harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\ell_m(\gamma) \leq nr_k(I)M^{-k}+ n(M-2n+1) \sum_{j=k}^m r_j(y)M^{-j}& \leq nr_k(I)M^{-k}+ n(M-2n+1)r_k(y)M^{-k}\sum_{j=0}^{m-k}\left(\frac{M-2n+1}{M}\right)^j \leq nr_k(I)M^{-k}+ n(M-2n+1)Rr_k(I)M^{-k}\left( \frac{1}{1-(M-2n+1)/M}\right).
=-\left(\frac{\partial_{xx}^{2}\eta}{1+\left(\partial_{x}u\right)^{2}}\,+\,\left(\,a\left(x,\,t\right)\partial_{xx}^{2}u+b\left(x,\,t\right)\,\right)\,\partial_{x}\eta\right)\left(\partial_{x}u\right)
\dot Q_{B}= \frac{q^2\;\dot c\; (p^2 - m^2)}{1 + q^2}(1 - q^2)\quad \mbox{and}\quad\dot Q_{r}= \frac{d}{d\tau}(\lambda\;e)\;\frac{p^2-m^2}{1+q^2}(1-q^2).
I ={\frac{1}{1 6 \pi G}}\int d^{4}x \sqrt{- g}\biggl \{R -{\frac{1}{2 n^{2}}}\sum_{i = 1}^{n}\sum_{j = i + 1}^{n}( \partial \sigma_{i}- \partial \sigma_{j})^{2}-{\frac{1}{n}}\sum_{i = 1}^{n}e^{- \sigma_{i}}F_{( i )}^{2}\biggr \},
\sup_{t> \xi}\dfrac{\| k^{\alpha ,\beta}_t \|_{q,u}}{\|k^{\alpha,\beta}_t \|_{p,v}}=\bigg( \sup_{t> \xi}\dfrac{\| k^{q\alpha ,q\beta}_t \|_{1,u}}{\|k^{q\alpha,q\beta}_t \|_{p/q,v}}\bigg)^{1/q}.
\gamma_{3}=(a_{n}b_{1},\dots,a_{n}b_{n-1}), &\gamma_{4}=\frac{1}{|\beta|}\bigl((a_{1}^{2}-a_{n}^{2}b_{n})b_{1},\dots, (a_{n-1}^{2}-a_{n}^{2}b_{n})b_{n-1}\bigr).
\frac{d^{2}\phi ( t )}{d t^{2}}+ m^{2}( t ) \phi ( t ) + \frac{\lambda}{6}\phi^{3}( t ) + \frac{\lambda}{2}\phi ( t ) \int \frac{d^{3}k}{( 2 \pi )^{3}}( - i ) G_{k}( t , t ) = 0
\partial^\mu (\sqrt{-g}R_{\mu})\,=\,\frac{1}{192\pi^2}\left[(N_c^2-1) -\frac{1}{3}\sum_i (-1 +c_i^{(0)}) \right]\epsilon_{\mu\nu\lambda\delta}R^{\mu\nu\sigma\rho}R^{\lambda\delta}_{\sigma\rho}\; ,
\zeta_{AV,2}(s_1,s_2,s_3) = \sum_{m \leq at_3}\sum_{n<m}\frac{1}{m^{s_1}n^{s_2}(m+n)^{s_3}}+\begin{cases}O ( t_3^{\frac{1}{2}-\sigma_1-\sigma_3}) (\sigma_2>\frac{3}{2}) O (t_3^{\frac{1}{2}-\sigma_1-\sigma_3}\log t_3 ) (\sigma_2=\frac{3}{2}) O ( t_3^{\frac{3}{2}-\sigma_1-\sigma_2-\sigma_3}) (\sigma_2<\frac{3}{2}), \end{cases}
\lambda_0^n(f_1,f_2) := \inf\{t \geq 0: |f_1(t) - f_2(t)| = 1/n, (f_1(s_1) - f_2(s_1))(f_1(s_2) - f_2(s_2)) > 0 & s_1,s_2 \in (t,t+1)\}\lambda_1^n(f_1,f_2) := \inf\{t > \lambda_0^n : f_1(t) = f_2(t)\}.
\mu \times \mu ((T^p\times T^q)^{t_n}(E)\cap F) \geq \mu \times \mu ((T^p\times T^q)^{t_n}(A\times B)\cap (C\times D)) -\mu\times\mu(A\times B\setminus E)-\mu\times\mu(C\times D\setminus F) &>\frac{1}{16}\mu (C)\mu (D)-\frac{1}{32}\mu (A)\mu (B) -\frac{1}{32}\mu (C)\mu (D) &=0.
0 = Q^6 b_1 + b_1^4 0 = Q^{10}b_1 + (Q^4 b_1)^2 Q^6 b_2 = Q^8 b_1 + b_1^2 Q^4 b_1 0 = Q^{10}b_2 + b_1^2 Q^6 b_2
\|T(t_{n_k})x-T(t_{n_\ell})x\|=\|T(q_{n_k}t)T(r_{n_k})x-T(q_{n_\ell}t)T(r_{n_\ell})x\| &\quad\leq \|T(r_{n_k})(T(q_{n_k}t)-T(q_{n_\ell}t))x\|+\|T(q_{n_\ell}t)(T(r_{n_k})x-T(r_{n_\ell})x)\| &\quad\leq M\|(T(q_{n_k}t)x-T(q_{n_\ell}t)x)\|+M\|(T(r_{n_k})x-T(r_{n_\ell})x)\|\to 0\quad\mbox{as $k,\ell\to\infty$},
U(x,t) = \frac{1}{\sqrt{4\,\pi\,K_{\alpha}\,t^{\alpha}}}\,\sum\limits_{j=0}^{\infty}\frac{(-k_\ast\,t^{\alpha})^j}{j!}\,H^{2,0}_{1,2}\left[ \frac{x^2}{4\,K_{\alpha}\,t^{\alpha}}\left|\begin{array}{ll}\displaystyle \left( 1-\frac{\alpha}{2}+ \alpha\,j,\alpha\right) [10pt](0,1) \left( \frac{1}{2}+ j,1\right) \end{array}\right.\right].
