harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
P(N_t=0) =E\left(e^{-q(X_0|Y_0)\gamma_1}e^{-q(X_{\gamma_1}|Y_{\gamma_1})({\gamma_2}-\gamma_1)}...e^{-q(X_{\gamma_{j_t}}|Y_{\gamma_{j_t}})(t-\gamma_{j_t})}\right)\geq e^{-qt}.
\left| \frac{\partial ( a_{1}^{1}, a_{1}^{2}, a_{1}^{3}, a_{2}^{1}, a_{2}^{2}, a_{2}^{3})}{\partial ( r , \Theta , \Phi , \varphi^{1}, \varphi^{2}, \varphi^{3})}\right| = \frac{1}{8}\: r^{5}\operatorname{s i n}( 4 \Theta ) \; .
\partial_t \|u (\cdot, t)\|^2_{L^2_{\varLambda^q}({\mathbb R}^n)}+ \mu \, \sum_{j=1}^n\|\partial_j u (\cdot, t)\|^2_{L^2_{\varLambda^{q}}({\mathbb R}^n)}= (B_q (w,u)(\cdot, t),u (\cdot, t))^2_{L^2_{\varLambda^{q}}({\mathbb R}^n)}.
\widetilde{G}_n(x, n +z) = \frac{n^{nd}}{(n!)^d}\det((1 + z_j/n - x_i/n)^{n-1}e^{-(n + z_j - x_i)})_{i, j= 1}^d = \frac{n^{nd}}{(n!)^d}\det(e^{(n-1)\log (1+z_j/n - x_i/n)}e^{-(n + z_j - x_i)})_{i, j= 1}^d.
k(M\backslash A) - k(M) - f(M\backslash A) + k(M/A^c) = k((M\backslash A)^*) - k(M^*) - v((M\backslash A)^*) + k((M/A^c)^*) = k(M^*/A^*)-k(M^*) - v(M^*/A^*) + k(M^*\backslash (A^*)^c) \leq 0,
c^{\ast}_{\varepsilon}= \liminf_{|x|\to \infty}\inf_{t\geq T_{\varepsilon}}f(x,t)=\lim_{R\to \infty}\inf_{|x|\geq R}\inf_{t\geq T_{\varepsilon}}f(x,t)\leq \lim_{R\to \infty}\inf_{|x|\geq R, t\geq R}f(x,t)= c^{\ast}.
-\langle \triangle^2 u , |u|^{2p}u \rangle \leqslant ||u||_{3}||u||_{1+s'}||u||_1^{2p}&\leqslant C||u||_{3}^{1+\frac{1+s'}{3}}||u||_0^{\frac{2-s'}{3}}||u||_1^{2p}&\leqslant \delta ||u||_{3}^2+C(\delta)(||u||_0^{\frac{2-s'}{3}}||u||_1^{2p})^{\frac{2-s'}{6}},
\sum_{k=0}^n \binom{a}{k}\binom{b}{n-k}\binom{k}{c}\binom{n-k}{d}H_{b-n+k}={}\binom{a+b-c-d}{n-c-d}\binom{a}{c}\binom{b}{d}(H_{a+b-n}-H_{a+b-c-d}+H_{b-d})
\mathbf{U},\mathbf{V},\mathbf{W}\in\mathbb{C}^{n\times n\times n}\,\mbox{and}\,\left[\mbox{Prod}\left(\mathbf{U},\mathbf{V}^{\top^{2}},\mathbf{W}^{\top}\right)\right]_{i,j,k}=\begin{cases}\begin{array}{cc}1 \mbox{if}0\le i=j=k<n 0 \mbox{otherwise}\end{array}\end{cases},
U_{j}=\frac{1}{\alpha}\left[\frac{a-1}{a-b}B^{a-2}-\frac{b-1}{a-b}B^{b-2}\right]p^{\alpha}_{j}q^{-\alpha}_{j}&V_{j}=\frac{1}{\alpha}\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]
\frac{\partial^{2}\eta}{\partial t^{2}}-gh\frac{\partial^{2}\eta}{\partial x^{2}}-gh\frac{\partial^{2}}{\partial x^{2}}(\frac{3}{2}\frac{\eta^{2}}{h}+\frac{1}{3}h^{2}\frac{\partial^{2}\eta}{\partial x^{2}})=0.
