harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Delta(\underline{k},m,z) := \det\left(\begin{array}{ccc}A_{k_1,m}(z) B_{k_1,m}(z) C_{k_1,m}(z) A_{k_2,m}(z) B_{k_2,m}(z) C_{k_2,m}(z) A_{k_3,m}(z) B_{k_3,m}(z) C_{k_3,m}(z) \end{array}\right),
G(\xi,u) :=\Pi_{\varepsilon,\xi}^{\perp}\left\{W_{\varepsilon,\xi}+\Pi_{\varepsilon,\xi}^{\perp}u-i_{\varepsilon}^{\ast}\left[ b(x)f\left(W_{\varepsilon,\xi}+\Pi_{\varepsilon,\xi}^{\perp}u\right) +\omega^{2}b(x)g\left( W_{\varepsilon,\xi}+\Pi_{\varepsilon,\xi}^{\perp}u\right)\right] \right\}& \qquad+\Pi_{\varepsilon,\xi}u.
-\frac{1}{H^{2+\sigma}}B=|\nabla f|^2\frac{\partial}{\partial t}U_{\sigma}-\left(2C_0-\frac{2+\sigma}{H}\langle\nabla H,\nabla f\rangle-2H|\nabla f|\right)U_{\sigma}-\frac{8L_{22}}{H^{\frac{2+\sigma}{2}}|\nabla f|^{3}}\sqrt{U_\sigma},
J_{13}= \int -\partial^\zeta_x(u\cdot\nabla u)_t \cdot \partial^\zeta_x u_t \leq C\|u_t\|_2\|u\|_3| u_t|_{D^2}+\int -\big(u\cdot \nabla\big) \partial^\zeta_x u_t \cdot \partial^\zeta_x u_t \leq C\Big(1+| u_t|^2_{D^2}+|\nabla u|_\infty|\partial^\zeta_x u_t|^2_2\Big) \leq C\Big(1+| u_t|^2_{D^2}\Big),
\mathbf{b}_R(\psi_1,\theta_1) = \frac{1}{\sqrt{N_1}}\left[ e^{j2\pi n_1 \frac{d_R}{\lambda}\cos(\theta_1)\sin(\psi_1)}\right]^T_{n_1\in \mathcal{I}(N_1)}\otimes \frac{1}{\sqrt{N_2}}\left[ e^{j2\pi n_2 \frac{d_R}{\lambda}\sin(\theta_1)}\right]^T_{n_2\in \mathcal{I}(N_2)}
s_{\bullet}=\frac{G(M_{\bullet}+M_{\odot})\sqrt{ln\Lambda}}{V^{2}/2}\approx\frac{M_{\bullet}}{M_{\odot}}\frac{V_{\odot}^{2}}{V^{2}}R_{\odot}>[s_{Hill},s_{Loss}]_{max}=(1-4000)R_{\odot},
\phi_{0}^{(k)}(z) =\frac{1}{z-a_{k}^{(0)}-\sum_{j=1}^{p}a_{k}^{(j)}\,\phi_{j-1}^{(k+1)}(z)}\phi_{j}^{(k)}(z) =\phi_{0}^{(k)}(z)\,\phi^{(k+1)}_{j-1}(z)1\leq j\leq p.
\Gamma^{\mu \nu}= ( p^{2}+ A ) \left( \eta^{\mu \nu}-{\frac{p^{\mu}p^{\nu}}{p^{2}}}-{\frac{\bar{\theta}^{\mu}\bar{\theta}^{\nu}}{\bar{\theta}^{2}}}\right) + ( p^{2}+ A + B ){\frac{\bar{\theta}^{\mu}\bar{\theta}^{\nu}}{\bar{\theta}^{2}}}+{\frac{p^{\mu}p^{\nu}}{\xi}}
r(t) := -\frac{1+\mu(t)(\mu^{\sigma}(t))^{\dag}}{\mu^{2}(t)(\mu^{\sigma}(t))^{\dag}},s(t) := \frac{p(t)-\mu(t)[2Q_{x}(t,0)+\mu(t)P_{xx}(t,0)]}{\mu^{2}(t)(\mu^{\sigma}(t))^{\dag}},
S_{\mathrm{D B I}}^{( \mathrm{M 1 0})}= - \int_{R^{9 + 1}}|{\hat{k}}|^{3}\sqrt{| \mathrm{d e t}\left({\hat{\Pi}}+ |{\hat{k}}|^{- 1}{\hat{\cal F}}\right) |}{\hat{R}}({\hat{T}}, \partial{\hat{T}}, \dots ) .
\phi(s_\alpha s_\alpha^*) = S_{i_1}S_{i_2}\cdots S_{i_k}S_{i_k}^* S_{i_k}S_{i_k}^* S_{i_{k-1}}^* \cdots S_{i_1}^* = S_{i_1}S_{i_2}\cdots S_{i_k}S_{i_k}^* S_{i_{k-1}}^* \cdots S_{i_1}^*.
H\left(\sum_{i=1}^k q_i \mu_i,\mathcal{E}\right) \leq \sum_{E \in \mathcal{E}}\sum_{i=1}^k \phi\left(q_i \mu_i(E)\right) & = \sum_{i=1}^k q_i \left(\sum_{E \in \mathcal{E}}\phi(\mu_i(E))\right) + \sum_{i=1}^k \phi(q_i)\left( \sum_{E \in \mathcal{E}}\mu_i(E)\right) & =\sum_{i=1}^k q_i H(\mu_i,\mathcal{E}) + H(q).