h_{\ell}^{\prime \prime}( i t , r ) = - 2 t \xi_{\ell}( t r ) h_{\ell}^{\prime}( i t , r ) + \left[ \sigma ( r ) - 2 t^{2}\left. \frac{d \xi_{\ell}( \tau )}{d \tau}\right|_{\tau = t r}\right] h_{\ell}( i t , r )
P_{A,\theta}\left( \sigma(x^1) = y^1,\ldots,\sigma(x^p) = y^p \right) = (-1)^{\theta_1 g_1 + \theta_2 g_2}\prod_{i = 1}^k w_{A}(x^i,y^i) \det \limits_{i,j = 1}^k \left( K^{-1}_{A,\theta}( y^i,x^j) \right).
{\delta_{\rho}}{\mathcal{F}_D^0}={{Q^2}\over{8\pi}}\int{d^2}\xi\sqrt{\hat{g}}\hat{R}\rho+{{Q^2}\over{8\pi}}\oint{d}\hat{s}(\xi)\int{d^2}\xi'\rho(\xi){\partial_{\hat{n}}}{\hat{G}_D}(\xi,\xi')\sqrt{\hat{g}(\xi')}\hat{R}(\xi').
\left(\theta_{f(n)}-\theta_{f(n+1)}\right)\sum_{j=f(n)}^{f(n+1)}\lambda_j \le\frac{2^{d-1}dq(d+1-p)}{(1-p)^2}n^{-\frac{1}{2}}+\frac{2^pq(d+1-p)}{1-p}n^{-\frac{d(p+q)}{1-p}-1}\le\frac{2^{d-1}dq(d+1-p)}{(1-p)^2}n^{-\frac{1}{2}}+\frac{2^pq(d+1-p)}{(1-p)^2}n^{-\frac{1}{2}}\le\frac{2^ddq(d+1-p)}{(1-p)^2}n^{-\frac{1}{2}}.
\|(H_{q}[\omega,0]\upharpoonright\Lambda-E)\widetilde\psi\| =(|\widetilde\psi({\nu-M-1})|^{2}+|\widetilde\psi({\nu+M+1})|^{2})^{\frac{1}{2}}&< 2e^{-(\min\gamma-2\varepsilon_{1})(M+1)}\|\psi\upharpoonright\Lambda\|^{-1}<4e^{-(\min\gamma-2\varepsilon_{1}) M}< 4\delta_{0}
{\mathbf T}_j(\theta,\ell) = t_{0,j}+ h t_{1,j}(\theta) +O(h^2) (1\le j \le k+1 ),{\mathbf Y}_j(\theta,\ell) =y_{0,j}+ h y_{1,j}(\theta) +O(h^2) (1\le j \le k),{\mathbf Y}_t(\theta,\ell) = 1+h y_{1,t}(\theta) +O(h^2) .
|\langle S, h \rangle| = |\langle g, \sum_{j}P_j^{\psi}( h P_{j}^{\phi}f) \rangle| & \lesssim \| g \|_{\operatorname{BMO}}\| \sum_{j}P_j^{\psi}(h P_{j}^{\phi}f ) \|_{\mathcal H^1},
(v\cdot b)_{(0)}\otimes_B a_{(2)}\otimes S(a_{(1)}) (v\cdot b)_{(1)}a_{(3)}=v_{(0)}\cdot b_{(2)}\otimes_B a_{(2)}\otimes S(a_{(1)})S(b_{(1)}) v_{(1)}b_{(3)}a_{(3)}& = v_{(0)}\otimes_B b_{(2)}a_{(2)}\otimes S(b_{(1)}a_{(1)}) v_{(1)}b_{(3)}a_{(3)}
A_{2k-2}^{\left[ j\right]}\left( \theta\right) =\frac{4}{2k\left(2k-1\right)}\frac{d^{2}}{d\theta^{2}}~A_{2k}^{\left[ j\right]}\left(\theta\right) +\left( -4\right)^{j-k+1}\times t\left( 2+2j,2k\right)\times\frac{\left( 2k-2\right) !}{\left( 2j\right) !}~A_{2j}^{\left[j\right]}\left( \theta\right) \ ,
\phi(-q^{1/2}x_3) \phi(-q^{1/2}x_1) \phi(-q^{1/2}x_2) & = \phi(-q^{1/2}x_3) \phi(-q^{1/2}x_2)\phi(-q x_1 x_2) \phi(-q^{1/2}x_1) & =\phi(-q^{1/2}x_2)\phi(-q x_3 x_2)\phi(-q^{1/2}x_3) \phi(-q x_1 x_2) \phi(-q^{1/2}x_1) & =\phi(-q^{1/2}x_2)\phi(-q x_3 x_2) \phi(-q x_1 x_2) \phi(-q^{3/2}x_3 x_1 x_2) \phi(-q^{1/2}x_3) \phi(-q^{1/2}x_1)
|\frac{\sin(2(v_{1}+v_{2}+v_{3}+v_{4}))}{2r^{2}}\left(\cos(2Q_{\frac{1}{\lambda(t)}})\left(\cos(2v_{5})-1\right)-\sin(2Q_{\frac{1}{\lambda(t)}})\sin(2v_{5})\right)| &\leq C \frac{|v_{1}+v_{2}+v_{3}+v_{4}|}{r^{2}}\left(v_{5}(t,r)^{2}+ \frac{r \lambda(t)}{r^{2}+\lambda(t)^{2}}|v_{5}(t,r)|\right)
s_1(z, x) = \zeta^{-1}\varphi_1(\zeta,x,\alpha_0), c_1(z, x) = \vartheta_1(\zeta,x,\alpha_0), & s_2(z, x) = \zeta^{-1}\vartheta_2(\zeta,x,\alpha_0), c_2(z, x) = \varphi_2(\zeta,x,\alpha_0), z=\zeta^2, \; x \geq 0.
