harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\|R_Ax-R_Ay\|^2 =4\|J_Ax-J_Ay\|^2-4\langle J_Ax-J_Ay,x-y\rangle+\|x-y\|^2 &\leq (4\delta-4)\langle J_Ax-J_Ay,x-y\rangle+\|x-y\|^2 &\leq \left(4(\delta-1)\left(\tfrac{\delta}{2}+\tfrac{(1-\alpha-\delta/2)^2-\alpha^2+(\delta/2)^2}{2(1-\alpha-\delta/2)}\right)+1\right)\|x-y\|^2
\frac{1}{\varphi_{a}}+\frac{t_{a}^2}{\varphi_{a+1}\varphi_{a}}=\frac{1}{\varphi_{a}}\left(1+\frac{t_{a}^2}{\varphi_{a+1}}\right)=\frac{1}{\varphi_{a}}\frac{\varphi_{a}}{\varphi_{a+1}}=\frac{1}{\varphi_{a+1}}.
\frac{u_{j}^{n+1}-u_{j}^{n}}{k}=\frac{1}{2}(\frac{u_{j+1}^{n+1}-2u_{j}^{n+1}+u_{j-1}^{n+1}}{h^{2}}+\frac{u_{j+1}^{n}-2u_{j}^{n}+u_{j-1}^{n}}{h^{2}}).
J^{\mu}= \frac{1}{A ( S^{n - 1}) ( n - 1 ) !}\epsilon^{\mu \mu_{1}. . . \mu_{n}}\epsilon_{\hat{\phi}_{1}. . . \hat{\phi}_{n}}\partial_{\mu_{1}}\hat{\phi}^{a_{1}}. . . \partial_{\mu_{n}}\hat{\phi}^{a_{n}}
\limsup_{s\to\infty}R(t_s) \leq \limsup_{s\to\infty}\hat{\lambda}^{s-\hat{s}}R(t_{\hat{s}})+\sum_{\tau=\hat{s}}^{s-1}c(\tau)t_{\tau+1}^{\beta}\hat{\lambda}^{s-\tau-1}\cr =\lim_{s\to\infty}c(s)t_{s+1}^{\beta},
\omega_A(TS \pm ST) \leq \sqrt{{\big\|TT^{\sharp_A}+ T^{\sharp_A}T\big\|}_A}\sqrt{{\big\|SS^{\sharp_A}+ S^{\sharp_A}S\big\|}_A}&\leq 2\sqrt{\big\|TT^{\sharp_A}\big\|_A^2+\big\|T^{\sharp_A}T\big\|_A^2}\,\omega_A(S) &= 2\sqrt{\big\|T\big\|_A^2+\big\|T\big\|_A^2}\,\omega_A(S)(\,\eqref{diez}).
\Pi_{r e}= \{\frac{1}{2}\Pi_{0}^{l}+ ( 2 n + 1 ) \delta \varepsilon_{0}\}\cup \{\Pi_{0}^{s}+ 2 n \delta \varepsilon_{0}\}\cup \{\Pi_{0}^{l}+ 4 n \delta \varepsilon_{0}\}, \n \in Z .
\sum_{\substack{V_{\underline{i}}\cap V_{\underline{j}}= \emptyset \rho(\underline{i})= r_1 , \, \rho(\underline{j}) =r_2 \; \sigma(\underline{i}) = s_1 , \, \sigma(\underline{j}) = s_2}}\left| \mathbb{E}\left[ X_N(\underline{i}) \, X_N(\underline{j}) \right]\right| \ \leq \D_k \, N^{r_1+r_2- s_1/2- s_2/2}\, ,
\omega_{\underline{d}}(x)=\bigcap_{\mu\in V_f(x)}S_{\mu}=\bigcap_{\mu\in K_i}S_{\mu}~\textrm{and}~\omega_{\overline{d}}(x)=\overline{\bigcup_{\mu\in V_f(x)}S_{\mu}}=\overline{\bigcup_{\mu\in K_i}S_{\mu}}.
Z(t) =\left( 1-p(t)\gamma_{2}(t)\right)^{-1}\left\{p(t)\left[\alpha_{2}(t)X(t)+\beta_{2}(t)Y(t)+L_{2}(t)\right] +q(t)X(t)+\nu(t)\right\}& =\left( 1-p(t)\gamma_{2}(t)\right)^{-1}\left[ \left( \alpha_{2}(t)p(t)+\beta_{2}(t)p(t)^{2}+q(t)\right) X(t)+p(t)\beta_{2}(t)\varphi(t)+p(t)L_{2}(t)+\nu(t)\right] .
{\cal L}= \sum_i\frac{1}{2}\partial_{\mu}\Phi_{i}\partial^{\mu}\Phi_{i}- \sum_i \frac{1}{2}m_{i}^{2}\Phi_{i}^{2}+\frac{g}{\sqrt{2}}\left( \Phi_{1}^{2}+\Phi_{2}^{2}\right) \Phi_{3}.
g(x+h) +g(x-h)-2g(x) \leq f(y)+\langle G(y), x+h-y \rangle + \varphi_y(x+h) & + f(y)+\langle G(y), x-h-y \rangle + \varphi_y(x-h) & -2 \left( f(y)+ \langle G(y), x-y \rangle + \varphi_y(x) \right) + 2 \varepsilon & = \varphi_y(x+h) + \varphi_y(x-h)- 2\varphi_y(x) + 2 \varepsilon & \leq C_R |h|^2 +2 \varepsilon,
\begin{aligned}(\xi\smallsmile\eta)_0=\xi_0\wedge\eta_0,\quad(\xi\smallsmile\eta)_1=\xi_1\wedge\eta_1\quad &(\xi\smallsmile\eta)_{01}=(-1)^{p+q}\xi_0\wedge\eta_{01}+\xi_{01}\wedge\eta_1.\end{aligned}
\frac{1}{1 + H^{2}r^{2}}\left( \frac{d r}{d \xi^{1}}\right)^{2}- ( 1 + H^{2}r^{2}) \left( \left( \frac{d t}{d \xi^{1}}\right)^{2}+ c_{1}^{2}\right) + r^{2}\left( \left( \frac{d \phi}{d \xi^{1}}\right)^{2}+ c_{2}^{2}\right) = 0 .