\boldsymbol{V}_{A\rightarrow B}^{ext}=\left((\overline{\boldsymbol{V}}_{A}^{post})^{-1}-(\boldsymbol{V}_{A}^{pri})^{-1}\right)^{-1}, \boldsymbol{x}_{A\rightarrow B}^{ext}=\boldsymbol{V}_{A\rightarrow B}^{ext}\left((\overline{\boldsymbol{V}}_{A}^{post})^{-1}\boldsymbol{x}_{A}^{post}-(\boldsymbol{V}_{A}^{pri})^{-1}\boldsymbol{x}_{A}^{pri}\right),
H \: = \: r^2h + h_{0}+ r^{-2}h_{2}+ \cdots + r^{-2i}h_{2i}+ \cdots, f \: = \: - \frac14 r^2 + f_{0}+ r^{-2}f_{2}+ \cdots + r^{-2i}f_{2i}+ \cdots.
{\bar\theta}^{\dot 2}= Y (x){\bar\theta}^{\dot 1},\quad{\bar\theta}^{\dot \alpha}={\bar\theta}^{\dot 1}{\bar\psi}^{\dot\alpha},\quad{\bar\zeta}^{\dot 2}= Y (x){\bar\zeta}^{\dot 1},\quad{\bar\zeta}^{\dot \alpha}={\bar\zeta}^{\dot 1}{\bar\psi}^{\dot\alpha}.
\frac{\partial^{n}}{\partial a^{n}}Q_{\nu}(a,b)=n!(-a)^{n}\sum_{k=0}^{[n/2]}\frac{(-2a^{2})^{-k}}{k!(n-2k)!}\sum_{p=0}^{n-k}(-1)^{p}(\begin{matrix}n-k\\p\end{matrix})Q_{\nu+p}(a,b),
\times \exp\left\{\frac{1}{(1-x)^3(1-y)^5}\sum_{n=1}^{\infty}(-2n^2x^{n+2}y^{n+1}+2n^2x^{n+2}y^{n+2}-n^2xy^{n+1}) \frac{z^n}{n^4}\right\}
{\mathcal F}^{r,\bullet}=[f^{r}(\underline{M}\otimes_K\Omega^0_X)\longrightarrow f^{r-1}(\underline{M}\otimes_K\Omega^1_X)\longrightarrow f^{r-2}(\underline{M}\otimes_K\Omega^2_X)\longrightarrow\ldots].
{\bf P}\left( \hat{a}^{\dagger}_{a_1}\ldots \hat{a}^{\dagger}_{a_s+1}A^{a_1 \ldots a_{s+1}}_{\;\;\quad \quad b_1 \ldots b_s}\hat{a}^{b_1}\ldots\hat{a}^{b_s}\right) = \hat{a}^{\dagger}_a K^a
as_{\lambda}(F_1, F_2, \psi) = \int_{X_{\psi^*}}(F_1\circ\psi\circ\nabla \psi^*(y)^{1-\lambda}(F_2\circ\psi^*(y))^{\lambda}(\det \,\nabla^2 \psi^*(y))^{1-\lambda}dy.
V_t \leq e^{ct}\prod_{s \leq t}\left( 1 + M \sum_{0 \leq j_1+\ldots+j_l \leq Q}\prod_{i=1}^{l}\left|\Delta L^{(i)}_s\right|^{j_i}\right) \leq e^{ct}\prod_{i=1}^l \prod_{s \leq t}\left( 1+M \sum_{j=1}^Q \left|\Delta L^{(i)}_s \right|^j \right)
\begin{aligned}\limsup_n\frac{\alpha_n}{2}|\hat{x}_n-\hat{y}_n|^2 \leq \limsup_n (u(\hat{x}_n)-v(\hat{y}_n))-(u(x^\ast)-v(x^\ast)) &\leq u(\hat{x})-v(\hat{y})-(u(x^\ast)-v(x^\ast)),\end{aligned}
\left( \prod_{1 \le j \le m}\alpha_j^{\binom{\nu_j}{2}}\right)\prod_{1 \le i < j \le m}(\alpha_i - \alpha_j)^{\min \left\{\nu_i^2,\, \nu_j^2 \right\}}\underset{\mathbb{Z}[\alpha_1, \ldots, \alpha_m]}{\bigg|}\widehat{D}(\overline{\alpha}).
g_{\tau \tau}= g_{\xi \xi}= \frac{\widetilde{a}^{2}( r )}{\left( \cosh \tau - \cos \xi \right)^{2}}, \qquad g_{s s}= \frac{\widetilde{a}^{2}( r ) \sin^{2}\xi}{\left( \cosh \tau - \cos \xi \right)^{2}},
\mathbb E [ X_{t}^{(u)}] \, =\, \mathbb E [ \widetilde{X}_{t}^{(u)}] \, =\, \mathbb E [ X_{t}^{\bullet}] \, =\, \mathbb E [ X_{t}^{\dagger}] \, =\, 0 \, , t \ge 0 \, , \, \, u \in [0, 1] \, ,
D ={(1-ax)(1-bx)(1-cx)(1-dx)\over (1-x^2)(1-q x^2)}\left(T_{q,x}^{+1}-1\right) &+{(1-a/x)(1-b/x)(1-c/x)(1-d/x)\over (1-1/x^2)(1-q /x^2)}\left(T_{q,x}^{-1}-1\right),
U_{i(k)}^{m,2}(x,t):= -\int\limits^t_{A}\int\limits_{\mathbb{R}^{3}_{+}}G_{ij}^{m}(x,y,t-\tau)\Big[\frac{\partial}{\partial y_{l}}F^{k}_{jl}(y,\tau)+\frac{\partial}{\partial{y_j}}p^{1(k)}(y,\tau)\big]dyd\tau.
E_{t}(x) := \int_0^t \nabla \cdot e^{(t-s)\Delta}\langle \mu^N_{s}, V^N (x-\cdot) \left( F\big( K\ast u^N_s(x)\big)-F\big( K\ast u^N_s(\cdot)\big)\right)\rangle \ds, M^N_{t}(x) :=\frac{1}{N}\sum_{i=1}^N \int_0^t e^{(t-s)\Delta}\nabla V^N (x-X_s^{i,N})\cdot dW^i_s.