\rho(e_0)=\begin{pmatrix}0 -1 1 0\end{pmatrix}, \; \rho(e_1)=\begin{pmatrix}0 i i 0\end{pmatrix},\; \rho(e_1)=\begin{pmatrix}0 j j 0\end{pmatrix},\; \rho(e_1)=\begin{pmatrix}0 k k 0\end{pmatrix}.
\langle \zeta^{a}\left( z\right) \zeta^{b}\left( w\right) \rangle = \frac{i}{\pi}\theta^{ab}\mathcal{A}\left( z,w\right) - \frac{1}{\pi}G^{ab}\mathcal{B}\left( z,w\right) + \frac{1}{2\pi}g^{ab}\mathcal{C}\left( z,w\right) ,
\Phi_k(\sigma_1,\sigma_2)=\sum_{s \in{\bf Z}}\sum_{j=1}^{m}E \left( \frac{m}{n}\left(\frac{\sigma_2}{2\pi}+k+ns\right)+j, i \sigma_1 \right) \widetilde{\phi}_j\left(\frac{\sigma_2}{2\pi}+k+ns\right).
B_{q_1, q_2}(p; x_1, x_2) - p(x_1, x_2) = \sum_{k = 0}^{q_1}\sum_{l = 0}^{q_2}\Bigg( \sum_{i = 0}^{n_1}\sum_{i = 0}^{n_2}a_{i,j}\delta_{k,l}^{q_1, q_2}(i, j) \Bigg) b_{k, l}^{q_1, q_2}(x_1, x_2).
D_{\mu \nu}( k ) = \frac{- i}{k^{2}- \theta_{i n}^{2}}\left( P_{\mu \nu}\, \, - \, \, i \theta_{i n}\epsilon_{\mu \nu \rho}\frac{k^{\rho}}{k^{2}}\right) - i \lambda \frac{k_{\mu}k_{\nu}}{k^{4}}f ( k^{2}) ,
\int_{0}^{t}e^{-|\xi|^{2\alpha}(t-s)}|H(\xi,s)|ds\leq C|\xi|\int_{0}^{t}(s+1)^{-\frac{5-4\alpha}{2\alpha}}ds \leq C\frac{2\alpha}{6\alpha-5}|\xi|((t+1)^{\frac{6\alpha-5}{2\alpha}}-1) \leq C\frac{2\alpha}{6\alpha-5}(t+1)^{-\frac{1}{2\alpha}}((t+1)^{\frac{6\alpha-5}{2\alpha}}-1) \leq C.
O_{val_{K}}=\left\{x\in K(t^{H},\beta)\,\vert\,val_{K}(x)\geq0\right\}= &=\left\{x\in K(t^{H},\beta)_{\delta}\,\vert\,h_{0}>0\,\,\,\,h_{0}=0\,\wedge\,v(a_{h_{0}})\geq0,\,h_{0}=min\,supp(x)\right\}= &=\left\{x\in K[t^{H},\beta]_{\delta}\,\vert\,x\in(t)\,\,\,\,v(a_{0})\geq0\right\}= &=O_{K}+(t),
-\frac{1}{2}\left(\frac{\delta S_0}{\delta \phi}\frac{\delta S_1}{\delta \rho}+\frac{\delta S_0}{\delta \rho}\frac{\delta S_1}{\delta \phi}\right)-\frac{1}{2i}\frac{\delta^2 S_0}{\delta \rho\delta \phi}+\frac{1}{2}\left( e^{-iS_1}\frac{-\delta^2 (e^{iS_1})}{\delta f^2}+{f^\prime}^2\right)=0.
(NP)^\tau_\varepsilon N_\varepsilon^*(\tau) := \displaystyle\min_{f\in L^{\infty}((0,\tau);U)}\left\lbrace \|f\|_{L^{\infty}((0,\tau);U)}\ |\ \mathbb{T}_t^\varepsilon \psi+\Phi_t^\varepsilon f\in B(0,r)\right\rbrace;
U = \hat{U}_{1 2}\cdot \cdot \cdot \hat{U}_{1 n}\hat{U}_{2 3}\cdot \cdot \cdot \hat{U}_{2 n}\cdot \cdot \cdot \hat{U}_{i , i + 1}\cdot \cdot \cdot \hat{U}_{i n}\cdot \cdot \cdot \hat{U}_{n - 1 , n}\ \ ,
\le\sum_{a^{n}\in T_{\delta}^{A^{n}}}Pr\{E_{a^{n}}\}Pr_{S}\{\exists E_{b^{n}}:b^{n}\in T_{\delta}^{p^{n}},b^{n}\ne a^{n},E_{a^{n}}^{\dagger}E_{b^{n}}\in N(S)\}
M R_{P}M^{- 1}= \left( \begin{array}{c c}{{- \cos 2 ( \phi_{P}+ \phi_{C})}}&{{- \sin 2 ( \phi_{P}+ \phi_{C})}}\\{{- \sin 2 ( \phi_{P}+ \phi_{C})}}&{{\cos 2 ( \phi_{P}+ \phi_{C})}}\\ \end{array}\right) .