\Psi ( \{x^{i}( \tau ) \}, \{d x^{j}( \tau ) \}) = \Psi ( \{x^{i}e^{- \tau}\}, \{d x^{j}e^{- \tau}\}) = e^{- \tau H}\Psi ( \{x^{i}\}, \{d x^{j}\}) \in \Lambda R_{\mathrm{a}}^{n},
{\phi}_{( 2 )}^{\mathrm{s e l f}}( x_{2}, t ) = e_{2}\int_{0}^{\mathrm{L}}d y \int_{0}^{\mathrm{L}}d z D ( x_{2}, y | \mathrm{L}) \bar{\Phi}( z , t | y , t ) ({\gamma}^{0}\otimes{\gamma}^{0}){\Phi}( z , t | y , t ) ,
\theta_\lambda^{(2)}(\zeta,\mu) = \int_0^1 s D^2 \bar J_\lambda(V + s\phi)[\phi^2] \,ds . &=\int_0^1 s \left[\int_{\Omega_\varepsilon}\vert\nabla \phi\vert^2 - \varepsilon^2 \lambda \phi^2-5 (V + s\phi)^4 \phi^2\right]\,ds
\frac{\omega_{c}^{\prime}}{i\omega}\rightarrow\frac{1}{Q_{1}(\frac{i\omega}{\omega_{01}}+\frac{\omega_{01}}{i\omega})}+\frac{1}{Q_{2}(\frac{i\omega}{\omega_{02}}+\frac{\omega_{02}}{i\omega})}+\cdot\cdot\cdot
A_1 =\frac{1}{|z|}\big| D_t \tilde h(z+t_0(X_1)) - D_t \tilde h(t_0(X_1)) - (D_t \tilde h(z+t_0(X_2)) - D_t \tilde h(t_0(X_2)) ) \big| |b(X_1,z+t_0(X_1))| A_2 = \frac{1}{|z|}\big| (D_t \tilde h(z+t_0(X_2)) - D_t \tilde h(t_0(X_2)) ) (b(X_1,z+t_0(X_1)) -b(X_2,z+t_0(X_2)) )\big|
\int \, 2 \, R[g] \, \sqrt{-g}\, d^Dx = \int \, \exp \left[(D-2)\alpha \,\phi \right] \, \left(2 R[\overline{g}] -\alpha^2 \left( D-1\right) \left( D-2\right) \partial^\mu \phi \partial_\mu \phi \right) \sqrt{-\overline{g}}\,d^Dx
l = - \frac{i m k ( c o s \, \mu \, s i n \, \lambda + e^{2 i a k}c o s \, \lambda \, s i n \, \mu ) + m E ( e^{2 i a k}- 1 ) s i n \, \lambda \, s i n \, \mu}{m^{2}( e^{2 i a k}- 1 ) s i n \, \lambda \, s i n \, \mu + k^{2}c o s ( \lambda + \mu ) + i E k \, s i n ( \lambda + \mu )}
q = (r_1^{x_1},r_2^{x_2},\ldots,r_k^{x_k}):=(\underbrace{r_1,r_1,\ldots,r_1}_{x_1},\underbrace{r_2,r_2,\ldots,r_2}_{x_2},\ldots,\underbrace{r_k,r_k,\ldots,r_k}_{x_k}).
\phi_{opt}(t,z,\theta )=\int_{-\infty}^{+\infty}dt'\int_{S^d}d\mu (\theta ')e^{dz'\over \beta_H}\left( G\partial_{z'}\varphi (z',t',\theta')-\partial_{z'}G\varphi (z',t',\theta')\right)_{z'=Z}\ ,
\begin{array}{c}{\widetilde{\tau}= \displaystyle \frac{\tau}{\cosh \omega}+ \Theta ( \widetilde{\tau}, x ) \tanh \omega , \hfill}\\{\widetilde{\Theta}= \displaystyle \frac{\Theta ( \widetilde{\tau}, x )}{\cosh \omega}- \tau \tanh \omega , \hfill}\\ \end{array}
\widetilde{P}_{2,1,k}^{(0^{2})}(z; \lambda_1) = -4k ( k+4 ) P_{k}^{( 2,1)}(z) \left( \lambda_{1}-(z+1)^{2}+\frac{2}{3}(z+1)^{3}-\frac{1}{8}(z+1)^{4}\right) \hphantom{\widetilde{P}_{2,1,k}^{(0^{2})}(z; \lambda_1) =}{}+( k+4) (1-z)^{3}(1+z)^{2}P_{k-1}^{( 3,2)}(z), k\ne 0, \widetilde{P}_{2,1,0}^{(0^{2})}(z; \lambda_1) = 1.
Q_{w_1}^{(1)}T_{w_1}Q_{w_1}^{(2)}\ldots Q_{w_s}^{(1)}T_{w_s}Q_{w_s}^{(2)}(P_{w_{s+1}}^\perp T_{w_{s+1}}P_{w_{s+1}}) \ldots (P_{w_{d}}^\perp T_{w_{d}}P_{w_{d}}).
