pi}\frac{d \sigma_{+}}{\pi}\exp \left[ 4 i n \frac{e_{\mu}X_{L}^{\mu}( \sigma_{+})}{e_{\mu}P_{L}^{\mu}}\right] \xi_{i}^{\alpha}\partial_{+}X_{L}^{i}( \sigma_{+}) ,
\mathfrak{B}_{s_1s_2\ldots s_p}=\left\{c_1c_2\ldots c_p\big|\begin{array}{ll}c_1c_2\ldots c_p \hbox{is a valid label of equivalent classes in graph}\mathfrak{F}_{i1i2\ldots i_k}&\hbox{and there are}s_{\ell}\hbox{$\ell$'s in the label}c_1c_2\ldots c_p \end{array}\right\}
\max_{\{x_{k_j}\}}\;\; \mathcal{U}_k=\max \left\{\omega_k \log\left(\sum_{j=1}^J x_{k_j}c_{k_j}\right) - \sum_{j=1}^J x_{k_j}p_{j_k}, 0 \right\}\mbox{s.t.}\;\; \sum_j x_{k_j}\leq 1,
\sim\int \frac{d^3p}{(2 \pi)^3}D_{00}( p^0=0, \vec{p}) e^{i\vec{p}\cdot \vec{R}}= \int \frac{d^3p}{(2 \pi)^3}\frac{e^{i\vec{p}\cdot \vec{R}}}{\vec{p}^2 + m_{el}^2}= \frac{e^{-m_{el}R}}{4 \pi R}\;.
\begin{aligned}\max\limits_{\alpha\in\mathfrak{m}_{i}}|f_{i}(\alpha)|\ll N_{i}^{1-1/18+\varepsilon}, &\max\limits_{\alpha\in\mathfrak{m}_{i}}|S_{i}(\alpha)|\ll N_{i}^{1/3-1/42+\varepsilon}.\end{aligned}
L(z_l,{\dot z}_l)=\frac{1}{2i}\sum_k \gamma_k\big(\bar z_k\dot z_k-z_k\dot{\overline{z}}_k\big)-\frac{1}{2}\sum_n\sum_{k\neq n}\gamma_n\gamma_k|z_n-z_k|, z_l\in\mathbb{C},
\bigg|\varepsilon_k \frac{|\nabla u_k|^2}{2}+ \frac{W(u_k)}{\varepsilon_k}- 2|\nabla w_k|\bigg| = \Bigg( \sqrt{\frac{\varepsilon_k}{2}|\nabla u_k|}- \sqrt{\frac{W(u_k)}{\varepsilon_k}}\Bigg)^2 = |\xi_k|,
\left. \begin{array}{l l}\nu_E = x_1(y_2\!-\!y_3)+x_2(y_3\!-\!y_1)+x_3(y_1\!-\!y_2) \nu_1=x_2(y_3\!-\!y_E)+x_3(y_E\!-\!y_2)+x_E(y_2\!-\!y_3) \nu_2=x_3(y_1\!-\!y_E)+x_1(y_E\!-\!y_3)+x_E(y_3\!-\!y_1) \nu_3=x_1(y_2\!-\!y_E)+x_2(y_E\!-\!y_1)+x_E(y_1\!-\!y_2)\end{array}\right.