= \mathrm{T r}( \gamma_{\mu}\gamma_{\alpha}\gamma_{\nu}\gamma_{\beta}) \int_{x = 0}^{1}d x \frac{1}{( 4 \pi )^{\omega}\Gamma ( 2 )}\Bigg ( \frac{1}{2}\delta_{\alpha \beta}\frac{\Gamma ( 1 - \omega )}{[ m^{2}+ k^{2}x ( 1 - x ) ]^{1 - \omega}}
f_{t_i}(x_i) =K_i(1+x_i)[(a_i-d_i)(d_j+s\rho_j)-d_js\rho_i]-K_ia_i(d_j+s\rho_j) f_{b_i}(x_i) =[d_i-(a_i-d_i)x_i][K_i(d_j+s\rho_j)+r_i\rho_j(1-s)(1+x_i)(1+K_i)]+d_jK_is(1+x_i)\rho_i.
L_{ij}[\psi] =- \frac{1}{2}\phi_{ij}(\nabla^k\nabla^l\psi_{kl}-\square\psi-\Lambda\psi)- \square \psi_{ij}- \Lambda \psi_{ij}- \frac{1}{2}\nabla_i\nabla_j \psi + \nabla^k\nabla_{(i}\psi_{j)k},
\mathrm{L i}_{2}( e^{2 \pi i p / q}) = \frac{1}{q^{2}}\sum_{k = 1}^{q - 1}e^{2 \pi i k p / q}\psi^{( 1 )}\left( \frac{k}{q}\right) + \frac{\pi^{2}}{6 q^{2}},
\mu = \mu^{}+ \frac{1}{4\pi i}\sum_{\gamma}\Omega(\gamma) \! \int_{\ell_{\gamma}}\! \frac{d\zeta}{\zeta}\Big(\frac{Z_{\gamma}}{\zeta}- \bar{Z}_{\gamma}\zeta \Big) \ln (1 - \mathcal{X}_{\gamma}(\zeta)) \rlap{.}
-M^3 \int d^5 x \sqrt{g}{\cal R}+ \int d^5 x \sqrt{g}\left(\frac{1}{2}\nabla \phi \nabla \phi -V(\phi) \right)- \int d^4x \sqrt{g_4}\lambda_{P}(\phi) - \int d^4x \sqrt{g_4}\lambda_{T}(\phi),
S^{(b)}_{q,i}(b^k)=\frac{1}{q}\sum_{m\equiv \frac{\pm ql}{2l+1}}\zeta_q^{-im}\prod_{p=0}^{k-1}\left(1-\zeta_q^{b^pm}\right)+\frac{1}{q}\sum_{m\not\equiv \frac{\pm ql}{2l+1}}\zeta_q^{-im}\prod_{p=0}^{k-1}\left(1-\zeta_q^{b^pm}\right)
\displaystyle\lim_{\varepsilon \to 0+}\int_0^\infty{\rm e}^{-s t}\langle \widetilde w_\varepsilon(t),\eta\rangle \,{\rm d}t = \int_0^\infty{\rm e}^{-s t}\langle \widetilde w_0(t),\eta\rangle \,{\rm d}t \qquad(s>0,\ \eta\in X).
\frac{\alpha(\gamma - \beta)}{\beta(\gamma - \alpha)}\lambda + \frac{\gamma(\beta - \alpha)}{\beta(\gamma - \alpha)}\mu = \frac{1}{\beta}\frac{\gamma - \beta}{\gamma - \alpha}\tau(TX) + \frac{1}{\beta}\frac{\beta - \alpha}{\gamma- \alpha}\tau(TY)= \frac{1}{\beta}\tau(TZ) \in V_\beta(T),
\begin{aligned}\|x^{k+1}- q \|^2 = \|\beta_k(v^{k}- q) + (1 - \beta_k)(Tz^k - q \|^2 \leq \beta_k\|v^{k}- q \|^2 + (1 - \beta_k)\|z^k - q \|^2. \end{aligned}
d \psi(g^{y_1}g_1 (g^{y_2})^{-1}, g^{y_2}g_2 (g^{y_0})^{-1}) = \omega_0(g^{y_1}g_1 (g^{y_2})^{-1}, g^{y_2}g_2 (g^{y_0})^{-1}).
P\left({{\bf X}_0 \to{\bf \hat X}_0}|\textbf{H}_0 \right) \le \exp\left({- \frac{\rho{\left\|{\sum\limits_{i=0}^{N_v-1}\textbf{H}_{0,i}\textbf{F}_{0,i}\left({{\bf X}_0 -{\bf \hat X}_0}\right)}\right\|_F^2}}{4\left(N_v\sigma_{n}^2+\sigma_{MUI}^2\right)}}\right).
\alpha_{2}= - \frac{1}{2}C_{a}^{*}f_{\; \; b c}^{a}C^{b}C^{c}- C_{a}^{* \mu}f_{\; \; b c}^{a}\left( C^{b}C_{\mu}^{c}+ \eta^{b}A_{\mu}^{c}\right) .
\lim_{\begin{array}{l l}{x , y \to \infty}\\{x - y = O ( 1 )}\\ \end{array}}K_{\mathrm{\tiny o r i g i n}}( x , y ) \ = \K_{\mathrm{\tiny b u l k}}( x , y ) \ = \c^{\prime}\ \frac{\sin ( x - y )}{x - y}\ .
D_x\phi(\hat x, \hat y) = \frac{1}{\epsilon}a(z)(\hat x-\hat y) + \varphi(z,v(z))\beta(z) - \delta Dd(\hat x) + 2\delta (\hat x-z), -D_y\phi(\hat x, \hat y) = \frac{1}{\epsilon}a(z)(\hat x-\hat y) + \varphi(z,v(z))\beta(z) + \delta Dd(\hat y).