\Sigma_{2b}= \sum \limits_{\delta = \pm 1}\sum \limits_{\substack{D_2 | D_1 m_1 D_2^2 = n_2 D_1}}\frac{\tilde{S}(\delta m_2, n_2, n_1, D_2, D_1)}{D_1 D_2}\tilde{\mathcal{J}}_{\delta}\left(\sqrt{\frac{m_2 n_1 n_2}{D_1 D_2}}\right),
Q/Q(2) \downarrow L_{1235}= (1,1,01) + (0,1,00), Q(2)/Q(3) \downarrow L_{1235}= (1,0,01) + (0,0,10), Q(3)/Q(4) \downarrow L_{1235}= (0,1,10), Q(4) \downarrow L_{1235}= (1,0,00).
\Vert U_{\sigma}^{-1}\Vert_{\rm HS}^2 \leqslant \left(\vert\nu^c\vert-n+1+{\rm Tr}\left(A^{-1}B\right)\right)\cdot{\rm Tr}\left(A^{-1}\right) -{\rm Tr}\left(A^{-2}B\right)\left(\vert\nu^c\vert-n+{\rm Tr}\left(A^{-1}B\right)\right) &=\left(m-r-n+1+r\right)\cdot n-r\cdot\left(m-r-n+r\right) &=(m-n)\cdot (n-r) +n.
I = 2 \left({\frac{\alpha}{\sinh^{2}A + \sin^{2}{\frac{\gamma}{2}}}}-{\frac{\sinh{\frac{2 A}{\alpha}}}{\sinh{2 A}}}{\frac{1}{( \sinh^{2}{\frac{A}{\alpha}}+ \sin^{2}{\frac{[ \gamma ]}{2 \alpha}})}}\right) .
\left( \begin{array}{c}{x^{\prime}}\\{t^{\prime}}\\ \end{array}\right) = \left( \begin{array}{c c}{\frac{1}{\sqrt{1 - \beta^{2}}}}&{\frac{\beta}{\sqrt{1 - \beta^{2}}}}\\{\frac{\beta}{\sqrt{1 - \beta^{2}}}}&{\frac{1}{\sqrt{1 - \beta^{2}}}}\\ \end{array}\right) \left( \begin{array}{c}{x}\\{t}\\ \end{array}\right)
|\Phi(\xi_1,\eta_1, \xi, \eta)| = r_1 r | \cos \theta_1 \cos \theta (r_1 \cos \theta_1 + r \cos \theta) +\sin \theta_1 \sin \theta (r_1 \sin \theta_1 + r \sin \theta) | \geq r_1 r \, ( r_1+r ) \, (1- 2^{-1}\sin 2\theta_1) |(\cos \theta_1 + \sin \theta_1)| & \geq 2^{-10}M^{-1}N_1^3,
S_{r}^{\alpha}={\frac{1}{\sqrt{2}}}\left( \begin{matrix}{f_{I}^{\Lambda}+ \mathrm{i}h_{\Lambda I}}&{\bar{f}_{I}^{\Lambda}+ \mathrm{i}\bar{h}_{\Lambda I}}\\{f_{I}^{\Lambda}- \mathrm{i}h_{\Lambda I}}&{\bar{f}_{I}^{\Lambda}- \mathrm{i}\bar{h}_{\Lambda I}}\\ \end{matrix}\right)
\frac{\ddot{b}}{b}= \frac{\beta^{\prime \prime}}{\beta}, \Biggl [ \frac{d}{d t}\left( \frac{\dot{c}}{c}\right) + \frac{\dot{b}}{b}\frac{\dot{c}}{c}\Biggr ] = \Biggl [ \frac{d}{d x}\left( \frac{\kappa^{\prime}}{\kappa}\right) + \frac{\beta^{\prime}}{\beta}\frac{\kappa^{\prime}}{\kappa}\Biggr ] .
\begin{array}{c}\displaystyle E(L)=-\displaystyle\frac{\pi \tilde{c}(l)}{6L}=-\displaystyle\frac{\pi}{6L}\left( c-12(\Delta +\bar{\Delta})\right) +\ldots P(L)=\displaystyle\frac{2\pi}{L}(\Delta -\bar{\Delta})+\ldots \end{array}
F(z) = \sum_{k=0}^\infty \frac{(-1)^k \Gamma\left(\frac{k+1}{2}\right)}{k!}z^k = \sum_{l=0}^\infty \frac{\Gamma\left(l+\frac 12 \right)}{(2l)!}z^{2l}+\sum_{l=1}^\infty \frac{\Gamma\left(l \right)}{(2l-1)!}z^{2l-1},
\begin{aligned}\left< \varphi,f_N^{(1)}(t) - \mathcal{T}(t-T_1) f_N^{(1)}(T_1) \right> = \int_{T_1}^t \left<\varphi, \mathcal{T}(t-\tau) C_2 f_N^{(2)}(\tau) \right> d\tau\end{aligned}
\left. c_{2}( \beta ) \left( \frac 1 2 R_{\mu \nu \lambda \rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}- \frac 1 4 R_{\mu \nu}n_{i}^{\mu}n_{i}^{\nu}\right) \right] .
{\cal L_Q}=-{1\over 4}\Bigl(1+{ae^2\over \pi}\Bigr)F^2 +{e^2\over{2\pi}}A^2+{1\over 2}\partial_\mu\Sigma\partial^\mu\Sigma -{e\over{\sqrt\pi}}\partial_\mu\Sigma A^\mu -{e\over{\sqrt\pi}}\widetilde{\partial}\cdot A\chi.