I_p(a,b) :=\int_0^{\frac{\pi_p}{2}}\frac{d\theta}{(a^p\cos_p^p{\theta}+b^p\sin_p^p{\theta})^{1-\frac1p}}, J_p(a,b) :=\int_0^{\frac{\pi_p}{2}}(a^p\cos_p^p{\theta}+b^p\sin_p^p{\theta})^{\frac1p}\,d\theta,
\frac{1}{\sqrt{(1+w^{2}-R^{2}\lambda(t)^{2})^{2}+4R^{2}\lambda(t)^{2}}+1+w^{2}-R^{2}\lambda(t)^{2}}= \frac{1+w^{2}-R^{2}\lambda(t)^{2}- \sqrt{(1+w^{2}-R^{2}\lambda(t)^{2})^{2}+4R^{2}\lambda(t)^{2}}}{-4R^{2}\lambda(t)^{2}}
\mathbf{r}^k \in\arg\min_{\mathbf{r}\in \mathcal{Y}}~ \mathbf{r}^{\top}\nabla f(\mathbf{y}^k) \mathbf{y}^{k+1}:=\mathbf{y}^k+\eta_k(\mathbf{r}^k-\mathbf{y}^k)
A_{0}(J_{\varepsilon}\textbf{u}^{\varepsilon})\partial_{t}\textbf{u}^{\varepsilon}+\sum_{j=1}^{d}J_{\varepsilon}A_{0}A_{j}(J_{\varepsilon}\textbf{u}^{\varepsilon})\partial_{x_{j}}J_{\varepsilon}\textbf{u}^{\varepsilon}= - \left( \begin{array}{c}0 \frac{(\mathbb{I}-\mathbb{P})v^{\varepsilon}}{\varepsilon}\end{array}\right),
L'' ={\rm LCM}\{{\rm q}[a_{21}a_{11}(L/L')^2/2],\, L'\}={\rm LCM}\{{\rm q}[a_{21}a_{11}(L/{\rm q}[a_{12}/(2 L a_{11})])^2/2], \hskip1in{\rm q}[a_{12}/(2 L a_{11})],\, L\},
p^i(\tilde x)|_A=p^i(\tilde y)|_A, \ (\tilde x, \tilde y\in \mathcal Pa(E), \A\in\mathcal F_{i}, \A\subset\Omega_i) \ \ \mbox{implies}\ \p^i\circ h(\tilde x)|_{A}=p^i\circ h(\tilde y)|_{A}\ \mbox{a.s.}
e_1(W_n)(T) + e_1(W_n)(0)=2 e_1(W_n)(0) - \int_0^T \langle C_{n n-1}w_{n-1}, w_n^{\prime}\rangle & \le 2 e_1(W_n)(0) +\frac{T\beta_n}{2}\int_0^T \langle C_{n n-1}w_{n-1}, w_{n-1}\rangle +\frac{1}{T}\int_0^T e_1(W_n) (t)\,dt \,.
|\phi^{N,i}(t,\boldsymbol{x}_t)-\Phi(t,\bar{\nu}_t^i)|=|\Phi(t,\nu^{N,i}_{\boldsymbol{x}_t})-\Phi(t,\bar{\nu}_t^i)|\leq C|\nu^{N,i}_{\boldsymbol{x}_t}-\bar{\nu}_t^i|.
\omega^{(t)}(E^{(t)}) =\frac{1}{\langle\omega^{(t)},\omega^{(t)}\rangle}\omega^{(t)}((\omega^{(t)})^\sharp)=\frac{1}{\langle\omega^{(t)},\omega^{(t)}\rangle}\langle\omega^{(t)},\omega^{(t)}\rangle=1.
L_{0}^{m}+ L_{0}^{g h}= \frac{1}{2}p^{i}p_{i}+ \sum_{n = 1}^{\infty}n ( N_{b n}+ N_{c n}+ \sum_{i = d + 1}^{2 5}N_{i n}) + \sum_{r = 1 / 2}^{\infty}\sum_{a = 0}^{d}r N_{a r},
\mathop{S}_{\r T}^{\d = 2}\mathrm{\LARGE \sl ~ S}_{\n r T}^{\d = 2}={\frac{1}{\lambda}}\int d^{3}x \, \varepsilon^{\mu \nu \rho}\, E_{\mu}^{\underline{{\alpha}}}\, T_{\nu \rho}^{\underline{{\beta}}}\, \eta_{\alpha \beta}.
\frac{\eta_{W}(q^{5})}{\eta_{W}(q)}=\frac{\eta_{W}(q^{2})^{4}}{\eta_{W}(q)^{4}}\frac{\vartheta_{01}(q^{5})}{\vartheta_{01}(q)}[\frac{5\vartheta_{01}(q^{5})^{2}}{4\vartheta_{01}(q)^{2}}-\frac{1}{4}]^{-1}
\nabla \tilde u \cdot \nabla h =-\psi(v)^{-\ell-1}\eta^{s}[\ell \psi'(v) v - \psi(v)] |\nabla \tilde u|^2+ s\psi(v)^{-\ell}\eta^{s-1}\tilde u \, \nabla\tilde u \cdot \nabla \eta .
\bar{\partial}_{A}\alpha = \bar{\partial}_{A}\beta = \bar{\partial}_{A}^{*}\beta = \alpha u = 0 , F_{A'}^{0,2}= \bar{\partial}^{*}u = \bar{\alpha}\beta = 0 , i \Lambda F_{A'}^{1,1}= \frac{s}{8}\left( |u |^2 + |\beta|^2 - |\alpha|^2 \right) .