N_{\gamma}\coloneqq \begin{pmatrix}V_0 V_1 -\gamma W_0 -\gamma W_1\end{pmatrix},\widetilde{N}_{\gamma}\coloneqq \begin{pmatrix}V_1 V_0 \gamma W_1 \gamma W_0\end{pmatrix},F_{\gamma}\coloneqq \begin{pmatrix}E_{\gamma}0 0 E_{\gamma}\end{pmatrix},
\begin{aligned}F(Y_t,\widetilde Y_t) \le - 4(1-\varepsilon)p\nu^2\left(\frac{\kappa}{\nu^2}+ 1 - \sqrt{2}\right) \int_0^t Z_s\,ds &+ 4p^2\varepsilon^{2p-1}\nu^2 \int_0^t \|X_s-\widetilde X_s\|^2\,ds.\end{aligned}
\lim_{z\rightarrow1}\mu_2\frac{\lambda_1}{\mu_1}\left(k_2\frac{A^{(1)}(z,\xi)}{D(z)}\right) - \mu_2\left(1-\frac{\lambda_1}{\mu_1}\right)\left(k_1\frac{\mu_2}{\mu_1}\left(\frac{A^{(1)}(z,\xi)}{D(z)}-\phi^{(1)}_{0,k_1+k_2}\right)-\sum_{h=1}^{k_1-1}(k_1-h)\phi^{(1)}_{1,h}\right)=0.
\Gamma_p^{-1}\left( D \overline{v}(p) \cdot X \right) = \overline{\nabla}_X v - \overline{\nabla}_X \widetilde{v(p)}&= \nabla_X v + \langle v(p), A_p(X) \rangle \eta(p) - \overline{\nabla}_X \widetilde{v(p)}
C_{m}^{l}( \tau ) = \sqrt{\frac{\tau}{i k ( k + 2 )}}\sum_{l^{'}= 0}^{k}\sum_{m^{'}= - k + 1}^{k}\mathrm{e}^{i \pi m m^{'}/ k}\sin \left[ \frac{\pi ( l + 1 ) ( l^{'}+ 1 )}{k + 2}\right] C_{m^{'}}^{l^{'}}( - 1 / \tau )
\begin{aligned}\sigma^k \in \bigoplus_{\lambda \in I}\langle x \mapsto x^j \exp(\lambda x) : j = 0,\ldots,p \rangle \oplus \bigoplus_{(\mu,\nu) \in J}\langle x \mapsto x^j \exp(\mu x) \cos(\nu x), \qquad\qquad\quad\,\,\, x \mapsto x^j \exp(\mu x) \sin(\nu x) : j = 0,\ldots,p \rangle\end{aligned}
\theta(n+ud+kd) =\theta(n)+ud+kd=\theta(n+ud)+kd,\cr\theta(m+ud+kd) =\theta(m+\theta(n)+(u-l+k)d)\cr =\theta(m+\theta(n))+(u-l)d+kd\cr =\theta(m+\theta(n)+(u-l)d)+kd\cr =\theta(m+ud)+kd.
\delta \bar{S}_{\pm \mu \nu}= \bar{\tau}_{\pm ( T )}T_{( T ) \mu \nu}+ \bar{\tau}_{\pm ( L T )}T_{( L T ) \mu \nu}+ \bar{\tau}_{\pm ( L L )}T_{( L L ) \mu \nu}+ \bar{\tau}_{\pm ( Y )}T_{( Y ) \mu \nu},
X(\rho e^{\imath t})= \left( \big(\frac1{\rho}-\rho\big)\sin t \,,\, \big(\rho-\frac1{\rho}\big)\cos t \,,\, -\frac12\big(\rho^2+\frac1{\rho^2}\big)\sin 2t \,,\, \frac12\big(\rho^2+\frac1{\rho^2}\big)\cos 2t \right).
R^{- 1}P R = \sum_{j_{1}^{\prime}, j_{2}^{\prime}}U_{1}^{j_{1}^{\prime}}U_{2}^{j_{2}^{\prime}}R^{- 1}| \psi > < \psi | R U_{1}^{- j_{1}^{\prime}}U_{2}^{- j_{2}^{\prime}}
\mathbb{E}\sum_{i_1,\cdots,i_{k}\}\prod_{j=1}^k({\tau}^*_{i_j}-x_j)_+ = \mathbb{E}(nS(I)/4)^k(2\ln n)^{\frac{k}{2}}|\Sigma_k(G_n(x_1)/S(I),\cdots,G_n(x_k)/S(I))| = (S(I)/4)^k\int_{I^k}\phi_{k,n}(y_1,\cdots,y_k)dy_1\cdots dy_k.
S = \int d p [ A_{f}( \vec{p}) A_{f}^{\dagger}( \vec{p}) l n ( c h^{2}( g ( \vec{p}, \beta ) ) ) - A_{f}^{\dagger}( \vec{p}) A_{f}( \vec{p}) l n ( s h^{2}( g ( \vec{p}, \beta ) ) ) ] ,
\frac{\rho_i^{k+1}-\rho_i^k}{\Delta t}=\frac{1}{\Delta x^2}\{\sum_{j\in N(i)}\rho_j^k( F_j(\rho^k)- F_i(\rho^k))_+-\sum_{j\in N(i)}\rho_{i}^k( F_i(\rho^k)-F_j(\rho^k))_+\}.
{d \widetilde g^K_{i}(t)\over dt}< \bar p \log K \sum_{\ell=-\lfloor 1/2\delta_K\rfloor}^{1/\delta_K-1-\lfloor 1/2\delta_K\rfloor}h_{K}G(h_{K}\ell) \left[\widetilde g^K_{\ell+i}(t) - \widetilde g^K_{i}(t)\right]^+ \, e^{h_{K}L|\ell|}.