{\cal A}_{n + 1}( x ) = \frac{1}{n}{\cal D}{\cal A}_{n}( x ) = \frac{1}{n !}{\cal D}^{n + 1}\ln{\cal A}_{0}( x ) \; \; \, m b o x{w i t h}\ \ \{\cal A}_{0}( x ) = \frac{x - 1}{x \ln x}\,
h_j c_{ji}^\alpha h_\alpha X_i(\phi) + h_i h_j X_j(X_i(\phi)) = \underbrace{h_j c_{ji}^{\bar\ell}h_{\bar\ell}X_i(\phi)}_{1}+ \underbrace{h_j c_{ji}^\ell h_\ell X_i(\phi) + h_i h_j X_j(X_i(\phi))}_{2},
\mathbf{T}_{N}= \begin{bmatrix}\mathbf{1}_{32}\mathbf{0}\dots \mathbf{0}\mathbf{0}\mathbf{1}_{32}\dots \mathbf{0}\vdots \vdots \ddots \vdots \mathbf{0}\mathbf{0}\dots \mathbf{1}_{32}\end{bmatrix}.
\begin{matrix}&P(S^{t}|O^{0}\wedge\cdot\cdot\cdot\wedge O^{t})\\ =&P(O^{t}|S^{t})\times\sum_{S^{t-1}}[P(S^{t}|S^{t-1})\times P(S^{t-1}|O^{0}\wedge\cdot\cdot\cdot\wedge O^{t-1})]\end{matrix}
\Pi(S_n^c) = \Pi\left( \sum_{r=0}^{r_n}e^{-r^2v^2}\left|\sum_{j=1}^J w_j e^{2i\pi{jr\over J}}\right|^2 \leq c' \sum_{r=r_n +1}^{\infty}e^{-r^2v^2}\left|\sum_{j=1}^J w_j e^{2i\pi{jr\over J}}\right|^2 \right)
V(\psi) = \frac{h}{4}\sum_{j}\left[ \frac{4}{\lambda^{2}h^{2}}\sin^{2}\frac{\lambda}{2}(\psi_{j+1}-\psi_{j}) +\frac{m^{2}}{\lambda^{2}}\sin^{2}\frac{\lambda}{2}(\psi_{j+1}+\psi_{j}) \right]
\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\left(t_{k},T_{j}\right)\left|\mathcal{F}_{i}\right.;\tilde{A_{i}}\left(T_{j}\right)\right]=\mathbb{E}^{\mathbb{P}}\left[\tilde{A}\left(t_{k},T_{j}\right)\left|\mathcal{F}_{i}\right.\right]+M\left(t_{i},T_{j}\right)
P(0,0) =2, P(0,1) =\dfrac{x^{2}}{y}-\dfrac{y}{x^{2}}=\dfrac{4q}{K}, P(1,0) =xy^{2}-\dfrac{q^{2}}{xy^{2}}=K, P(1,-1) =\dfrac{y^{3}}{x}+\dfrac{q^{2}x}{y^{3}}=K-2q+\dfrac{4q^{2}}{K}.
{\cal L}= - \frac{1}{4 g^{2}}F_{\mu \nu}^{k}F_{\mu \nu}^{k}+ \frac{1}{g^{\prime}}F_{\mu \nu}^{k}\tilde{F}_{\mu \nu}^{k}+ \frac{2}{g^{\prime}}\left( D_{\mu}\cal{B}_{\mu}\right)^{k}\! \left( D_{\nu}\cal{A}_{\nu}\right)^{k}+{\cal L}_{F P}\, ,
e^{2 \Phi^{\prime}}= 1 +{\frac{( P_{1}+ P_{2}- Q_{1}- Q_{2})}{2 r}}+{\cal O}( r^{- 2}) , \ \Psi ={\frac{q ( P_{2}- P_{1})}{2 r^{2}}}+{\cal O}( r^{- 3}) ,
{\cal A}= \pi i \gamma \sum_{r = 1}^{N}\left[ \left( 2 R^{2}\frac{\bar{z}_{r}}{1 + | z_{r}|^{2}}+ \tilde{b}_{r}\right) d z_{r}- \left( 2 R^{2}\frac{{z}_{r}}{1 + | z_{r}|^{2}}+ \bar{\tilde{b}}_{r}\right) d \bar{z}_{r}\right] .
\begin{aligned}x =F_x(\varphi_1,\dot\varphi_1,\ldots,\varphi_1^{(r_1-1)},\varphi_2,\dot\varphi_2,\ldots,\varphi_2^{(r_2-1)}) u =F_u(\varphi_1,\dot\varphi_1,\ldots,\varphi_1^{(r_1)},\varphi_2,\dot\varphi_2,\ldots,\varphi_2^{(r_2)})\,.\end{aligned}
\|\varphi\,\big|\,\mathbb{W}_p^1(\Omega)\,\|:=\left[\|\varphi\,\big|\,\mathbb{L}_p(\Omega)\,\| +\sum\limits_{j=1}^n\|\partial_j\varphi\,\big|\,\mathbb{L}_p(\Omega)\|_p\right]^{1/p}
\eta_{s_i}\eta_{s_{i+1}}\eta_{s_i}= \eta_{s_i}\eta_{s_{i+1}}\eta_{s_i}, \eta_{s_j}\eta_{s_k}= \eta_{s_k}\eta_{s_j}, \eta_{s_i}C_{s_i}= C_{s_i}, \xi_{s_i}\xi_{s_{i+1}}\xi_{s_i}= \xi_{s_i}\xi_{s_{i+1}}\xi_{s_i}, \xi_{s_j}\xi_{s_k}= \xi_{s_k}\xi_{s_j}, \xi_{s_i}C_{s_i}= C_{s_i}.
\Phi_1^c=-\Lambda_A+(F'_A(\omega_B)-1)\Lambda_B+\frac{1}{2}F''_A(\omega_B) \Lambda_B^2+O(\Lambda_B^3)\,, & \Phi_2^c=-\Lambda_B+(F'_B(\omega_A)-1)\Lambda_A+\frac{1}{2}F''_B(\omega_A) \Lambda_A^2+O(\Lambda_A^3)\,.