P_{R}= \frac{\partial L}{\partial \dot{R}}= - \frac{3 \pi R \dot{R}}{2 G}, P_{\phi}= \frac{\partial L}{\partial \dot{\phi}}= \frac{2 \pi^{2}}{G_{\phi}}R^{3}\dot{\phi}, P_{a}= 2 \pi^{2}m R^{3}\dot{n}_{a}
\begin{array}{ll}R_{n+1}^{(1)}(z) = (z+1)R_{n}^{(1)}(z) - 4 \, d_{n+1}^{(1)}z R_{n-1}^{(1)}(z), begin{align*}-0.5ex] n \geq 1, begin{align*}-0.5ex] R_{n+1}^{(2)}(z) = (z+1)R_{n}^{(2)}(z) - 4 \, d_{n+1}^{(2)}z R_{n-1}^{(2)}(z), \end{array}
H^2(e_1,e_1) = 2\, \frac{a^4p^2+b^4q^2}{a^4b^4}, H^2(e_2,e_2) = 2\, \frac{q^2}{a^2\mu^2}= H^2(e_3,e_3), H^2(e_4,e_4) = 2\, \frac{p^2}{b^2\mu^2}= H^2(e_5,e_5).
(a)_{i}= (a^{-1}q^{1 - i})_{i}(- a)^{i}q^{i (i - 1) / 2}, \;\; (a)_{i + j}= (a)_{i}(a q^{i})_{j}, \;\; (a)_{i - j}= \frac{(a)_{i}}{(a^{-1}q^{1 - i})_{j}}(- a^{-1})^{j}q^{\left\{j (j + 1) / 2 \right\}- i j}.
\left\langle f, R_{\mathcal{S}_p(1)}(e_{\zeta}\circ D_t)\right\rangle_{\mathcal{S}_p(1)}= D_tf(\zeta) = f(\widehat{\rho}_t(\zeta))t^{1/2\mu}= \left\langle f, t^{1/2\mu}R_{\mathcal{S}_p(1)}(e_{\widehat{\rho}_t(\zeta)})\right\rangle_{\mathcal{S}_p(1)}.
\frac{1}{\rho}\frac{d}{d \rho}\left( \rho \frac{d \psi_{4}}{d \rho}\right) - \frac{a_{+}^{2}}{\rho^{2}}\psi_{4}- g_{1}\eta^{2}\psi_{4}( \psi_{4}^{2}+ \psi_{6}^{2}- 1 ) - g_{2}\eta^{2}\psi_{4}( \psi_{4}^{2}- \psi_{6}^{2}- c ) = 0 ,
(X_{i_{k}^*}^\top M_{\mathcal{S}_{k-1}}Y)^2 \geq \frac{c_\beta^2\tau_{\min}^2}{2}n^{2-2\xi_\beta}-\tau_{\max}\cdot n^{1+\xi_\epsilon}&= \frac{c_\beta^2\tau_{\min}^2}{2}n^{2-2\xi_\beta}\left[1-\frac{2\tau_{\max}}{c_\beta^2\tau_{\min}^2}n^{\xi_p+3\xi_s^*-1}\right] &\geq \frac{c_\beta^2\tau_{\min}^2}{4}n^{2-2\xi_\beta},
I(C:M)_{(\Phi\otimes\mathbb{I}_M)(\hat{\rho}_{AM})}= S(\Phi(\hat{\rho}_A)) - S(C|M)_{(\Phi\otimes\mathbb{I}_M)(\hat{\rho}_{AM})}&\le n\;g\left(\eta\,E + |1-\eta|\,E_0 + \frac{\eta+|1-\eta|-1}{2}\right) &\phantom{\le}- n\ln\left(\eta\,\mathrm{e}^{-g(E)}+ \left|1-\eta\right| \mathrm{e}^{S_0}\right)\;,
\theta = dz - \sum_{i=1}^n p_i dx_i = - \left(d\tilde{z}- \sum_{i=1}^n x_i dp_i\right), \omega = f(x, z) dx_1 \wedge \cdots \wedge dx_n - g(p, \tilde{z}) dp_1\wedge \cdots \wedge dp_n,
\frac{\partial J}{\partial y_{m}}=L_{y}(t_{m},y_{m},\frac{y_{m+1}-y_{m}}{\Delta t})\Delta t+L_{y^{\prime}}(t_{m-1},y_{m-1},\frac{y_{m}-y_{m-1}}{\Delta t})-L_{y^{\prime}}(t_{m},y_{m},\frac{y_{m+1}-y_{m}}{\Delta t}).