\int_{\mathcal{I}_j}|\widehat{d_j}(r,\varphi)|^2 \,dr= \int_{\mathcal{I}_j}|\widehat{h}_j(r\sin(\varphi))|^2 \,dr \lesssim 2^{js}\|\widehat{h}_j\|^2_\infty \lesssim 2^{js(1-\alpha)}\|h_j\|^2_2 .
\begin{cases}v_1\partial_y f^L = \mathcal{L}[f^L]\,,y\in[0,\infty) \,, f^L|_{y=0,v_1>0}= \phi^x(x_2,x_3)\left(\phi^v(v)-\eta\right)\,, f^L|_{y=\infty}= 0\,;\end{cases}
Z^S_{ab}= \left(\begin{array}{cccc}0 z_+^2 0 0 z_+^2 0 0 0 0 0 z_H^2 0 0 0 0 z^2_\chi \end{array}\right)\qquad\begin{array}{ccc}Z^R_{f_i}= \tilde z_{f_i}^2 Z^L_{f_i}= z_{f_i}^2\end{array}
\hat{B}_{\mathrm{x},i}(x)\!=\!\begin{cases}(1+w_{\mathrm{x}}^i)\left(-\ln(z_i(x))+\ln(d_{\mathrm{x}}^i)\right) z_i(x) > \delta begin{align*}0.1cm] (1+w_{\mathrm{x}}^i)\left(\,\beta(z_i(x);\delta) +\ln(d_{\mathrm{x}}^i)\right) z_i(x) \leq \delta \, \end{cases}
{\it L}_{gh}= \overline c^\mu \biggl(g_{\mu \nu}\nabla^2 + R_{\mu \nu}- \frac{2 \xi\varphi}{\alpha(\varphi)}(\nabla_\nu\varphi) \nabla_\mu - \frac{2 \xi\varphi}{\alpha(\varphi)}(\nabla_\mu \nabla_\nu \varphi) \biggr) c^\nu
A^{g}~ = ~ \int_{M_{g}}\frac{1}{( \det \mathrm{I m}\tau )^{2}}\int \prod_{i = 1}^{2 g}d^{2}x_{i}| \det{\omega_{i}( y_{j})}\det{\omega_{i}( u_{j})}|^{2}\langle \prod_{k = 1}^{3 g - 3}| ( \mu_{k}G^{-}) |^{2}\rangle_{\mathrm{t o p}},
\Delta^{(2)}{\cal V}= - \varepsilon^2 \frac{3 m^3}{2{\sqrt{2}}\lambda}~ m^2 t^2 \int \frac{dz}{\cosh^2 z}= - \varepsilon^2\frac{3 m^5}{\lambda{\sqrt 2}}t^2 \equiv \frac{M V^2}{2}.
\widetilde{w}_{(j,1)}\sigma_{(i,1)}=\sigma_{(i,1)}\widetilde{w}_{(j,1)}\Longleftrightarrow(\widetilde{w}_{(i,1)}\widetilde{w}_{1^{*}}\widetilde{w}_{(i,1)}\widetilde{w}_{1}\widetilde{w}_{(j,1)}\widetilde{w}_{1})^{2}=1.
[\sum_{i=2m,i>0}^{\infty}\nu_i^{(m)}u^iY_i,u^{2k+1-a}q_a] =\sum_{i=2m,i>0}^{\infty}\nu_i^{(m)}u^{2k+1+b}a\left(b\frac{\partial}{\partial q_b}\right).
\frac{\partial V_F}{\partial z^a}(S, \bar S_0, z_0^a, \bar z_0^{\bar a}) =0, \qquad\qquad V_F(S_0, \bar S_0, z_0^a, \bar z_0^{\bar a})=0, \qquad\qquad F_S(S_0, z_0^a) \ne 0,
h^0(X, kH)\geq{}\sum_{i=2}^k h^0(H, iH|_H) \geq \sum_{i=k_{n-1}}^{\lfloor k/2\rfloor}h^0(H, 2iH|_H) \geq{}\sum_{i=k_{n-1}}^{\lfloor k/2\rfloor}\delta_{n-1}(2i)^{n-1}(H^{n}) > \frac{\delta_{n-1}}{2^{n+2}n}k^n(H^{n}).
\Big\| \Big(\sum_{k \le s}[E_{\tau_j(k)}(|\Delta_{k-\ell}f|)]^2 \Big)^{1/2}\Big\|_{L^q(\mu)}\lesssim \Big\| \Big(\sum_{k \le s}|\Delta_k f|^2 \Big)^{1/2}\Big\|_{L^q(\mu)}\lesssim \mu(Q^*)^{1/q}.
I_3 = |x|^{\frac{d+1}2-\nu}\bigg(\int_{B'}\frac{|\bar z|^{\frac{1-d}2}\, d\bar z}{(\frac1{|x|^2}+|e_1-\bar z|^2)^{\frac{\nu}{2}}}+ \int_{B_{2}\backslash (B_{1/2}\cup B')}\frac{|\bar z|^{\frac{1-d}2}\, d\bar z}{(\frac1{|x|^2}+|e_1-\bar z|^2)^{\frac{\nu}{2}}}\bigg) &<C|x|^{\frac{d+1}2-\nu}(|x|^{\nu-d}+C) &<C|x|^{\frac{1-d}2}\,.