\|T_{\alpha,q}^{-1}u\|_{X(\mathbb{R}^n)}= \|u\|_{\tilde{X}(B_R)}\|\nabla (T_{\alpha,q}^{-1}u)\|_{Y(\mathbb{R}^n)}= \|L_{p,\theta}u\|_{\tilde{Y}(B_R)}.
p(\boldsymbol{\mathcal{A}}) := \sum_{k=0}^{{\ell}}\boldsymbol{\mathcal{A}}^{k_*}\times \boldsymbol{\alpha}_k , p^{D}(\boldsymbol{\mathcal{A}}) := \sum_{k=0}^{{\ell}}\boldsymbol{\alpha}_k^H \times \boldsymbol{\mathcal{A}}^{k_*},
|\mu_{t}(tg^N)| = \mu_t(-tg^N) = \mu_t \left(\sum_{i > n_0}-tg^N|_{[i]}\right) &\leq \sum_{i > n_0}\sup(-tg^N|_{[i]}) \mu_t[i] &\leq C_t \sum_{i > n_0}\sup(-tg^N|_{[i]}) \exp(\sup(tf|_{[i]})) \leq C_t \sum_{i > n_0}\sup(-tf|_{[i]}) \exp(\sup(tf|_{[i]}))\,,
I = \{1, 2, 3, 4, 5, 9, 10, 13, 15, 18, 20, 21, 22, 23, 24, 25, 26, 30, 33, 36, 37, 40, 41, 44, 45, 48, 51, 52, 57, 63, 65, 66, 67, 68, 70, 72, 74, 75, 78, 82, 86, 93, 98, 101, 102, 103, 104, 106, 109, 110, 111, 112 \}.
\begin{aligned}\ \O(p{\rm log}\, m + p{\rm log}\, n + p^3 + npl + nd^2 + d^3) = O\left(\frac{k^{8}}{\epsilon^{4}\delta}{\rm log}\, m + \frac{k^{16}}{\epsilon^{8}\delta^2}{\rm log}\, n + \frac{k^{24}}{\epsilon^{12}\delta^3}+ \frac{k^{8}}{\epsilon^{4}\delta}nl + nd^2 + d^3 \right). \end{aligned}
\left( \begin{array}{c}{\hat{a}_{i}( t )}{\hat{a}_{i}^{\dag}( t )}\end{array}\right) = \sum_{j = 1}^{2}\left( \begin{array}{c c}{\alpha_{i j}( t )}{\beta_{i j}( t )}{\beta_{i j}^{\ast}( t )}{\alpha_{i j}^{\ast}( t )}\end{array}\right) \left( \begin{array}{c}{\hat{a}_{j}}{\hat{a}_{j}^{\dag}}\end{array}\right) \; ,
c\mbox{$\sum\limits_{k\in I}v_k\otimes v_{\mathbf i}\otimes v_{pq}\otimes v_{\mathbf j}\otimes \bar v_k x_{r_1}+d \sum\limits_{k\in I}$}v_k\otimes v_{\mathbf i}\otimes v_{pq}\otimes v_{\mathbf j}\otimes \bar v_k=0
\begin{aligned}\nabla T =\dfrac{\partial T}{\partial x_i}\vec e_i=\dfrac{\partial X_j'}{\partial x_i}\dfrac{\partial T}{\partial X_j'}\vec e_i=\dfrac{\partial (x_j'-y_j')}{\partial x_i}\dfrac{\partial T}{\partial X_j'}\vec e_i=\alpha_{ij}\dfrac{\partial T}{\partial X_j'}\vec e_i=\dfrac{\partial T}{\partial X_j'}\vec e_j'=\nabla' T.\end{aligned}
T_{\alpha}, \quad G_{a}= \frac{1}{1 2}\sigma_{a}^{\alpha \dot{\beta}}( R_{\dot{\beta}\alpha \gamma}^{\gamma}+ R_{\alpha \dot{\beta}\dot{\gamma}}^{\dot{\gamma}}) , \quad W_{\alpha \beta \gamma}= \sigma_{( \alpha \dot{\delta}}^{a}\sigma_{\beta}^{b \dot{\delta}}T_{a b \gamma )}
J_1(\beta,N,T) := N\int_0^T\int_0^T \int_{\mathbf{B}_2(0)}f^{(1,1,\alpha)}_{t,s}(z)dz ds dt =N\int_0^T\int_0^T \frac{1}{(2\pi (t-s))^{1/2}}\int_{|z|<2}e^{-\frac{(z-\beta^{1/3}N^{1/3}(t-s))^2}{2(t-s)}}dzds dt.
g \, \langle \langle F_{0 k}( z_{j}) \rangle \rangle^{\mathrm{S R}}= \frac{2}{3}\alpha_{s}\left( \frac{r_{j i}^{k}}{r_{j i}^{3}}+ \frac{r_{j n}^{k}}{r_{j n}^{3}}\right) ,
F'(\varphi)(\psi) \,(\xi)= \tfrac{\partial f}{\partial z}(\xi,\varphi(\xi)) \psi(\xi), \xi \in \mathcal{O}, F''(\varphi)(\psi_1, \psi_2) \,(\xi)= \tfrac{\partial^2 f}{\partial z^2}(\xi,\varphi(\xi)) \psi_1(\xi) \psi_2(\xi), \xi \in \mathcal{O}
D(\mathcal{A}) = \Big\{(\varphi, \psi)' \in \mathcal{V}\,\big|\,\exists (f_{1}, f_{2})' \in \mathcal{H}\; \forall (\tilde{\varphi}, \tilde{\psi})' \in \mathcal{V}:B(\varphi, \psi; \tilde{\varphi}, \tilde{\psi}) = \int_{\Omega}f_{1}\tilde{\varphi}+ f_{2}\tilde{\psi}\mathrm{d}x \Big\}
v_{N,l}^F:=\begin{cases}1-\gamma^{N^F}\left(A^{N^F}_{\varepsilon_N}\right) , 1-\gamma^{N^F}\left( B^F(T_{N,l})^l\times A^{(N-l)^{F}}_{\varepsilon_{N,l}}\right) ,\end{cases}v_N^F:=\max\{v_{N,l}^F \ |\ 0 \leq l \leq n_N-1 \}.