\mu \llcorner_{B_{2r}(z)}\, = \, \int_{Q( B_{2r}(z) )}\mu^{\alpha}q(d\alpha), \mu^{\alpha}\ll \mathcal{H}^{1}\llcorner_{\gamma^{\alpha}}, q\textrm{-a.e.}\, \alpha \in Q( B_{2r}(z) ),
\max_{x>0}g(x)=g(x_1) =\frac{N(p-2)-4s_1}{2N(p-2)}\left(\frac{2ps_1}{C_{N,p,s_1}N(p-2)}\right)^{\frac{4s_1}{N(p-2)-4s_1}}\|u_0\|_2^{\frac{-2N(p-2)-4ps_1}{N(p-2)-4s_1}}&=\mathcal{E}(\phi)M(\phi)^{\sigma_c}\|u_0\|_2^{-2\sigma_c}.
|e_i(\gamma_1,\ldots,\gamma_N)| \leq \sum_{|I| = i}|\gamma_I| & \leq \sum_{|I| = i}\left(\prod_{n=1}^N \max\{1,|\gamma_{n}|\}\right) & = \binom{N}{i}\cdot\prod_{n=1}^N \max\{1,|\gamma_{n}|\}& = \binom{N}{i}\cdot\frac{M(\gamma)}{|a_N|}
\alpha_1=a_0+a_2,\alpha_2=a_1+a_3, \beta_1=b_0+b_2,\beta_2=b_1+b_3, \gamma_1=c_0+c_2,\gamma_2=c_1+c_3, \delta_1=d_0+d_2,\delta_2=d_1+d_3.
\phi^{i}( x ) = \varphi^{i}( x ) + \sum_{j k}a_{j k}^{i}\varphi^{j}( x ) \varphi^{k}( x ) + \sum_{j k l}a_{j k l}^{i}\varphi^{j}( x ) \varphi^{k}( x ) \varphi^{l}( x ) + \cdots .
C_t = \sup \{||U(\cdot,0)u_0||_{L^{q_1}(0,t;W^{2,p_1})};||u_0||_{H^2}\le 1 \}&+\sup \left\{\left| \left| \int_0^{\cdot}U(\cdot,s)f(s)ds \right| \right|_{L^{q_1}(0,t;W^{2,p_1})};||f||_{L^{q'_2}(0,t;W^{2,p'_2})}=1 \right\}.
\sum_{k=\max(1,\ell-n_p)}^{\ell-1}w_{\ell-1}\dots w_k\cdot \frac{\vert\lambda_\ell-\lambda_{j(\ell)}\vert}{\vert \lambda_k-\lambda_{j(k)}\vert}\cdot\big\vert s_{k,\ell}^{(n_p)}\big\vert,
p\left({{d_{p2}}\leq \delta}\right) = M\int_{\mathbf{\Omega}}\prod\limits_{i = 1}^k{{{\left({{{\sin}^2}{\theta_i}}\right)}^{n - 2k}}}\,\,\, \times \prod\limits_{1 \leq i < j}^k{{{\left({{{\sin}^2}{\theta_i}-{{\sin}^2}{\theta_j}}\right)}^2}}d{{\sin}^2}{\theta_1}\cdots d{{\sin}^2}{\theta_k}
u(x)\leq \left\lbrace\begin{array}{ll}\displaystyle{\delta(x)}^s \hbox{for}0\leq\beta< s, \displaystyle{\delta(x)}^s\log\frac{1}{\delta(x)}\hbox{for}\beta=s, \displaystyle{\delta(x)}^{-\beta+2s}\hbox{for}s<\beta<1+s.\end{array}\right.
\|s_1\|_{L^{\infty}(\partial D_{R_0})}= \sup_{z \in \partial D_{R_0}}|g(z)|\|s_0(z)\|^d\le \|s_0\|^d_{L^{\infty}(\partial D_{R_0})}JR_0^m\max\{1,(2R_0)^J\}\max_{0\le j < J}|a_j| .