\partial_t \mu_{\varepsilon}= - \nu \Delta \partial_t \varphi_{\varepsilon}+ \beta_{\varepsilon}'(\varphi_{\varepsilon})\partial_t \varphi_{\varepsilon}+ \pi'(\varphi_{\varepsilon})\partial_t \varphi_{\varepsilon}- \gamma \partial_t \chi_{\varepsilon}.
z^r \frac{\partial^r}{\partial z^r}G^{m,n}_{p,q}\left(\begin{array}{c}a_1,\, \cdots,\, a_p b_1,\,\cdots,\, b_q \end{array}\Bigg|\, z \right)= G^{m,n+1}_{p+1,q+1}\left(\begin{array}{c}0,\,a_1,\, \cdots,\, a_p b_1,\,\cdots,\, b_q,\,r \end{array}\Bigg|\, z \right).
\big\langle \alpha_1[\xi_1]\beta_1, \alpha_2[\xi_2]\beta_2\big\rangle \,=\, \big\langle\ell^{-1}(\alpha_2^*\circ\alpha_1)\xi_1 r^{-1}(\beta_2^*\circ\beta_1),\xi_2\big\rangle_{L^2R}.
\mathrm{R e}\, \Pi^{\mathrm{( n o n \mathrm{-}p l a n a r )}}= \frac{2 \pi^{\frac{n + 1}{2}}}{( 2 \pi )^{n}}\left( | \bar{\theta}| \right)^{\frac{3 - n}{2}}\int_{0}^{\infty}d z \, \frac{1}{z^{2}- p^{2}}\, z^{\frac{n - 3}{2}}J_{\frac{n - 3}{2}}( \frac{| \bar{\theta}| z}{2})
S_{\mathrm{C}}= \frac{1}{e^{2}}\int d^{4}x \sqrt{g}\Bigr \{( \nabla^{\mu}T_{\mu \rho}^{-}- i T_{\mu \rho}^{-}V^{\mu}) ( \nabla_{\nu}T_{+}^{\nu \rho}- i V_{\nu}T_{+}^{\nu \rho}) \Bigr \},
\frac{1}{N}\int\sum_{i=1}^{N}\left|\nabla_{i}f(x)\right|dx \le\frac{1}{\sqrt{N}}\int\left|\nabla f(x)\right|dx\le\frac{1}{\sqrt{N}}\left[\int\frac{\left|\nabla f(x)\right|^{2}}{f(x)}dx\,\int f(x)dx\right]^{\frac{1}{2}}.
\varphi(\sigma + it) = \frac{\zeta^*(2-2\sigma - 2it)}{\zeta^*(2\sigma + 2it)}= \frac{\zeta^*(-1+2\sigma + 2it)}{\zeta^*(2\sigma + 2it)}\ll_{\sigma}\frac{\big|\Gamma(\sigma-\frac12+it)|}{\big|\Gamma(\sigma +it)|}\ll_{\sigma}(1 + |t|)^{-1/2}.
\frac{d}{d\xi}\left(e^{-\frac{\xi}{2}}\xi^{\frac{k}{2}}F(\xi)\right)=e^{-\frac{\xi}{2}}\left(-\frac{\xi^{\frac{k}{2}}F(\xi)}{2}+\frac{k\xi^{\frac{k}{2}-1}F(\xi)}{2}\xi^{\frac{k}{2}}\frac{dF}{d\xi}(\xi)\right),
\tau(q,r)=4\lambda^{-2}\left(\frac{\partial^2q}{\partial u\partial v}+\Gamma^\alpha_{\alpha\alpha}\frac{\partial q}{\partial u}\frac{\partial q}{\partial v},\frac{\partial^2r}{\partial v\partial u}+\Gamma^\beta_{\beta\beta}\frac{\partial r}{\partial u}\frac{\partial r}{\partial v}\right),
c_{k}^{\pm}= \pm\frac{1}{2}f(t)\xi (1+(\xi/k-t)^2)^{-1/4}(1+(\xi/(k\pm 1)-t)^2)^{-3/4}, d_{k}^{\pm}= \pm \frac{1}{2}g(t)\xi \frac{k}{k\pm 1}(1+(\xi/k-t)^2)^{1/4}(1+(\xi/(k\pm 1)-t)^2)^{-3/4}.