Q(n,u) =h_u^{s,(n-1)}&+ \sum_{i=1}^{n-2}\int_s^u \cdots \int_s^{v_{n-i+1}^*}\Phi(u-v_{n-1}) \prod_{k=n-i+1}^{n-1}\Phi(v_k-v_{k-1}) h_{v_{n-i}}^{s,(n-i-1)}dv_{n-i}\ldots dv_{n-1}&+ \mu \int_s^u \int_s^{v_{n-1}}\cdots \int_s^{v_2}\Phi(u-v_{n-1}) \prod_{k=2}^{n-1}\Phi(v_k-v_{k-1}) dv_1\ldots dv_{n-1}.
\varpi_2^l(h): \: f(Z) \mapsto \bigl( \varpi_2^l(h)f \bigr)(Z) =\frac{f \bigl( (aZ+b)(cZ+d)^{-1}\bigr)}{N(cZ+d)^2 \cdot N(a'-Zc')}, \varpi_2^r(h): \: f(Z) \mapsto \bigl( \varpi_2^r(h)f \bigr)(Z) =\frac{f \bigl( (aZ+b)(cZ+d)^{-1}\bigr)}{N(cZ+d) \cdot N(a'-Zc')^2},
\sum_{\substack{j=3 j\equiv_2 1}}^\infty \vert b_j(p) \vert<\frac{8\pi_pc_p}{\pi^2}\sum_{\substack{j =3 j\equiv_2 1}}^\infty \frac{1}{j^2}=\frac{\pi_p^2 c_p(\pi^2-8)}{\pi^2 \pi_p}.
\int_{\Omega_{T}}-u\left( x,t\right) v\left( x\right) \partial_{t}c\left(t\right) +\left( \int_{\mathcal{Y}_{n,m}}a_{0}\left( x,t,y^{n},s^{m}\right)dy^{n}ds^{m}\right) \cdot \nabla v\left( x\right) c\left( t\right) dxdt =\int_{\Omega_{T}}f\left( x,t\right) v\left( x\right) c\left( t\right) dxdt
u\left( b\right) -u\left( a\right) =u\left( r\left( 1\right) \right)-u\left( r\left( 0\right) \right) ={\displaystyle\int\limits_{0}^{1}}\nabla u\left( r\left( t\right) \right) \cdot\nabla r\left( t\right)dt=0.
W ( N , n ) = \frac{1}{2 \pi i}\, \oint_{| z | = \rho}\, d z \, \bigl ( f ( z ) \bigr )^{N}\, z^{- n - 1}= \frac{1}{2 \pi i}\, \oint_{| z | = \rho}\frac{d z}{z}\, \exp \{N ( \ln f ( z ) - \mu \ln z ) \}\, ,
\Big[ \int_0^1\int_{\Omega}\sup_{R\in\mathcal{D}_{\mu}}\Big(\frac{1}{|R|}\int_{R}\sum_{k=\mu}^{\infty}{2^{skq}\big| \Pi_k T_{[a]}f_L^{\omega,t,\mu}(x)\big|^q}dx\Big) d\lambda dt\Big]^{1/q}&\gtrsim L^{-(s+d-d/q)/2d}\big( \log{L}\big)^{1/2}.
(\begin{matrix}\alpha\\ \beta\end{matrix})=(\begin{matrix}\alpha_{1}\\ \beta_{1}\end{matrix})(\begin{matrix}\alpha_{2}\\ \beta_{2}\end{matrix})\cdot\cdot\cdot(\begin{matrix}\alpha_{n}\\ \beta_{n}\end{matrix})=\frac{\alpha!}{\beta!(\alpha-\beta)!}
2\ell^{\prime}\left( 0,a,\chi\right) =-\overline{E}_{0,\overline{\chi}}\left( 0\right) \log a-\frac{1}{a}\overline{E}_{1,\overline{\chi}}\left(0\right) +\frac{\chi\left( -1\right)}{\left( 2k\right)^{2}}\int\limits_{0}^{2k}\overline{E}_{1,\overline{\chi}}\left( t\right)\zeta\left( 2,\frac{t+a}{2k}\right) dt.
-(A^{2,0}a_0)_0 = 2^{2m}n^{2m}P_3(m) + n^{2m-2}2^{2m -2}P_1(m)\ (2m - 2) \alpha_p(3) -(A^{2,0}a_0)_1 = -2^{2m+1}n^{2m+1}P_4(m) -n^{2m-1}2^{2m-1}P_2(m) (2m-1)\alpha_p(3) - n^{2m-3}2^{2m-3}\Big((2m-3)(2m-4)\frac{\alpha^2_p(3)}{2!}+ (2m-3)\alpha_p(5) \Big).
C\ge (-K_{F_{\alpha,i, \beta}})^{r_{\alpha,i}}=(-m_\alpha K_{X_\alpha})^{r_{\alpha,i}}\cdot F_{\alpha,i, \beta}= (H_\alpha+ D_\alpha)^{r_{\alpha,i}}\cdot F_{\alpha,i, \beta}\ge H_\alpha^{r_{\alpha,i}}\cdot F_{\alpha,i, \beta}.
(\partial^4_y \tilde u_0)(x, 0)+\epsilon(\partial^4_y \mu_\epsilon)(x, 0))= \big[\partial_yu^s_0(0) + (\partial_y\tilde{u}_0)(x, 0)+\epsilon(\partial_y \mu_\epsilon)(x, 0)\big] &\times\big[(\partial_y\partial_x\tilde{u}_0)(x, 0)+\epsilon(\partial_y \partial_x\mu_\epsilon)(x, 0)\big],
[z^n u^{r-1}] Q_d(z,u) = 2^{r-1}\,[v^{2\ell-2r+2}] (1-v^2) (1+v^2)^{n-r}= 2^{r-1}\,[v^{\ell-r+1}] (1-v) (1+v)^{n-r}= 2^{r-1}\binom{n-r}{\ell-r+1}- 2^{r-1}\binom{n-r}{\ell-r}= 2^{r-1}\binom{n-r}{n-\ell-1}- 2^{r-1}\binom{n-r}{n-\ell}.