H_{2,1}(F_{f}/2) \leq\frac{1}{2304}\left(|-3\tau^4_{1}+12(1-\tau^2_{1})\tau^2_{1}\tau_{2}-8(1-\tau^2_{1})(2+\tau^2_{1})\tau^2_{2}| \right. &\left.+24\tau_{1}(1-\tau^2_{1})(1-|\tau^2_{2}|)\right) &=\frac{1}{96}\tau_{1}(1-\tau^2_{1})\left(|A+B\tau_{2}+C\tau^2_{2}| +1 -|\tau_{2}|^2\right),
\mathrm{LC}_{s=1}^r [\mathcal{G}_{D,m}(z,s)]^{\mathrm{hol}}= -\sum_{n|m}\biggl(\frac{D}{m/n}\biggr) n \sqrt{D}\mathrm{LC}_{s=1}^r[1] q^{-n^2D}+ \sum_{0 > d \equiv 0, 1 (4)}\mathrm{Tr}_{d, D}(F_{0,m,r}) q^{-d}.
\begin{aligned}\sigma_{-2m}(VW\Delta^{-m}) =-\frac{m(m+1)}{3||\xi ||^{2m+4}}V^{a}W^{b}\mathrm{Ric}_{cd}\xi_{a}\xi_{b}\xi_{c}\xi_{d}&+\frac{2m}{3||\xi ||^{2m+2}}V^{a}W^{b}\xi_{c}(\xi_{a}\mathrm{Ric}_{bc}+\xi_{b}\mathrm{Ric}_{ac}-R_{cadb}\xi_{d})+o(\mathbf{1}).\end{aligned}
~\begin{aligned}J_3 =\iint_{[-\infty,\infty]\times[-\infty,y_{0}-\epsilon]}\biggl[\frac1{(a+xw_1+yw_2)^k}&\quad-k\frac{w_1}{(a+xw_1+yw_2)^{k+1}}P_{1}(x) &\quad-k\frac{w_2}{(a+xw_1+yw_2)^{k+1}}P_{1}(y) &\quad+k(k+1)\frac{w_1w_2}{(a+xw_1+yw_2)^{k+2}}P_{1}(x)P_{1}(y)\biggl]dxdy.\end{aligned}
{\renewcommand{\arraystretch}{1 . 3}\begin{array}{c | c | c | c | c}{{A_{+ -}^{i 4}}}&{{A_{+ +}^{i 4}}}&{{A_{+ -}^{i 5}}}&{{A_{+ +}^{i 5}}}&{{A_{+ +}^{i j}}}\\ \hline{{\sqrt{M^{2}{+}M^{\prime}{}^{2}}}}&{{M}}&{{M}}&{{\sqrt{M^{2}{+}M^{\prime}{}^{2}}}}&{{\sqrt{4 M^{2}{+}M^{\prime}{}^{2}}}}\\ \end{array}}
(A{\mathcal H}x)(\zeta) = \begin{bmatrix}0 1 1 0 \end{bmatrix}\frac{\partial}{\partial \zeta}\left( \begin{bmatrix}\frac{1}{\rho(\zeta)}0 0 T(\zeta) \end{bmatrix}x(\zeta) \right); D(A{\mathcal H}) =\left\{x\in{\cal W}^{1,2}((0,1);\mathbb F^2)\mid \hat W_B ({\mathcal H}x)(0,t) =0\right\},
\tilde{D}_{\alpha \beta}^{\mu \nu}(k)=\frac1{k^2}\left[P_{\alpha\beta}^{\mu\nu}+\frac1{k^2}\tilde{\Pi}_{\alpha\beta}^{\mu\nu}(k)+\left(\frac1{k^2}\right)^2\tilde{\Pi}_{\rho\sigma}^{\mu\nu}(k)\tilde{\Pi}_{\alpha\beta}^{\rho\sigma}(k)+\dots\right]
|D^k\tilde{v}(x)|\leq C\int_{\mathbb{R}^n}\frac{e^{n\tilde{u}(y)}}{|x-y|^k}dy \leq C\int_{\mathbb{R}^n}\frac{e^{n\left(\tilde{v}(y)+p\left(\frac{y}{|y|^2}\right)+q(y)\right)}}{|x-y|^k|y|^{n\left(2+\beta+\frac{\Lambda}{\gamma_{n}}\right)}}dy.