v_i^{\prime}\cdot v_i^{-1}\cdot T_1 = v_i^{\prime}\cdot v_i^{-1}\cdot (1 - v - vw) = v_i^{\prime}\cdot v_i^{-1}\cdot (1 - v_iv_mv_f - v_iv_mv_f w ) = &= v_i^{\prime}\cdot v_i^{-1}- v_i^{\prime}v_mv_f - v_i^{\prime}v_mv_f w.
( 1 . g ) = - \ \frac{i}{6 ( 2 \pi )^{4}}\mathrm{t r}\int d^{d}k \ \frac{\stackrel{\displaystyle ( S k )}{\scriptscriptstyle ( - p ) \phantom{+}}\stackrel{\displaystyle ( S ( k + q ) )}{\scriptscriptstyle ( - q ) \phantom{+ q q q q q q q}}\stackrel{\displaystyle ( S ( k - p ) )}{\scriptscriptstyle ( p + q ) \phantom{+ q q q q q q q}}}{k^{2 l}( k - p )^{2 l}( k + q )^{2 l}}
R(X,Y)Z =\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z \mathrm{Ric}(X,Y) =\mathrm{tr}(Z\longmapsto R(Z,X)Y) R =\mathrm{tr}(Z\longmapsto \mathrm{Ric}(Z)),
B(\alpha) := \frac{1}{32\pi^3\alpha^2}\int_0^\infty\! d\tau\frac{\alpha\sin(\pi/\alpha)\,[\, \cos(\pi/\alpha) -\cosh(\tau)+\tau\sinh(\tau)\, ] - \pi\, [\, \cos(\pi/\alpha)\cosh(\tau)-1\, ]}{[\,\cosh(\tau)-\cos(\pi/\alpha)\,]^2\cosh^4(\alpha\tau/2)}.
X = \lim_{\longleftarrow}\{\nu^{n+1}_n: G/G_{n+1}\to G/G_n \mid n\geq 0\}\phantom{--------}= \left\{(g_0 G_0 ,g_1 G_1,\ldots) \in \prod_{n \geq 0}G/G_n \mid \nu^{n+1}_n (g_{n+1}G_{n+1}) = g_nG_n \right\}
\tilde{F}^i(x)=\lambda^{\frac{1}{2}-\alpha}F^i(x_0+\lambda x), \tilde{g}_1(x)=\lambda^{\frac{3}{2}-\alpha}g_1(x_0+\lambda x), \tilde{g}_2(x)=\lambda^{\frac{3}{2}-\alpha}g_2(x_0+\lambda x).
\int_{- \pi / 2}^{+ \pi / 2}~ \frac{d \xi^{1}}{\pi}~ O_{M}^{c l}( \{\sigma_{+}+ \xi^{1}\}; \{\sigma_{-}- \xi^{1}\}) \equiv \int_{- \pi / 2}^{+ \pi / 2}~ \frac{d \xi^{1}}{\pi}~ O_{M}^{c l}( \{\sigma^{0}\}; \{\sigma^{1}+ \xi^{1}\})
0 . 7 \times 0 . 4 \times 0 . 5 = 0 . 1 4 ( m^{3}) = 1 4 0 ( d m^{3}) = 1 4 0 ( d m^{3}) = 1 4 ( m^{3}) = 1 4 0 ( d m^{3}) = 1 4 0 c
[\Delta^{2},i\Gamma_{\varphi_{R}}]=8\partial^{2}_{kl}(\partial^{2}_{lm}\varphi_{R})\partial^{2}_{mk}+4\partial_{k}(\partial^{2}_{kl}\Delta\varphi_{R})\partial_{l}+2\partial_{k}(\Delta^{2}\varphi_{R})\partial_{k}+\Delta^{3}\varphi_{R}.