\Sigma_1 : = p^{(w-1) h_p}\sum_{m=0}^{k_p-h_p-1}\sum_{n=0}^{\infty}\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}}\right)}{p^{m(2-\alpha-w)}p^{n(1-\beta)}}
|c\alpha + d|^2 = (c\alpha + d)(c\bar{\alpha}+ d) &= c^2 \alpha \bar{\alpha}+ cd (\alpha + \bar{\alpha}) +d^2 &= c^2 \left(- \frac{b}{c}\right) + cd \,\frac{a-d}{c}+ d^2 &= -bc + ad - d^2 + d^2 &= ad-bc &= 1 .
\left\{\begin{array}{c}I_c^{xx}I_c^{xy}I_c^{yy}\end{array}\right\}= \int\limits_{\Omega_c}\left\{\begin{array}{c}(x-x_c)^2 (x-x_c)(y-y_c) (y-y_c)^2 \end{array}\right\}~\mathrm{d}\Omega
|A_n|= |\{d\in \mathbb{N}:d\leq \sqrt{2n}\}|+2|\{k\in \mathbb{N}: k<\frac{n+1}{\sqrt{2n}+1}k\nmid n\}|+ |\{k\in \mathbb{N}: k<\frac{n+1}{\sqrt{2n}+1}k|n\}|+\theta(n) & =\lfloor \sqrt{2n}\rfloor+2\left\lfloor \frac{n+1}{\sqrt{2n}+1}\right\rfloor-\sum_{\substack{d|n d<\frac{n+1}{\sqrt{2n}+1}}}1 +\theta(n).
D F ( \xi , x , y )=\left(\begin{matrix}\tau ' (\xi) 0 0 0 (1 / \tau ') ( \rho_{k(\xi)}(x) ) 0 0 ((y \lambda ' + g') / \tau ') ( \rho_{k(\xi)}(x) ) \lambda ( \rho_{k (\xi)}(x) )\end{matrix}\right) ,
\frac{1}{1-c^2b_1^hb_2^v}(gradf_{_2})^v =c^2b_1^h\hskip -0.3cm\sum_{j=m_1+1}^{m_1+m_2}\hskip -0.2cm\left(\frac{a_j^v}{\|u'_j\|(1-c^2b_1^hA_j^v)}\right)^2T_j^v+\hskip -0.3cm\sum_{j=m_1+1}^{m_1+m_2}\hskip -0.2cm\left(\frac{a_j^v}{\|u'_j\|^2 (1-c^2b_1^hA_j^v)}\right)e_j^v.
\liminf_{n\to\infty}\frac{1}{a_c^{(n)}}\log P(S_n(\kappa_n(0))\leq\lfloor\kappa_n(0)\rfloor-a_n) \geq\liminf_{n\to\infty}\frac{n-a_n}{a_c^{(n)}}\log(1-\pi_n(\kappa_n(0))) &=\liminf_{n\to\infty}\left(-\frac{n}{a_c^{(n)}}\pi_n(\kappa_n(0))+\frac{n}{a_c^{(n)}}o(\pi_n(\kappa_n(0)))\right) &=-r^{-1}(1-r^{-1})^{r-1}\alpha^r=-J(0).
S_{\rm eff}= - \left({m\over 2\sqrt{\pi}}\right)^{d-1}{B\over 2\pi}\int_0^\infty{ds\over s^{(d+1)/2}}\, e^{-s}\, \left[\coth\left({Bs\over m^2}\right)-{m^2\over Bs}\right] \theta_4{\left({i\beta\mu\over 2}\left|{i\beta^2m^2\over 4\pi s}\right. \right)}
\gamma = \frac{1}{m\theta_s\theta_x}\left(\frac{2\beta}{s}+ 2 - \theta_x\right)(2s - \theta_su) = \frac{1}{m\theta_s\theta_x}\left[4\beta - (2-\theta_x)\theta_s u + 2(2-\theta_x)s - \frac{2\beta\theta_su}{s}\right] = \frac{1}{m\theta_s\theta_x}\left[ A(s,u) + B \right],
\overline{\mathcal{M}^\Delta_{\mathbb{P}^1,\mathcal{O}(t)}}(Q,\mathbf{a}) \cong \mathbb{P}^q \times \prod_{j=1}^{i'}\emph{Gr}\Big(s_j,d_3-a_j+t+1-\sum_{k=1}^{j-1}s_{k}(a_k-a_{j}+1)\Big) &\qquad\times \prod_{j=i'+1}^{m}\emph{Gr}\Big(s_j,a_j-d_1+t+1-\sum_{k=j}^{m-1}s_{k}(a_k-a_{j}+1)\Big)
E_1:=\int_{(\eta,T)}\zeta_K(s) \frac{x^s}{s}\ds &E_2: = \frac{1}{2\pi i}\left( \int_{\eta+iT}^{c+iT}\zeta_K(s) \frac{x^s}{s}\ds -\int_{\eta-iT}^{c-iT}\zeta_K(s) \frac{x^s}{s}\ds \right) &E_3:= O \left(\sum_{n=1}^\infty \left( \frac{x}{n}\right)^c c_K(n) \min\left(1,\frac{1}{T|\log \frac{x}{n}|}\right) \right).