S^{(D)}_{0}(k)=\frac{1 - \lambda \, e^{i \pi \nu}\, \Xi (-2 \nu) \, \left(E/\mu^{2}\right)^{\nu}}{1 - \lambda \, e^{-i \pi \nu}\, \Xi (-2 \nu) \,\left(E/\mu^{2}\right)^{\nu}}\; ,
V =\frac{1}{\nu (G,w)}\left\{\int_0^1\frac{B^2(t)}{g^2(Q_G(t))}w(t)\mathrm{d}t- \left[\int_0^1 \frac{B(t)}{g(Q_G(t))}w(t) \mathrm{d}t\right]^2\right\}\ \ \ -\left[\frac{1}{\nu(G,w)}\int_0^1 \frac{B(t) Q_G(t)}{g(Q_G(t))}w(t)\mathrm{d}t-\frac{\mu_1(G,w)}{\nu(G,w)}\int_0^1\frac{B(t)}{g(Q_G(t))}w(t) \mathrm{d}t\right]^2\ ,
\Lambda_{n}^{\mu}=\begin{cases}\displaystyle\sqrt{\frac{2}{\Gamma(\frac{n+1}{2}+\mu)(\frac{n}{2})!}}(-1)^{n/2}n , \displaystyle\sqrt{\frac{2}{\Gamma(\frac{n}{2}+\mu+1)(\frac{n-1}{2})!}}(-1)^{\frac{n-1}{2}}n .\end{cases}
\beta F = - \frac{c}{\beta^{2}}\int d \phi \int_{r_{H}+ h}^{L}d r \frac{\sqrt{g_{3}}}{( - g_{t t}^{'})^{3 / 2}}= - c \int_{0}^{\beta}d \tau \int d \phi \int_{r_{H}+ h}^{L}d r \sqrt{g_{3}}\frac{1}{\beta_{l o c a l}^{3}},
H_{\{2,3,4\}}(5 x_2^2 x_3 + 7 x_2^2 x_3 x_4^5 + x_1x_2x_3x_4 + 4 x_4^3 x_5 + 13 x_2^6 x_3^8 x_4^7) = 7 x_2^2 x_3 x_4^5 + 13 x_2^6 x_3^8 x_4^7
\left. \begin{array}{l l}B( \textbf{x})\!\!\! = (x_E-a(\textbf{x}))^2 + (y_E-b(\textbf{x}))^2 +z_E^2 &~~- (x_1-a(\textbf{x}))^2 - (y_1-b(\textbf{x}))^2 - z_1^2.\end{array}\right.
\sum_{\delta\in\{0,1\}^{n}}u_{\delta}\,x_{1}^{\delta_{1}}\cdots\,x_{k-2}^{\delta_{k-2}}\cdot x_{n}^{\sum_{k-1}^{n}\delta_{i}}=\sum_{\substack{\delta\in\{0,1\}^{n}, \delta_{k-1}=\cdots=\delta_{n-1}=0}}v_{\delta}\,x_{n}^{\delta_{n}}\prod_{i=1}^{k-2}\left(x_{i}-x_{n}\right)^{\delta_{i}}
T \geq \frac{N_T -1}{2}g\biggl( \log_2 \Bigl(\frac{N_T-1}{8}\Bigr) \biggr) = \frac{N_T-1}{2}\left( -1 + \frac{1}{2d}2^{\frac{2}{d}\log_2 \bigl( (N_T-1)/8\bigr)}\right) & = - \frac{N_T-1}{2}+ \frac{N_T-1}{4d}\left( \frac{N_T-1}{8}\right)^{2/d}\geq \underbrace{- \frac{N_T-1}{2}}_{\geq - T /2}+ \frac{2}{d}\left( \frac{N_T-1}{8}\right)^{1+2/d}\,.
{[}\hat{D}^{(+)}_\lambda(\vec{k}),\hat{Q}^a]=-k_0\varepsilon_{\lambda \lambda'}\hat{C}^{a(+)}_{\lambda'}(\vec{k}),\;\; \varepsilon_{\lambda\lambda'}=-\varepsilon_{\lambda'\lambda},\;\; \varepsilon_{12}=-1,
\theta_2 \partial_1(a)b = \theta_1 \partial_2(a)b, \qquad\! \theta_2 \mu(\partial_1(a))v = \theta_1\mu(\partial_2(a))v, \qquad\! \theta_2 \alpha_1(\mu(a)v) = \theta_1 \alpha_2(\mu(a)v),\!\!\!\!
|\lambda_{\mathbf{d}}|=d\sum_{\mathbf{d}|\mathbf{r}}\frac{y_{\mathbf{r}}}{\phi_\omega(r)}=\frac{d W^k B^k \mathfrak{S}_{WB}(\mathcal{L})}{\phi_\omega(d)\phi(W B)^k}\sum_{\mathbf{d}|\mathbf{r}\in\mathcal{D}_k}\frac{1}{\phi_\omega(r/d)}F\Bigl(\frac{\log{r_1}}{\log{R}},\dots,\frac{\log{r_k}}{\log{R}}\Bigr).
\frac{R_{21}}{i-\omega(1,1)}u_1+\frac{R_{23}}{i-\omega(3,3)}u_3=-i(iR_{21}-iR_{23})=R_{21}-R_{23}=-1=u_2; \frac{R_{41}}{i-\omega(1,1)}u_1+\frac{R_{43}}{i-\omega(3,3)}u_3=R_{41}-R_{43}=-1-(-2)=1=u_4.