F^{-1 \sigma ~~\rho}_{~~~~\mu \nu ~\alpha \beta}F_{\rho~~ \omega}^{~\alpha \beta ~\eta \kappa}=\frac{1}{2}\delta^\sigma_\omega\left( \delta_\mu^\eta \delta_\nu^\kappa- \delta_\mu^\kappa \delta_\nu^\eta \right)
\begin{aligned}N \delta^{-\theta}\lambda_1^{2+\alpha}2^{4 \beta n}\|u\|_{L_{\infty}C^{\alpha/3, \alpha}_{x, v}(Q_{r_{n+1}})}\le \varepsilon_0^2 (A_{n+1}+ B_{n+1}) + N \varepsilon^{-\alpha}_0 \delta^{-\theta}\lambda_1^{(2+\alpha)(1+\alpha/2)}2^{(4 + 2 \alpha) \beta n}\|u\|_{L_{\infty}(Q_R),}\end{aligned}
q_{\infty}\left(\mathcal F_1^{c'},\mathcal F_2^{c'},\mathbf T^{'}\right)-q_{\infty}^{*}=\left(a_{n_2}-a_{n_3}\right)\left(f_{2,K_2^c,\infty}(T_{n_3}^{*})-f_{1,K_1^c+F_1^{b*},\infty}\right)> 0,
\pi_n = \inf_{|h| \leq \epsilon a_n}\P_h\left( P^{(n)}_T = 0, |S^{(n)}_T| \leq \epsilon, S^{(n)}_s \in I, s \leq T \right) = \inf_{|h| \leq \epsilon}\P\left( P^{(n)}_T = 0, |S^{(n)}_{T}+h| \leq \epsilon, S^{(n)}_s + h \in I, s \leq T \right),
= \;\left\langle \prod_{b=1}^N \; e^{-2 \sum_{b=1}^N \beta^{(b)}\epsilon^{(b)}\int_{\Lambda}d^2 x \; t(x) \;: \cos\big[ 2 \sqrt{\pi}\theta_2^{(b)}(x) \big] :_{C^2}}\right\rangle_{C^2}\; ,
[\hat n^\rho, v_1^\rho,\ldots, v_{r-1}^\rho] = [ n^\rho , v_1^\rho ,\ldots, v_{r-1}^\rho] \begin{pmatrix}1 0 \cdots 0 a_1 1 \cdots 0 \vdots 0 \ddots \vdots a_{r-1}\cdot \cdots 1\end{pmatrix}.
\begin{array}{l}<\phi_q^4>=\frac 1{128\pi^3C}\log \left[ 1+ \frac{\Lambda^2}{(m_r^2+(\xi_r-\frac 18)R-12\kappa_r\phi_c^2+15\lambda_r^2\phi_c^4+\frac QC)}\right] \end{array}
x ( \tau ) = \int_{0}^{1}\theta ( \tau - \tau^{\prime}) v ( \tau^{\prime}) d \tau^{\prime}= \int_{0}^{\tau}v ( \tau^{\prime}) d \tau^{\prime}, \; \; v ( \tau ) ={\dot{x}}( \tau ) , \; \; J = \mathrm{D e t}\; \theta ( \tau - \tau^{\prime})
f\left( t \right) -{\left.{D_{{0^+}}^\alpha \,f\left( t \right)}\right|_{t = b}}\frac{{{t^\alpha}}}{{\Gamma \left({\alpha + 1}\right)}}+ I_{{0^+}}^\alpha \,I_{{b^-}}^\alpha \left(\lambda + q \left(t\right) \right)f\left( t \right) = 0.
\phi_{\Phi_k, \beta}(x, y)=R_{k, \beta}\left(\frac{1}{(1-\delta)}\right)y^{\frac{1}{(1-\delta)}}+O_\delta\left( y^{\frac{1}{(1-\delta)}}\exp\left( -\sqrt{p_k \log y\log\log y}\right)\right),
C_{a b}^{b}= E_{i}^{b}( \xi_{b}^{j}\xi_{a , j}^{i}- \xi_{a}^{j}\xi_{b , j}^{i}) = A_{c}^{b}( A_{a , j}^{c}\xi_{b}^{j}- A_{b , j}^{c}\xi_{a}^{j}) .
\begin{aligned}\pi(\mathcal F_1) =\{F\in\pi(\mathcal F):[d-1]\subseteq F,F\cap\{d,d+1\}\neq\emptyset,d+2\in F\}&\cup\{F\in\pi(\mathcal F):1\notin F,[2,d-1]\subseteq F,\{d,d+1\}\subseteq F,d+2\in F\}.\end{aligned}
\sum_{J=1}^{2n}\Omega(\Phi_J[z_J]|B^{-\frac{\sigma_3}{2}}d\Xi)=d \sum_{J=1}^{2n}\Omega(\Phi_J[z_J]|B^{-\frac{\sigma_3}{2}}\Xi)+\Omega(B^{-\frac{\sigma_3}{2}}\Xi|d\Phi_J[z_J])
\omega''\cdot v =(\omega\cdot (u-u'))(\omega'\cdot v)-(\omega'\cdot (u-u'))(\omega\cdot v) &\geq (\omega\cdot (u-u'))(\omega'\cdot v)-(\omega'\cdot (u-u'))(\omega\cdot u) &>-(\omega'\cdot (u-u'))(\omega\cdot u)=\omega''\cdot u.
a_n = \frac{2}{\pi}\int_{0}^{\infty}\bigg( \frac{\sin x}{x}\bigg)^{n + 1}\cos x \, dx. = \frac{2}{\pi}\bigg( \frac{n + 1}{n + 2}\bigg) \int_{0}^{\infty}\bigg( \frac{\sin x}{x}\bigg)^{n + 2}\, dx.
\sum_{i=1}^{n}\sum_{A\in A}(p_{d^{*}}(A)\cdot u_{i}(A))=\sum_{A\in A}p_{d^{*}}(A)\sum_{i=1}^{n}u_{i}(A)\le max_{A\in A:p_{d^{*}}(A)>0}\sum_{i=1}^{n}u_{i}(A)
R(z) \leq \frac{1}{(1-z)^2}- \left(\frac{z^{d_r}}{(1-z)^2}+ \frac{z^{d_{r-1}}+ z^{d_{r-2}}+ \cdots + z^{d_1}}{1-z}\right)= \frac{1 + z + \cdots + z^{d_r-1}}{1-z}- \frac{z^{d_{r-1}}+ z^{d_{r-2}}+ \cdots + z^{d_1}}{1-z}.