\Big(\frac{\partial^2 H^c}{\partial p_n\partial q_l}\,p_l-\frac{\partial^2 H^c}{\partial p_n\partial p_l}\,q_l\Big)\Big|_{z^\bullet}=0\quad\Big(\frac{\partial^2 H^c}{\partial q_n\partial q_l}\,p_l-\frac{\partial^2 H^c}{\partial q_n\partial p_l}\,q_l\Big)\Big|_{z^\bullet}=0.
f_{ij}(x) =f(\cdots,\,x_{i-1},\,x_{i}+0,\,x_{i+1},\cdots,\,x_{j-1},\,x_{j}-0,\,x_{j+1},\,\cdots) D_{ij}f(x) =(\nabla_{i}-\nabla_{j})f(\cdots,\,x_{i-1}\,x_{i}+0,\,x_{i+1},\cdots,\,x_{j-1},\,x_{j}-0,\,x_{j+1},\,\cdots)
\begin{array}{l}2n-2(n_1 + \cdots + n_j )+\beta^{(k)}_{ij}\geq a^{(j)}_1 \geq \cdots \geq a^{(j)}_{n_j -n_{j+1}}\geq 0, ~~~~1 \leq j \leq k-2, n-(n_1 + \cdots +n_{k-1}) \geq b_1 \geq \cdots \geq b_{n_{k-1}}\geq 0.\end{array}
\mathrm{d}s_{*}^{2}= r_{0}^{2}\left[ \mathcal{R}^{2}\left\{u^{2}\mathrm{d}x_{d}^{2}+ \left({\frac{\mathrm{d}u}{u}}\right)^{2}\right\}+ \mathrm{d}\Omega_{\tilde{d}+ 1}^{2}\right] \, .
\#\left\{i:\frac{d_{i+1,k}}{d_{i,k}}>\exp\left[{-\frac{1}{2\psi^{-1}(k)}}\right]\right\}>\frac{\log e^{-1/2}}{\log \exp(-k/(2\psi^{-1}(k))}=\frac{1}{k}\psi^{-1}(k).
K_1=\left(\begin{array}{llllc}1 0 \dots 0 -\frac{f_1}{f_\ell}0 1 \dots 0 -\frac{f_2}{f_\ell}\vdots \dots \vdots 0 0 \dots 1 -\frac{f_{\ell-1}}{f_\ell}0 0 \dots 0 -\partial+\frac{f_\ell^\prime}{f_\ell}\end{array}\right)\,.
J_{1}=\int_{F_{ji}}f_{ji}(x)g_{\epsilon}(x_{i}-x_{j})dS_{ji}(x)-\int_{F_{ij}}f_{ij}(x)g_{\epsilon}(x_{i}-x_{j})dS_{ij}(x) J_{2}=\sum_{k:k\neq i,\,j}\left[\int_{F_{jk}}f_{jk}(x)g_{\epsilon}(x_{i}-x_{j})dS_{jk}(x)-\int_{F_{kj}}f_{kj}(x)g_{\epsilon}(x_{i}-x_{j})dS_{kj}(x)\right]
\left\{\; \int_{\cal R}d^{n}p^{\prime}\, ( \tilde{\omega}_{p^{\prime}}| e_{p^{\prime}p}| \eta_{p}) \; , \; \int_{\cal R^{\prime}}d^{n}q^{\prime}\, ( \tilde{\omega}_{q^{\prime}}| f_{q^{\prime}q}| \eta_{q}) \; \right\}=
\mathcal{L}_k(\alpha+1) - \mathcal{L}_k(\alpha) = \tau_k(p^{\alpha+1}) \tau_{\ell}(p^{\alpha+1})X^{\alpha+1}+ \sum_{i=\alpha+2}^{\infty}(\tau_{k-1}(p^{\alpha+1}) \tau_{\ell-1}(p^i) + \tau_{\ell-1}(p^{\alpha+1}) \tau_{k-1}(p^i)) X^i.
\cal F^{-1}\Phi_0\cal F(\theta_{N,+}^{(s_0)}\theta_{N,-}^{(s_1)})= \cal F^{-1}\Phi_0\cal F\big(\sum_{k,l=1}^{N}2^{-ks_0-ls_1}e^{(i(2^{k}-2^l)x_{j_0})}\theta^2\big) =N\theta^2,
u^{ij}Q_{ij}={}- \frac{u^{ij}w_{ij}}{w^2}+ 2 \frac{u^{ij}w_i w_j}{w^3}+ Mn \ge{}-d^{1-2\theta}f + Mn \ge - \left( d^{1-2\theta}f - Mn\right)^+.