\int_{\Lambda^\infty}S_\lambda f^{m,v}\overline{S_\lambda f^{m',v}}\, dM =\sum_{\mu \in D_v}c^{m,v}_\mu \overline{c^{m',v}_\mu}\rho(\Lambda)^{d(\lambda)}M(Z(\lambda \mu)) &= \sum_{\mu \in D_v}c^{m,v}_\mu \overline{c^{m', v}_\mu}\rho(\Lambda)^{-d(\mu)}x^{\Lambda}_{s(\mu)}&= \delta_{m, m'},
\begin{alignedat}{1}(\dot u_h(t),v_h) + a(u_h(t),v_h) = (f(t),v_h) \forall \,v_h\in V_h (0<t\le T) (u_h(0),v_h) = (u_{0},v_h) \qquad\ \ \, \forall \,v_h\in V_h . \end{alignedat}
F[-i\frac{\delta}{\delta J}]G[J]=(-i)^{n}\frac{\partial^{k_{1}}}{\partial x_{1}^{k_{1}}}\frac{\delta}{\delta J(x_{1})}\cdot\cdot\cdot\frac{\partial^{k_{n}}}{\partial x_{n}^{k_{n}}}\frac{\delta}{\delta J(x_{n})}G[J].
\|f\|_{Y_{\alpha,\epsilon}(\mathcal{B}_R^+)}:=\sup_{\bar y\in P\cap \mathcal{B}_R}\left(\| d_G(\cdot, \bar y)^{-(1+2\alpha)}(f-P_{\bar y})\|_{L^{\infty}(\mathcal{B}_3^+(\bar y)\cap \mathcal{B}_R^+)}\right. \left. + [d_G(\cdot, \bar y)^{-(1+2\alpha)}(f-P_{\bar y})]_{\dot{C}^{0,\epsilon}_\ast(\mathcal{B}_3^+(\bar y)\cap \mathcal{B}_R^+)}+|P_{\bar y}|_1 \right),
u_1 =-\frac{1}{2}a^4+2a^3+a^2+2a-\frac{1}{2}+\frac{1}{2}(a+1)(a-1)^2\sqrt{a^2-6a+1}, u_2 =-\frac{1}{2}a^4+2a^3+a^2+2a-\frac{1}{2}-\frac{1}{2}(a+1)(a-1)^2\sqrt{a^2-6a+1},
Q_1^{\epsilon_1}Q_2^{\epsilon_2}\cdots Q_k^{\epsilon_k}= V_k \cdots V_2 V_1 t_{b_1}^{-\epsilon_1}t_{b_2}^{-\epsilon_2}\cdots t_{b_k}^{-\epsilon_k}= t_{f(a)}^{-1}ft_af^{-1}.
\int d^{10}\hat{x}\sqrt{|\hat{\jmath}|}\,\left[{\textstyle\frac{1}{2\cdot 5!}}\left(\hat{G}_{(5)}\right)^{2}+{\textstyle\frac{1}{2\cdot 5!}}\left(\hat{\overline{G}}_{(5)}\right)^{2}\right]\, .
\stackrel{\mathrm{H W}}{S^{\alpha \beta}}= \int \tau_{\mathrm{K G}}^{\alpha \beta 0}\, d^{3}x \, = \frac{1}{4 m i}\int \left[ \left( D^{0}\overline{{{\Psi}}}\right) \sigma^{\alpha \beta}\Psi - \overline{{{\Psi}}}\sigma^{\alpha \beta}\left( D^{0}\Psi \right) \right] \, d^{3}x \, .
\int_{\{\Psi<-t\}\cap D_0}|\tilde F|^2e^{-k\varphi-a\Psi}\le 2\int_{\{\psi<-t\}\cap D_0}|f|^2e^{-a\Psi}+2\int_{\{\Psi<-t\}\cap D_0}|\tilde F-fF^{2k}|^2e^{-k\varphi-a\Psi}< +\infty.
I(F;Y) \leq \frac{4\delta^2n}{m}\sum_{i=1}^m\min\Big\{2^{10}\log(md)d^{-1}H(Y^{(i)}),1\Big\}+4\log 2 &\leq \frac{4L^2\delta_n2^{j_n}\|\psi\|_\infty^2 n}{m}\sum_{i=1}^m\min\Big\{2^{10}\log(md)d^{-1}H(Y^{(i)}),1\Big\}+4\log 2.
\frac{i}2 (\sigma_{\mu}{}^{\rho})_{\alpha\beta}L_{\nu\rho}{}^{\alpha\beta}\ + \ \mathrm{h.c.}=2 \mbox{Re}\{b_1^2\}\sum_{k=0}^\infty \frac{t_{\mathrm{a}i}(k) \, [\mbox{a$i$}]_{\mu\nu}{}(k) + t_{\mathrm{b}j}(k) \, [\mbox{b$j$}]_{\mu\nu}{}(k)}{(k!)^2}
F_2\Big(\hbar\partial_{x_0}-\frac{n\hbar}{2x_0}+Q_0,\hbar\partial_{x_1}+Q_1,...,\hbar\partial_{x_n}+Q_n,y_{n+1}\Big)\cdot g\Big(\frac{x_1}{x_0},...,\frac{x_n}{x_0}\Big)=0
\varphi\left(x,\frac{p(x+\bar{h})}{1-\partial_{x_{1}}\varphi(x,p(x+\bar{h}))h+o(h)}\right)=1 \mbox{and}&\varphi\left(x+\bar{h},\frac{p(x)}{1+\partial_{x_{1}}\varphi(x,p(x))h+o(h)}\right)=1.
\eta_n(t)=||(u^n,\mathbf{w}^n,\mathbf{h}^n,\theta^n,\psi^n)(t)||_{H^1(\Omega)}^2 + \int_0^t ||(u_{yy}^n,\mathbf{w}_{yy}^n,\mathbf{h}_{yy}^n,\theta_{yy}^n)(s)||_{L^2(\Omega)}^2 ds.