\sum_{n=1}^{N}\frac{(-q)_n}{(q)_n}\frac{q^n}{1-q^n}- 2\sum_{n=1}^{N}\left[\begin{matrix}N n \end{matrix}\right] \frac{q^{n(n+3)/2}(-q)_{n-1}}{(q)_n (1-q^n)}= \sum_{n=1}^{N}\frac{q^n}{1-q^n}.
\|\omega_z\|_{q/2,u}=\bigg(\frac1{z^q}\int_0^zu(t)\,dt+\int_z^\infty u(t)\,\frac{dt}{t^q}\bigg)^{q/2}\le(1+b_{q/2}(u))^{2/q}\bigg( \frac1{z^q}\int_0^zu(t)\,dt\bigg)^{2/q}.
p^{7}=\left[\begin{array}{ccccc}1 1 1 2 2 2 2 2 3 3 3 3 3 1 1 \end{array}\right],p^{8}=\left[\begin{array}{ccccc}1 1 1 2 3 2 2 2 1 2 3 3 3 3 1 \end{array}\right],p^{9}=\left[\begin{array}{ccccc}1 1 1 2 1 2 2 2 1 3 3 3 3 3 2 \end{array}\right],
L' =-{1\over 4g^4}trF_{\mu\nu}F^{\mu\nu}+L_{MAT}-{1\over 2\alpha}tr(\partial_{\mu}A^{\mu})^2+{1\over\alpha}trcM\overline c+tr(\rho^{\mu}\Delta A_{\mu}+U\Delta\overline c)+Y^a\Delta\phi_a
m_{p,q}(c) =\frac{c-\alpha}{c}m_{p,q}(c)+\frac{\alpha}{c}_{p,q}m_{p,q}(c) &=\frac{c-\alpha}{c}m_{p,q}(\frac{c}{c-\alpha}(c-\alpha))+\frac{\alpha}{c}m_{p,q}(\frac{c}{\alpha}\alpha) &\leq m_{p,q}(c-\alpha)+m_{p,q}(\alpha),
\sum_{k=1}^{N}T_{\varepsilon,\eta,d,k,B}=\frac{(\varepsilon-\eta) K^2}{2\alpha^2 d^2}B^{2-\frac{1}{r}}+O_{\varepsilon,\eta}\left(\frac{K^2 B^{2-\frac{1}{r}}}{N d^2}\right)+O_{\sigma}\left(\frac{K^\sigma B^\sigma N}{d^\sigma}\right)
\tau^1_{\mu, \mu}= (1 - q^{-1}) \frac{\mathbf{z}^{\left(n_\alpha \lceil \frac{B(\alpha^\vee, \mu)}{n_\alpha Q(\alpha^\vee)}\rceil - \frac{B(\alpha^\vee, \mu)}{Q(\alpha^\vee)}\right) \alpha^\vee}}{1-q^{-1}\mathbf{z}^{n_\alpha \alpha^\vee}}
\lambda_{N,\vec{x}}- \tilde{\lambda}_{N,\vec{x}}= \frac{1}{N}\sum_{(b_1,\vec{b}_\ast):\;\mathcal{U}_{b_1,\vec{b}_\ast}\neq 0, \; \mathcal{U}_{b_1-1,\vec{b}_\ast}\neq 0}\Big(\mathcal{U}_{b_1,\vec{b}_\ast}-\mathcal{U}_{b_1-1,\vec{b}_\ast}\Big)\delta_{b_1,\vec{b}_\ast}
\begin{aligned}w \nabla_{ii}w \geq \, \nabla_{lii}u \nabla_l u + R^{k}_{iil}\nabla_{k}u \nabla_{l}u \geq \, \nabla_{l}u \nabla_l U_{ii}- \frac{w^2}{\phi}(a A^{ii}_{p_{k}}\nabla_{k}\phi + \phi A^{ii}_{u}) - \nabla_l u \nabla'_l A^{ii}.\end{aligned}
\frac{1}{2\omega_{p}}[\begin{matrix}\frac{1}{2}f_{0}\omega_{n}^{2}&-(\omega_{p}^{2}-\omega_{n}^{2})\\ \omega_{p}^{2}-\omega_{n}^{2}&-\frac{1}{2}f_{0}\omega_{n}^{2}\end{matrix}]
S_{D}= - \int d^{4}\sigma \sqrt{- \mathrm{d e t}\, \Big ( G_{\mu \nu}+ \frac{1}{\sqrt{1 + C_{0}^{2}}}( \tilde{F}_{\mu \nu}+ C_{\mu \nu}+ C_{0}b_{\mu \nu}) \Big )}+ \int \Omega_{D},
V_m^{(H)}=\sum_{i=2m}^{\infty}\nu_i^{(m)}u^{i-2m}L_i+\sum_{j=2m+3}^{\infty}\gamma_j^{(m)}ju^{j-2m-3}\frac{\partial}{\partial q_j}-\frac{u^2}{24}\delta_{m,-1}+\frac{1}{8}\delta_{m,0},
\Psi = \left( \begin{matrix}{{{\frac{i G_{\ell j}}{r}}\phi_{j m}^{\ell}}}\\{{{\frac{F_{\ell j}}{r}}{{\mathrm{\boldmath ~ \s i g m a ~}}\cdot{\bf{\hat{r}}}}\phi_{j m}^{\ell}}}\\ \end{matrix}\right)
P_{X(U_{n-k},...,U_{n})}d_k^* \left(\prod_{i=1}^{n-k-1}P_{X(U_{i},...,U_{k+1+i})}d_k d^*_k \right) P_{X(U_0,...,U_{k+1})}d_k P_{X(U_0,...,U_{k})}= \left(\prod_{i=1}^{n-k}P_{X(U_{i},...,U_{k+i})}d_k^* d_k \right) P_{X(U_{0},...,U_{k})}.