V = \left( \frac{b}{6}\right)^{2}\frac{| z_{t r}|^{4}}{( 1 - | z_{t r}|^{2})^{2}}\left( C_{2}| z_{t r}|^{2}+ a_{1}\Delta z + a_{2}\Delta z^{2}+ a_{3}\Delta z^{3}+ \ldots \right)
\Delta u_i+\sum\limits_{j\in J_0}k_{ij}^{\prime}e^{u_j}=4\pi\sum\limits_{\ell=1}^{N}\alpha_{\ell,i}\delta_{p_{\ell}}\ \ \mathbb{R}^2, \ \int_{\mathbb{R}^2}e^{u_i(y)}\mathrm{d}y <+\infty, \ \ \forall \i\in J_0,
\begin{pmatrix}b_s, ..., b_r\end{pmatrix}=\begin{pmatrix}a_0, 0, ..., a_{p-1},0\end{pmatrix}Y^*\tilde{N}_p(VV^*)^{-1}VN_{r-s+1}=:\begin{pmatrix}a_0, 0, ..., a_{p-1},0\end{pmatrix}Y^*\tilde{N}_pV^{\dagger}N_{r-s+1},
\frac{p_k}{q_k}:=\sum_{i=1}^{2n_{1}+\cdots +2n_{k}+k}\epsilon_i\Big(\frac{p}{q}\Big)^{i}+\Big(\frac{p}{q}\Big)^{2n_{1}+\cdots +2n_{k}+k+1}\sum_{i=0}^{\infty}(p/q)^{2i},
\begin{array}{c}{{6 \beta^{\prime}v^{2}v^{\prime}+ 2 \beta^{\prime}( D - 1 ) v^{3}+ 2 \beta^{\prime}v^{\prime}= \alpha^{\prime}}}\\{{v^{\prime}= I_{1}( \alpha^{\prime}) / I_{0}( \alpha^{\prime})}}\\ \end{array}
\Omega_R/dR \cong \mathbb{C}\omega_0 \oplus \bigoplus_{i=3}^{4}U_i^{\oplus \Upsilon_i(\epsilon_i, \nu_i)}\oplus \bigoplus_{j=1}^{k-1}V_j^{\oplus \frac{(1-(-1)^j)n}{k}},
\nu_{q ,{\bf n}}= ( q + n + 1 ) \left( \begin{matrix}{{q + n - 1}}\\{{n - 1}}\\ \end{matrix}\right) ={\frac{( q + n + 1 ) ( q + 1 ) \cdots \bigl ( q + ( n - 1 ) \bigr )}{( n - 1 ) !}}.
|\nabla_x \eta(t,s)| =|(\nabla_\xi p(t,s;x,\eta(t,s)))^{-1}\nabla_x p(t,s;x,\eta(t,s))| &\leq C|\nabla_x p(t,s;x,\eta(t,s))| &\leq C\rho^{\varepsilon_0}\langle s\rangle^{-1-\varepsilon_1},
\begin{aligned}Cov(\bar{X},\bar{Y}) =Cov(\frac{1}{|S|}\sum_{j\in S_k}X_j,\frac{1}{|S|}\sum_{j\in S_k}Y_j) &=\frac{1}{|S|^2}Cov(\sum_{j\in S_k}X_j,\sum_{j\in S_k}Y_j) &=\frac{1}{|S|^2}\sum_{j\in S_k}Cov(X_j,Y_j) &\approx \frac{1}{|S|}s_{XY}\end{aligned}
\frac{1}{a_{i+1}}< \theta - \sum_{j=1}^i \frac{1}{a_j}= \left( \theta - \sum_{j=1}^{i-1}\frac{1}{a_j}\right) - \frac{1}{a_{i}}& \leq \frac{1}{a_{i}-1}- \frac{1}{a_{i}}= \frac{1}{a_{i}(a_{i}-1)}
\inf_{(u(\cdot),y_{\epsilon}(\cdot),z_{\epsilon}(\cdot))}\liminf_{\mathcal{T}\rightarrow\infty}\frac{1}{\mathcal{T}}\int_0^{\mathcal{T}}(0.1u_1^2(t)+ 0.1u_2^2(t) -z_1^2(t))dt = V^*(\epsilon),
Q_2(t)=b_{0,\sup}(t)+\frac{\chi_2}{2d_3}\big(k{\bar r_1}+l{\bar r_2}\big)+\frac{l^2}{4\lambda d_3}\Big(\frac{\chi_1^2{\bar r_1}^2}{d_1}+\frac{\chi_2^2{\bar r_2}^2}{d_2}\Big)+\frac{a_{2,\sup}(t){\bar r_1}+b_{1,\sup}(t){\bar r_2}}{2}.