\langle \, 0 \, | \, \mathcal{O}( x ) \, | \, p_{1}, \dots , p_{n}\, \rangle^{i n}= e^{- i x ( p_{1}+ \dots + p_{n})}\, f_{n}^{\mathcal{O}}( \theta_{1}, \dots , \theta_{n}) ~ , \mathrm{f o r}~ ~ \theta_{1}> \dots > \theta_{n}
\begin{array}{lllll}\hat{f}(\xi) = \displaystyle \hat{u_*}+i\int_1^t\int e^{i\frac 12(\eta^2-2\xi\eta)s}e^{-s}{\hat{f}(\xi-\eta,s)\hat{v_*}(\eta,s)}d\eta ds & \displaystyle +i\int_1^t\int_1^s\iint e^{i\eta(\sigma-\xi)s}F(s,\xi,\sigma,\eta)dsd\eta d\sigma,\end{array}
\frac{\kappa_{(11)(00)}^{(\ell+1)}}{K_{(11)}^{(\ell)}K_{(00)}^{(\ell)}}=-\frac{1}{3}\xi (1+O(\ell^{-1}))+O(n^{-2}) \frac{\kappa_{(11)(11)}^{(\ell+1)}}{(K_{(11)}^{(\ell)})^2}= \frac{8}{3}\xi(1+O(\ell^{-1})) +O(n^{-2}) \frac{\kappa_{(11)(22)}^{(\ell+1)}}{K_{(11)}^{(\ell)}K_{(22)}^{(\ell)}}=\frac{2}{3}\xi(1+O(\ell^{-1}))+O(n^{-2}) ,
{\cal L}_{\mathrm{D B I}}= T_{3}q_{3}\biggl \{\frac{1}{2}e^{\psi}{| v |^{2}}- \frac{1}{4}( 2 \pi \alpha^{\prime})^{2}e^{- \Phi}{F_{\mu \nu}F^{\mu \nu}\biggr \}\ .}
||\mathcal{F}(\sqrt{\cdot}(N(u_{1})-N(u_{2}))(t,\cdot\lambda(t)))(\omega \lambda(t)^{2})||_{L^{2}(\rho(\omega \lambda(t)^{2})d\omega)}\leq \frac{C}{\lambda(t)}\frac{||y_{1}-y_{2}||_{Z}}{t^{4}\log^{\epsilon}(t)}
\Sigma = - 3 g^{2}C_{2}( G ) q_{m}q_{n}\Delta^{- 1}( q ) \int \frac{d^{4}s}{( 2 \pi )^{4}}\frac{( \delta_{m n}s^{2}- s_{m}s_{n})}{s^{2}( s^{2}+ s_{0}^{2}) ( s^{2}+ 2 s q + q^{2})}.
a_{n+1}\cdot \sum_{i=0}^{n}(-1)^{i}\sum_{\mathrm{card}(I)=i}\left(\prod_{j\in I}x_{j}\right)\cdot A\left(\prod_{k\in \left\{1, \ldots, n+1\right\}\setminus I}x_{k}\right)=0 \left(x\in R\right).
\tilde \nabla^a D \beta_a = \tilde \sigma^{ab}\tilde \nabla_a (\nabla^N_L \bar W_{Lb L \underline L}) = \tilde \sigma^{ab}\nabla_a \bar W_{Lb L \underline L}+\nabla^N_L \tilde \nabla^a \bar W_{La L \underline L}= 4 \nabla^N_L \bar W_{L\underline L L \underline L}.
\begin{aligned}{\rm dist}(\Gamma_{0,\alpha-2-u},\Gamma_{-1}+1-\alpha) \leq (\kappa-2^{4/3}a)/t^{1/3},{\rm dist}(-\Gamma_{0,\alpha-2-u},\Gamma_{-1}+2-\alpha) \leq (\kappa-2^{4/3}a)/t^{1/3}.\end{aligned}
\prod_{j=1}^{n+1}\left| \left(e^{- 2 i \vartheta}\left( u_0 - q + \sum_{l=1}^{j-1}[M_A v]_l \right)^2 - E - i \eta \right)^{-1}\right| & \leq C_{E,\vartheta}^{n-1}g(q,u_0,v) .
\mathfrak{M}\bigl(\frac{q}{\tau}\,|\,\frac{1}{\tau},\tau\lambda_1,\tau\lambda_2\bigr) (2\pi)^{-\frac{q}{\tau}}\,\Gamma^{\frac{q}{\tau}}(1-\tau) \Gamma(1-\frac{q}{\tau}) = \mathfrak{M}(q\,|\,\tau,\lambda_1,\lambda_2) (2\pi)^{-q}\times \times \Gamma^{q}(1-\frac{1}{\tau}) \Gamma(1-q).
\left. \kappa^{( 2 )}= \frac{1}{2}\frac{d F}{d r}\right|_{r = r_{s t}}= \frac{\sqrt{M^{2}- Q^{2}- a^{2}\cos^{2}\theta}}{2 M^{2}- Q^{2}+ 2 M \sqrt{M^{2}- Q^{2}- a^{2}\cos^{2}\theta}},
G(x,y) = \frac{e^{ik_{-}\vert x \vert}}{\sqrt{\vert x \vert}}G^{\infty}(\hat x, y) + G_{Res}(x,y), \nabla_y G(x,y) = \frac{e^{ik_{-}\vert x \vert}}{\sqrt{\vert x \vert}}H^{\infty}(\hat x,y ) + H_{Res}(x,y),
I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=\lambda^{2}+\frac{1}{\lambda^{2}}+1;I_{2}=\frac{1}{\lambda_{1}^{2}}+\frac{1}{\lambda_{2}^{2}}+\frac{1}{\lambda_{3}^{2}}=\frac{1}{\lambda^{2}}+\lambda^{2}+1
\vec L \omega = \sum_{i,j,k}\frac{\partial^2\omega_i}{\partial x^j\partial x^k}Z^{ij}\partial\phi^k + \sum_{i,j=1}^m \frac{\partial \omega_i}{\partial x^j}\partial Z^{ij}+ \sum_{i,j=1}^m \frac{\partial \omega_i}{\partial x^j}L\phi^i\partial \phi^j +\sum_{i,j=1}^m \omega_i\partial L \phi^i
\Big\Vert \psi \Big\Vert_{H^{-1,-\frac{1}{2}}\Big(\Omega\times\mathbb{R_+}\Big)}= \sup_{0\neq u \in H^{1,\frac{1}{2}}\Big( \Omega\times\mathbb{R_+}\Big)}\dfrac{\Big|\Big\langle \psi,u \Big\rangle_{\Omega\times\mathbb{R_+}}\Big|}{\Big\Vert u \Big\Vert_{\mathrm{H}^{1,\frac{1}{2}}\Big(\Omega\times\mathbb{R_+}\Big)}}.