N_{1}=\Sigma, N_{2}=F_{S_{A}}K^{\ast}\Sigma-S_{A}^{\dagger}H\Phi, N_{3}=\Sigma H^{\ast}E_{S_{A}}-\Psi KS_{A}^{\dagger}, N_{4}=S_{A}^{\dagger}-S_{A}^{\dagger}H\Phi H^{\ast}E_{S_{A}}-F_{S_{A}}K^{\ast}\Psi KS_{A}^{\dagger}+F_{S_{A}}K^{\ast}\Sigma H^{\ast}E_{S_{A}}.
\vec \psi (t;\zeta) =\left(\psi(t;\zeta),\partial_\zeta\psi(t;\zeta),\dots,\partial^{p-1}_\zeta\psi(t;\zeta)\right)^T\ , \vec \chi (t;\eta) =\left(\chi(t;\eta),\partial_\eta\chi(t;\eta),\dots,\partial^{p'-1}_\eta\chi(t;\eta)\right)^T\ ,
\Big( \Big( \frac{\partial}{\partial \overline{z}_i}\big|_{y_0}\Big)^{H}\tilde{g}\Big) (y_0, Z_N)=\Big( \frac{\partial}{\partial \overline{z}_i}\big( g \cdot z_N^{\otimes k}\big) \Big)(y_0) \exp \Big(- \frac{\pi}{2}|Z_N|^2 \Big).
\left( \ref{claim:dificil}\right) u=1\exists Z_{u=1,2}^{\ast}\mathbf{Z}_{min}\backslash \left[ \mathbf{O}_{min}\cup \left\{Z_{u=1,1}^{\ast}\right\}\right] \text{such that}Z_{u=1,2}^{\ast}Z_{u=1,1}^{\ast}
\bigg|\underbrace{\big\{[\sigma_0 x_0^{1/a_0}:\sigma_1 x_1^{1/a_1}:\sigma_2 x_2^{1/a_2}] \mid \sigma_i\in\mu^{a_i}\big\}}_{\pi^{-1}(|x_0:x_1:x_2|)}\bigg| < \underbrace{a_0a_1a_2}_{\deg\pi}\iff x_i=0i
\sum_{i,j=1}^nb^{ij}(x)\xi^{i}\xi^{j}\geq s_0 |\xi|^2,\,\,\,\,\,\,\,\,\,\, \forall \ (x,\xi)=(x,\xi^{1}, \cdots, \xi^{n}) \in \overline{G}\times \mathbb{R}^{n}.
\sqrt{-d^2/dx^2}\, u_\nu(x) = \widehat{(|k|\hat u_\nu)}(x) = \frac{2 \Gamma(\tfrac{1}{2}+\nu)}{\sqrt{\pi}\Gamma (\nu)}\;{}_2F_1\left(1,\tfrac{1}{2}+\nu;\tfrac{1}{2};-x^2\right).
\begin{array}{l}3(3,2)_{(1,-1,0,0,0^3)}+3(\bar{3},1)_{(-1,0,1,0,0^3)}+3(\bar{3},1)_{(-1,0,0,1,0^3)}+3(1,2)_{(0,1,0,0,\underline{-1,0,0})}+3(1,1)_{(0,0,-1,0,\underline{1,0,0})}+3(1,1)_{(0,0,0,-1,\underline{1,0,0})}+6(1,2)_{(0,1,-1,0,0^3)}+6(1,2)_{(0,-1,0,1,0^3)}+12(1,1)_{(0,0,1,-1,0^3)}. \end{array}
p(x) ={2^{65}\over 2^{127}+ 1}x_1^3 x_9 x_{13}^2 - \left({19 \over 7}\zeta_6 - 2^{129}\right) x_{21}^7\in \mathcal{P}\intertext{as the{\sc Python}tuple}\Big(\big((1, 3, 9, 1, 13, 2), \, \left(2^{65}/ (2^{127}+ 1), \, 0 \right)\big), \,\, \big(\left(21, 7\right), \, \left(2^{129}, \, -19/7\right)\big)\Big).
B(u_k) =\int_{\Omega}(|x|^{-\mu}*F(u_k))f^{'}(u_k) u_k^2 ~dx +\int_{\Omega}(|x|^{-\mu}*f(u_k)u_k)f(u_k)u_k ~dx \leq C(n, \mu) \left(\|f(u_k)u_k\|^2_{L^{2n/(2n-\mu)}(\Omega)}+ \|F(u_k)\|_{L^{2n/(2n-\mu)}(\Omega)}\|f^{'}(u_k) (u_k)^2\|_{L^{2n/(2n-\mu)}(\Omega)}\right).
d s^{2}= - \frac{1}{\kappa_{+}^{2}r^{2}}( r + r_{+})^{2}( r^{2}+ r_{-}^{2}) e^{- 2 \frac{r_{-}}{r_{+}}\arctan \frac{r}{r_{-}}}d U d V + r^{2}d \Sigma_{q}^{2}.
h\left( \{\psi^*_v\}_{v\in V^\ell}| I(\psi^*)\right) = \sum_{v\in V^\ell}h\left( \psi^*_v| \{\psi^*_w:w<_\sigma v \},I(\psi^*)\right) \leq \sum_{v\in U}h\left( \psi^*_v| I(\psi^*)\right)+ \sum_{v\in V^\ell\setminus U}h\left( \psi^*_v| \{\psi^*_w:w<_\sigma v \},I(\psi^*)\right),
S ={M}_{\mathrm{{P l}}}^{2}\int_{}^{}{d}^{4}x \sqrt{- g}( - 2 \Lambda + R + a{R}^{2}+ b{R}_{\mu \nu}{R}^{\mu \nu}+ c{R}_{\mu \nu \lambda \kappa}{R}^{\mu \nu \lambda \kappa}+ . . . )
\int_{\Omega}|u||u_k|^\beta\, dx\leq K^\beta\int_{\Omega}|u|+\int_{\{|u|\geq K\}}|u|^{p^*}\, |u_k|^{\beta}\, dx &\leq K^\beta\int_{\Omega}|u|+\left(\int_{\{|u|\geq K\}}|u|^{p^*}\, dx\right)^{1-\frac{p}{p^*}}\left(\int_{\Omega}|u|^{p^*}\, |u_k|^{\beta\frac{p^*}{p}}\, dx\right)^{\frac{p}{p^*}}.