\frac{1}{2}\frac{d}{dt}\int_{\mathbb T^2}\varphi^2(\theta)dx+\int_{\mathbb T^2}\nabla\cdot\left(\frac{1}{2}\varphi^2(\theta)u\right)\, dx +\nu\int_{\mathbb T^2}\left|\Lambda^{\frac{1}{2}}\varphi(\theta)\right|^2\, dx\leq\int_{\mathbb T^2}f\varphi(\theta)\, dx.
\mathcal{A}= \frac{1}{r^{2}}\begin{pmatrix}(\sigma r_{1}^{2}+\overline{\sigma}r_{2}^{2}) r_{1}r_{2}(\sigma-\overline{\sigma}) r_{1}r_{2}(\sigma-\overline{\sigma}) \overline{\sigma}r_{1}^{2}+ \sigma r_{2}^{2}\end{pmatrix}.
\frac{ig^2}4\int d^4xd^4x^{\prime}\left\{\phi^{+}(x)\phi^{+}(x^{\prime})\Delta_F^2\left( x,x^{\prime}\right) -2\phi^{+}(x)\phi^{-}(x^{\prime})\Delta_{-}^2\left( x,x^{\prime}\right) +\phi^{-}(x)\phi^{-}(x^{\prime})\Delta_D^2\left( x,x^{\prime}\right) \right\}
u(t) = \begin{cases}\tilde{K}_{1}(t) u_{0}+\tilde{K}_{2}(t) u_{0}+\tilde{K}_{3}(t) u_{1}\ \ 0<\nu<2, & E_{1}(t) u_{0}+E_{2}(t) u_{0}+ E_{3}(t)u_{1}\ \ \nu=2, & \tilde{J}_{1}(t) u_{0}+\tilde{J}_{2}(t) u_{0}+\tilde{J}_{3}(t) u_{1}\ \ \nu>2.\end{cases}
F = \sum_{i=1}^4 f_i V_i = 8\pi \left[(1-\alpha)^2 K_1 P_1 f_1+ \alpha(1-\alpha) K_2 P_2 f_2 + \alpha(1-\alpha) K_3 P_3 f_3 + \alpha^2 K_4 P_4 f_4 \right].
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{\left(\tau,v_{s+k+1},\omega_{k+1}\right) \in \mathcal{B}^+_{VII}}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_{d,\alpha}\left( s+k-1 \right) T R^d \theta^{(d-1)/2}\end{aligned}
\mathcal{K}_{euc}(\alpha)_{ij}=\frac{1}{2}\Big{(}\frac{a_{ij}+a_{ji}}{|z|^{m}}+\frac{-m z_{i}z_{p}a_{pj}-m z_{j}z_{t}a_{ti}}{|z|^{m+2}}\Big{)}+\frac{1}{m}\Big{(}\frac{-tr(a)}{|z|^{m}}+\frac{m z_{t}z_{p}a_{pt}}{|z|^{m+2}}\Big{)}\delta_{ij}
[ \hat{\mathrm{G}}_{\pm}^{a}( n ) , \hat{\mathrm{G}}_{\pm}^{b}( m ) ]_{-}= i e_{\pm}{\varepsilon}_{a b c}\hat{\mathrm{G}}_{\pm}^{c}( n + m ) \pm e_{\pm}^{2}n \delta_{n , - m}\delta^{a b}, \,
I(u) =\frac{1}{bp^2}M_u^p-\int_0^TF(t,u(t))dt-\frac{a^p}{bp^2}&\geq\frac{a^{p-1}}{p}\|u\|_{E^{\alpha,p}}^p-\frac{(1-\varepsilon)a^{p-1}}{pC_p^p}\int_0^T|u(t)|^pdt &\geq\frac{a^{p-1}}{p}\|u\|_{E^{\alpha,p}}^p-\frac{(1-\varepsilon)a^{p-1}}{p}\|u\|_{E^{\alpha,p}}^p &=\frac{\varepsilon a^{p-1}}{p}\|u\|_{E^{\alpha,p}}^p=\sigma,\ \ \forall u\in \partial B_\rho(0).
\hat{b}(\lambda):=\sum_{r=0}^{\infty}\star g^r_{\lambda+v\hbar}(\zeta)z^r\,\ \,\ \hat{c}(\lambda):=\sum_{r=0}^{\infty}\star g^r_{-\lambda-(w-v)\hbar}(\zeta)z^r &\hat{\Phi}:=\star H_{v, w}(-z), \,\H_{v, w}:=\frac{\prod_{i=1}^{v}\vartheta(z-t_i+\hbar) \prod_{i=v+1}^w \vartheta(z-t_i-\hbar)}{\prod_{i=1}^w \vartheta(z-t_i)}
I={1\over 16\pi G}\int d^4x\sqrt{-g}\biggl\{R-{1\over 2}\int_{0}^1 d\xi_1\int_{\xi_1}^1d\xi_2\left[\partial\sigma(\xi_1)-\partial\sigma(\xi_2)\right]^2-\int_{0}^{1}d\xi\;e^{-\sigma(\xi)}F(\xi)^2\biggr\},
L(q) = E_{2}(q) = 1-24\,\sum_{n=1}^{\infty}\sigma(n)q^{n}, M(q) = E_{4}(q) = 1+240\,\sum_{n=1}^{\infty}\sigma_{3}(n)q^{n}, M_{\chi}(q) = E_{4,\chi}(q) = C_{0}+ \sum_{n=1}^{\infty}\,\chi\,\sigma_{3}(n)\,q^{n}.
g(AdS;soliton) ={r^2}dt \otimes dt +{{L^2}\over{r^2}}{(1 -{{{r_0}^n}\over{r^n}})^{-1}}dr \otimes dr +{{r^2}\over{L^2}}{(1 -{{{r_0}^n}\over{r^n}})}d\phi \otimes d\phi +{r^2}\sum^{n-2}_{i=1}\; d\theta_i \otimes d\theta_i.