h_{X}(x)=\frac{\alpha (\theta_{1}+\theta_{2}+\theta_{3})(a+bx)\eta^{\alpha -1}(x)e^{-\eta^{\alpha}(x)}\left( 1-e^{-\eta^{\alpha}(x)}\right)^{-1}}{\left( 1-e^{-\eta^{\alpha}(x)}\right)^{\theta_{1}+\theta_{2}+\theta_{3}-1}-1},
\big|\max_{j\le n}|S_jv|\big|_{L^{p^*}(\mu)}=\big|\max_{j\le n}|S_j(v\circ\pi_\Delta)|\big|_{L^{p^*}(\mu_\Delta)}\le\big|\max_{j\le n}|S_j(\bar v\circ\bar\pi)|\big|_{L^{p^*}(\mu_\Delta)}+2|\psi|_{L^{p^*}(\mu_\Delta)}= \big|\max_{j\le n}|S_j\bar v|\big|_{L^{p^*}(\bar\mu_\Delta)}+2|\psi|_{L^{p^*}(\mu_\Delta)}\ll \|v\|_\eta \,n^{\max\{\frac12,\frac1p\}},
\begin{aligned}\| \nabla v\|_{C^0(\overline{D_T})}&\leq C\big(r^{-1-2\alpha}+r^{-2}+r^{-\frac{p}{2}(1+\alpha)}\big)\big(\|v_{0}\|_{C^{1+\alpha}}+\|f\|_{L^p(D_T)}+\|g \|_{C^{\alpha/2,\alpha}}\big).\end{aligned}
S_n = \sum_{i=0}^{n-1}\alpha_r(\delta,i) r^i(e) = \sum_{i=0}^{n-1}\alpha_r(\tau(e),i) r^i(e) = \sum_{i=0}^{n-1}\tau_{r^i}(e) r^i(e) = \sum_{i=0}^{n-1}\frac{e}{r^i(e)}r^i(e) = ne,
\left|{{dx}\over{d\lambda}}\right|=H^{-{2\over{(D-2)\Delta}}}H^{-{{2(p+1)}\over{(D-2)\Delta_p}}}_p\sqrt{H^{-{4\over{(D-2)\Delta}}}\left(E^2H^{{4(D-p-3)}\over{(D-2)\Delta_p}}_p-{{{\cal J}^2}\over{x^2H^{{4(p+1)}\over{(D-2)\Delta_p}}_p}}\right)-\epsilon},
D(17\mid\{1,2,2,3\}) =D(14\mid\{1,2,2,3\})+D(14\mid\{1,2,2\}) &=D(11\mid\{1,2,2,3\})+D(11\mid\{1,2,2\})+9 &=D(8\mid\{1,2,2,3\})+D(8\mid\{1,2,2\})+6+9 &=1+2+6+9=18.
I_{dyn}^{m}(\tilde{\rho}(\cdot,\,x)dx)=\frac{1}{2}\int_{0}^{T}\left\Vert \frac{\partial\tilde{\rho}}{\partial t}-\frac{1}{2}\nabla\cdot\left[D(\tilde{\rho})\nabla\tilde{\rho}\right]\right\Vert_{-1,\,A(\tilde{\rho})}^{2}dt.
G_{1}(\epsilon,v_1,v_2)=(-\Delta+\sigma)^{-1}[e^{v_2+h_{2\epsilon}}(1-e^{v_1+h_{1\epsilon}})+\sigma v_1-g_{1\epsilon}], & G_{2}(\epsilon,v_1,v_2)=(-\Delta+\sigma)^{-1}[e^{v_1+h_{1\epsilon}}(1-e^{v_2+h_{2\epsilon}})+\sigma v_2-g_{2\epsilon}],
U ( \lambda \mu \rho ) ={\frac{1}{d ( \rho )}}U_{k_{1}\, \, k_{2}}^{\, \, l_{1}\, \, l_{2}}\left( \begin{array}{c c c}{{\lambda}}&{{\mu}}&{{\rho}}\\{{k_{1}}}&{{k_{2}}}&{{m}}\\ \end{array}\right) \left( \begin{array}{c c c}{{\lambda}}&{{\mu}}&{{\rho}}\\{{l_{1}}}&{{l_{2}}}&{{m}}\\ \end{array}\right)^{\ast}.
\begin{aligned}\{{\eta}({\xi}_{1})+{\eta}({\xi}_{2})+\cdots+{\eta}({\xi}_{k})\}=\{\{\xi_{1}-p_{1}\eta\}+\{\xi_{2}-p_{2}\eta\}+\cdots+\{\xi_{k}-p_{k}\eta\}\}&=\{\xi_{1}+\xi_{2}+\cdots+\xi_{k}-(p_{1}+p_{2}+\dots+p_{k})\eta\}&=\{\xi_{1}+\xi_{2}+\cdots+\xi_{k}\},\end{aligned}
\mathbf{p}_i(k+1) := \min_{\mathbf{p}_c}f_i(\mathbf{p}_i,k+1) + (\boldsymbol{\lambda}^{[k]}- \boldsymbol{\lambda}^{[k-1]})^T\mathbf{p}_i + + \frac{\rho}{2}\|\mathbf{p}_i - \mathbf{p}_i^{[k]}\|^2 \underline{\mathbf{p}}_i^{[k+1]}\leq \mathbf{p}_i \leq \overline{\mathbf{p}}_i^{[k+1]}