\rho^{\rm tot}= \rho+{\rho^2\over2\lambda}+{\widetilde{\kappa}^{4}{\cal U}\over\kappa^6}+{\Lambda\over\kappa^2}, p^{\rm tot}= p+{\rho^2\over2\lambda}+{\rho p\over\lambda}-{\widetilde{\kappa}^{4}{\cal U}\over3\kappa^6}-{\Lambda\over\kappa^2}\,.
\int_\Omega (\sum_i |\nabla P_i|^2) u_1^2 \geq (\lambda_2 - \lambda_1) \int_\Omega (\sum_i P_i^2) u_1^2 = (\lambda_2 - \lambda_1) \int_\Omega h^2(\sigma(r_p)) u_1^2 .
\langle \textbf{K}\left[f \right](h),h \rangle =\int_E \textbf{K}\left[f\right](h)(x)h(x)dx &=\int_E\int_E \sqrt{f(x)}K(x,y)\sqrt{f(y)}h(y)h(x)dydx &=\int_E\int_E \sqrt{\frac{f(x)}{g(x)}}\sqrt{g(x)}K(x,y)\sqrt{g(y)}\sqrt{\frac{f(y)}{g(y)}}h(y)h(x)dydx &=\langle \textbf{K}\left[g \right]\left(l h\right), l h\rangle
x^T(t)P_2^{-1}x(t) \leq \left[\int_0^t x^T(s)(A^T P_2^{-1}+P_2^{-1}A+ \sum_{i=1}^m N_i^T P_2^{-1}N_i) x(s)\right. ds \ \ \ \ \ \ \ \ \ \left.+2\int_0^t x^T(s) P_2^{-1}B u(s) ds+\int_0^t x^T(s)P_2^{-1}x(s) \left\|u(s)\right\|_2^2 ds\right].
V^*\pi(J_2)^{*\gamma}\pi(J_1)^{*\beta}\pi(J_1)^\beta\pi(J_2)^\gamma V = T_2^{*\gamma}T_1^{*\beta}T_1^\beta T_2^\gamma &=V^*\pi(J_2)^{*\gamma}\pi(J_1)^{*\beta}VV^*\pi(J_1)^\beta\pi(J_2)^\gamma V
f_{{\bf{q}}}= ( \frac{\omega_{{\bf{q}}}+ \epsilon_{{\bf{q}}}+ \rho_{0}v_{{\bf{q}}}}{2 \mathrm{~}\omega_{{\bf{q}}}})^{\frac{1}{2}}d_{{\bf{q}}/ 2}({\bf{q}}) + ( \frac{- \omega_{{\bf{q}}}+ \epsilon_{{\bf{q}}}+ \rho_{0}v_{{\bf{q}}}}{2 \mathrm{~}\omega_{{\bf{q}}}})^{\frac{1}{2}}d_{-{\bf{q}}/ 2}^{\dagger}( -{\bf{q}})
-\mathrm{div}\,g(\mathbf{x})\nabla \widetilde{\Lambda}_1(\mathbf{x})+v(\mathbf{x})=0,\int_\Omega \widetilde{\Lambda}_1(\mathbf{x})\,d\mathbf{x}=0, &-\mathrm{div}\,g(\mathbf{x})\nabla \widetilde{\Lambda}_2(\mathbf{x})+\mathrm{div}\,g(\mathbf{x})\mathbf{A}(\mathbf{x})=0,\int_\Omega\widetilde{\Lambda}_2(\mathbf{x})\,d\mathbf{x}=0.