\hat{\hat{T}\,}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}{}^{\hat{\hat{\rho}}}= -\left(i_{\hat{\hat{k}}_{(n)}}\hat{\hat{C}\,}\right)_{\hat{\hat{\mu}}\hat{\hat{\nu}}}Q^{nm}\hat{\hat{k}}_{(m)}{}^{\hat{\hat{\rho}}}\, .
D_2 \approx \sum_{k=-\infty}^{M-1}2^{-k\frac{q}{p-q}}V_r(x_{k-1},x_k)^{\frac{pq}{p-q}}\sum_{i= k}^{M-1}\int_{x_i}^{x_{i+1}}u(s) \bigg(\int_{s}^{\infty}u\bigg)^{\frac{q}{p-q}}ds.
X^{n'}-1=(X-1)N_{i_1}N_{i_1q}\cdots N_{i_b}N_{i_bq}N_{i_{(b+1)}}N_{i_{(b+1)}q}\cdots N_{i_{u}}N_{i_{u}q}N_{j_1}N_{j_1q}\cdots N_{j_v}N_{j_vq}N_{-j_v}N_{-j_vq}.
\begin{array}{ll}\mbox{minimize}\sum_{t=0}^{N-1}\|u(t)\|_1 \mbox{subject to}x(t+1) = A x(t) + Bu(t), t=0,\ldots,N-1 & 0.2/T^\leq \|u(t)\| \leq 1/T^, t=0,\ldots,N & x(0) = (0, 0), x(N) = (1, 0).\end{array}
-A:D^2 v^{kl}= a\left(-B:D^2 v^{kl}_B\right) +\bar{b}_{kl}\left(- A:D^2 w\right) = a(b_{kl}- \bar{b}_{kl}) + \bar{b}_{kl}(a-\bar{a}) = a_{kl}- \bar{a}_{kl}
\|f\|^2_{2,I-c}= \int_{\mathbb R^n_+}x^{I-2c}f^2(x) \, dx = \sum_{k=1}^\infty a_k \int_0^\infty t^{-1}e^{-t}(R_q f)\bigl(\tfrac{p_k}{t}\bigr) \, dt = 0.
\pi y =\pi(1+x^2)(1+x)(1+x^4) y =\pi(1+x^2)y(1+x^5)(1+x^6) =-\pi\cdot(1+x^2)(1+x)(1+x^5)(1+x^6), &=\pi(-1+x^2+2x^3+2x^4+x^5)
\frac{1}{2}\langle NP^* y , (Id +(M_{ij}))NP^* y \rangle = \frac{N}{2}\langle y , PNP^* y + P(M_{ij}) NP^* y \rangle_Y & = \frac{N}{2}\langle y , (Id + P(M_{ij})NP^* y \rangle_Y.
\left.\begin{array}{ll}\displaystyle(1+2\Lambda)\dfrac{a_1}{4m_2}\int_{\mathbb{R}^N}|\nabla\eta|^2dx+\dfrac{a_1}{2}\omega_2\int_{\mathbb{R}^N}\eta^2dx-\dfrac{1}{2}\left(1+\dfrac{N}{2}\Lambda\right)a_1a_2\int_{\mathbb{R}^N}\xi^2\eta dx=0.\end{array}\right.
\Vert \rho(h_k)(f)\Vert^2 =\sum_{g\in G}^{}\vert \rho(h_k)(f)(g)\vert^2 w(g) =\sum_{g\in G}^{}\vert f(g+h_k)\vert^2 w(g) &=\sum_{n=1}^{+\infty}2^{-n}\sum_{g\in G}^{}\vert f(g+h_k)\vert^2 m^{\ast n}(g)
q(k)=\sum_{n=1}^{\infty}\frac{Sc(n)}{2^{4n-3}}(\frac{1-\sqrt[4]{1-k^{2}}}{1+\sqrt[4]{1-k^{2}}})^{4n-3}=k^{2}\{\frac{1}{2}+[\sum_{n=1}^{\infty}\frac{Sc(n+1)}{2^{4n+1}}k^{2n}]\}^{4}
\frac{\partial S}{\partial t}+ H [ f , \delta S / \delta f , t ] - 4 C P^{- 1 / 2}\sum_{a , b}\int d x \, \left( K_{x}^{a b}\frac{\delta^{2}P^{1 / 2}}{\delta f_{x}^{a}\delta f_{x}^{b}}+ \frac{\delta K_{x}^{a b}}{\delta f_{x}^{a}}\frac{\delta P^{1 / 2}}{\delta f_{x}^{b}}\right) = 0 .
\Biggl [ \cdots \biggl [ \Bigl [ [j_+{}^n, (j_+{}^m)^\dagger ], (j_+{}^{i_{1}})^\dagger \Bigr ], (j_+{}^{i_{2}})^\dagger \biggr ],\cdots (j_+{}^{i_{k+1}})^\dagger \Biggr] \left |{\cal E}_N,2N \right \rangle =0 ,
\sigma_{x}\sigma_{p}\ge\frac{\hbar}{4}\sqrt{3+\frac{1}{2}(\frac{\Omega^{2}}{\omega^{2}}+\frac{\omega^{2}}{\Omega^{2}})-(\frac{1}{2}(\frac{\Omega^{2}}{\omega^{2}}+\frac{\omega^{2}}{\Omega^{2}})-1)}=\frac{\hbar}{2}.
\mathcal{E}_\pi(T_1, \ldots, T_n) = \left(\prod^{p_1}_{q=1}\mathcal{E}_{\pi|_{V_{1,q}}}\left((T_1, \ldots, T_n)|_{V_{1,q}}\right)\right) \cdots\left(\prod^{p_m}_{q=1}\mathcal{E}_{\pi|_{V_{m,q}}}\left((T_1, \ldots, T_n)|_{V_{m,q}}\right)\right).