( D_{L})_{\beta \gamma}= \delta_{\beta , \gamma}( D_{L})_{\beta}, \quad ( D_{L})_{\beta}= - i \sum_{\Upsilon \in \Delta_{l}, \ \beta \cdot \Upsilon = 2}\left[ g_{l_{1}}z ( \Upsilon \cdot q ) + g_{l_{2}}z^{( 1 / 2 )}( \Upsilon \cdot q ) \right] .
G_1^{(m,k)}=\Phi(\vec{x}_r,\vec{x}_s)\left(\frac{1}{\mathcal{S}_+}-\frac{1}{\mathcal{S}_-}\right)=\Phi(\vec{x}_r,\vec{x}_s)\left(\frac{\sinh \mathcal{U}}{\cosh \mathcal{U}-\cos\mathcal{T}}\right)\;.
\left( \frac{m^{2}}{\Lambda^{2}}\right)^{( d - 3 ) / 2}\Gamma{\left( \frac{3 - d}{2}\right)}\sim{\frac{2}{( 3 - d )}}+ \gamma -{\ln}\left( \frac{m^{2}}{\Lambda^{2}}\right) + O ( d - 3 ) ,
=\frac{1}{|W^{\mathrm{abs}}_{M_S}|}\sum\limits_{w \in W^{\mathrm{abs}}_{M_S}}\sum\limits_{\gamma \in \Gamma}\gamma(w(\mu))=\frac{1}{|W^{\mathrm{abs}}_{M_S}|}\sum\limits_{\gamma \in \Gamma}\sum\limits_{w \in W^{\mathrm{abs}}_{M_S}}\gamma(w(\mu)).
\langle e_5,e_5\rangle_{L^2_{p,\mathcal{I}}}= \frac{1}{k_BT_0^2\left\{\widetilde{e}\right\}^{\prime}(T_{0})}\int_{\mathbb{R}^3}\int_0^\infty\left\{cp^0\left(1+\frac{\mathcal{I}}{mc^2}\right)-\widetilde{e}(T_0)\right\}^2 F_E^0\phi(\mathcal{I})\,d\mathcal{I}dp\cr =1.
\psi^{c}| 0 > = \int d^{3}p ~ e x p \{\frac{i}{2}\mathrm{\boldmath ~ \gamma ~}\cdot \mathrm{\boldmath ~ \lambda ~}({\bf p}) \}|{\bf p}> ~ e^{i \mathrm{\boldmath ~ p ~}\cdot \mathrm{\boldmath ~ x ~}}
\epsilon=\sum_{i=1}^{3}\sum_{j=1}^{3}\hat{\epsilon}_{ij}\hat{e}_{i}\otimes\hat{e}_{j}\Rightarrow\underline{\underline{\hat{\epsilon}}}=[\begin{matrix}\hat{\epsilon}_{11}&\hat{\epsilon}_{12}&\hat{\epsilon}_{13}\\ \hat{\epsilon}_{12}&\hat{\epsilon}_{22}&\hat{\epsilon}_{23}\\ \hat{\epsilon}_{13}&\hat{\epsilon}_{23}&\hat{\epsilon}_{33}\end{matrix}]
\bar{Y}_s^{1n}=g^1(X_T^{t,x})+\int_t^T C_2(1+|\varphi_n (X_r^{t,x})|)|\bar Z_r^{1n}| &+C_h(1+|\varphi_n(X_r^{t,x})|^\gamma +|\bar Y_r^{1n}|)dr -\int_s^T \bar Z_r^{1n}dB_r.
d_{>n}(t_1, \dots, t_\ell)\in \Big\{\sum a_{i_1,\dots, i_\ell}t_1^{i_1}\cdots t_\ell^{i_\ell}\, |\,a_{i_1,\dots, i_\ell}=0 i_1+\cdots+i_\ell \leq n \Big\}, d_{\ge n}(t_1, \dots, t_\ell)\in \Big\{\sum a_{i_1,\dots, i_\ell}t_1^{i_1}\cdots t_\ell^{i_\ell}\, |\,a_{i_1,\dots, i_\ell}=0i_1+\cdots+i_\ell < n \Big\}.
\gamma B_{a}^{0i}=\partial_{j}\eta_{a}^{(2)ij},\;\gamma B_{a}^{ij}=\eta_{a}^{(1)ij},\;\gamma H_{i}^{a}=C_{i}^{(1)a},\;\gamma H_{0}^{a}=-\partial^{i}C_{i}^{(2)a},
C_{p,q}^{(\nu)}(t) \coloneqq \begin{cases}C e^{-td^\nu}(t\ge 1) C t^{-\mu_\nu}(0<t\le 1), \end{cases}\mu_\nu \coloneqq \max \Big\{\frac{d}{\nu}\Big(\frac{1}{\min\{q,q'\}}-\frac{1}{\max\{p,p'\}}\Big) , 0\Big\},
Y_{j,(1)}= \begin{cases}\log(\frac{[T_{j,(1)}^{\flat}]}{T_{j,(1)}}) \mbox{if}1 \leq j\leq e \log(\frac{[S_{j,(1)}^{\flat}]}{S_{j,(1)}}) \mbox{if}e+1 \leq j \leq d\end{cases}
\mathrm{E}_{1}\left( \eta_{1}, \cdots , \eta_{p}\right) = \eta_{1}^{a_{1}}\cdots \eta_{p}^{a_{p}}h^{a}( \phi ) \partial_{a}\partial_{a_{1}}^{( 1 )}\cdots \partial_{a_{p}}^{( p )}F_{p}( \phi , \cdots , \phi )