T_{\xi_{1}}\circ\dots \circ T_{\xi_{\ell}}(I)\cap T_{\omega_{1}}\circ\dots\circ T_{\omega_{s}}(I)=\emptyset\mbox{and}&T_{\xi_{1}}\circ\dots \circ T_{\xi_{\ell}}(I)\cup T_{\omega_{1}}\circ\dots\circ T_{\omega_{s}}(I)\subset J.
\int_{\mathcal{B}{\epsilon , k}}\frac{\overline{\widehat{\varphi}(kp)}\widehat{\eta}(kp)}{(p^2 +1)^2}\mathrm{d}p= \pi |\overline{\widehat{\varphi}(\theta)}\widehat{\eta}(\theta)|^2+\mathcal{O}(k^2 )\,,
\int_DU'(y(T,x,z))dx\mathbb{E}[\delta_Z(z)|\mathcal{F}_T]=\tilde{p}(0,z) \exp(-\int_0^T \frac{a_0(s,z)}{b_0(s,z)}dB(s) - \frac{1}{2}\int_0^T (\frac{a_0(s,z)}{b_0(s,z)})^{2}ds).
d s^{2}= - ( d T + J d \phi )^{2}+ d R^{2}+ \alpha^{2}R^{2}d \phi^{2}= - ( d T + J d \phi )^{2}+ \alpha^{2}r_{0}^{2}( \zeta^{\alpha - 1})^{2}( d \zeta^{2}+ \zeta^{2}d \phi^{2}) ,
\left|\left\{g\in\mathcal G:\big|\left[b, R_j\right]\psi(g)\big|>{A_5\over 2}M^{1+{1\over Q}}\right\}\right| \leq\theta_b\int_{\mathcal G}{2|\psi(g)|\over A_5 M^{1+{1\over Q}}}\Big(1+\log^+\Big({2|\psi(g)|\over A_5 M^{1+{1\over Q}}}\Big)\Big)dg &\leq A_8\theta_bM^{-{1\over Q}},
\frac{\partial \mathrm{In_1}(\alpha ,\beta ,c,p,\mu_1,\mu_1)}{\partial \beta}=-\frac{2\beta^{\mu_1}}{\alpha^{\mu_1-1}\left(2p+\alpha^2\right)}\mathrm{exp}\left[\frac{-2p\cdot \beta^2+c^2}{2(2p+\alpha^2)}\right]I_{\mu_1-1}\left(\frac{\alpha \beta c}{2p+\alpha^2}\right),
\exp\left( -ikR\cos\theta\right) =\sum_{l_{1}=0}^{\infty}\left( -i\right)^{l_{1}}\left( 2l_{1}+1\right) j_{l_{1}}\left( kR\right) D_{00}^{\left(l_{1}\right)}\left( 0,-\theta,-\phi\right) , number
\lim_{n\to \infty}\Big[ \int_{\mathbb{R}^N}|\nabla v_n|^2dx+\int_{\mathbb{R}^N}A(|v_n(x)|)dx\Big]= \int_{\mathbb{R}^N}|\nabla \varphi|^2dx+\int_{\mathbb{R}^N}A(|\varphi(x)|)dx.
\sum_{j=0}^{d-1}B_{\tilde{n}}\left(\frac{j}{d}\right)=\frac{B_{\tilde{n}}(0)}{d^{\tilde{n}-1}}, \sum_{0\le j_i\le d_i-1,\atop(i=1,2)}B_{\tilde{n}}\left(\left\{\frac{j_1}{d_1}+\frac{j_2}{d_2}\right\}\right)=\frac{B_{\tilde{n}}(0)}{(d_1d_2)^{\tilde{n}-1}}.
L_B=-{1\over 4}H^2{}_{\mu\nu}(B) +{m\over 4!}\epsilon^{\mu\nu\rho\omega}\, B_\mu\,\partial_{\,[\,\nu}A_{\rho\sigma\,]}-{1\over 2\cdot 3!}F^2{}_{\mu\nu\rho}(A)
\begin{aligned}\left(\sup_{0 \le s \le \tau}\int_0^\tau \mathbf{1}_{s\le t}\|S_{Z}(t,s)\|^2 dt \right)^{1/2}\le e^{C_0}\left(\int_0^{\tau}e^{(t-\tau_1) \epsilon |x_0|}dt\right)^{1/2}\le \frac{e^{C_0}}{\sqrt{\epsilon}}|x_0|^{\epsilon - 1/2}\end{aligned}