z^{\mu}J_{\nu-\mu}^{(3)}(2z;q)=\sum_{m=0}^{\infty}\frac{\left(q^{\mu};q\right)_{m}}{\left(q;q\right)_{m}}J_{\nu+m}^{(3)}(2zq^{m/2};q)q^{\binom{m+1}{2}}\left(-\frac{q^{\nu-\mu}}{z}\right)^{m}
\vec{\Gamma}_{\nu}\partial_{\mu}\vec{J}^{\mu}= - \vec{\Gamma}_{\nu}\vec{q}\frac{d \Lambda}{d q_{0}}= - \partial_{\nu}\alpha \sin \alpha \frac{d \Lambda}{d q_{0}}= \partial_{\nu}q_{0}\frac{d \Lambda}{d q_{0}}= \partial_{\nu}\Lambda .
\widetilde{g^{\varepsilon}}(x,c,t):=\begin{cases}g^{\varepsilon}(x,c,t)\,\,\mbox{if}\,\,(x,c,t)\in\Omega\times\left(-\|u_{0}\|_{L^{\infty}\left(\Omega\right)},\|u_{0}\|_{L^{\infty}\left(\Omega\right)}\right)\times(0,T), 0\,\,\,\,\mbox{Otherwise}.\end{cases}
E^{\underline{{{a}}}}= ( E^{a}, E^{i}) = d X^{\underline{{{m}}}}u_{\underline{{{m}}}}^{\underline{{{a}}}}= ( d X^{\underline{{{m}}}}u_{\underline{{{m}}}}^{a}, d X^{\underline{{{m}}}}u_{\underline{{{m}}}}^{i}) \qquad{\underline{{{a}}}= 0 , \ldots , ( D - 1 ) ; ~ ~ a = 0 , \ldots , p}
B_{mn}(x):=(-1)^{m+n}\left( \sum_{k=1}^{N}\alpha_{k}^{-}e^{2\tau_{k}x}\frac{\left( \frac{1}{2}-\tau_{k}\right)^{m+n}}{\left( \frac{1}{2}+\tau_{k}\right)^{m+n+2}}+\frac{1}{2\pi}\int_{-\infty}^{\infty}s^{-}\left(\rho\right) e^{-2i\rho x}\frac{\left( \frac{1}{2}+i\rho\right)^{m+n}}{\left( \frac{1}{2}-i\rho\right)^{m+n+2}}d\rho\right) ,
m(\varprojlim E^{<n_k}) = \sum_{(\lambda,\nu) \in E(n_k)^*}e^{-\beta|\lambda|}\varepsilon(Z(\nu, k)) = \sum_{\nu \in E^{<n_k}}\sum_{\lambda \in E^* r(\nu)}e^{-\beta|\lambda|}\varepsilon(Z(\nu, k)) = \sum_{\nu \in E^{<n_k}}\int_{x \in Z(\nu, k)}y(x)\,d\varepsilon(x) = \int_{x \in \varprojlim E^{<n_k}}y(x)\,d\varepsilon(x).
\Lambda'(a-b)(\gamma(a)-\gamma(b)) =(a-b)^{p-1}(\gamma(a)-\gamma(b)) &= (a-b)^{p-1}\int_{b}^{a}\gamma'(t) dt &= (a-b)^{p-1}\int_{b}^{a}(\Gamma'(t))^{p}dt &\geq \left(\int_{b}^{a}(\Gamma'(t)) dt\right)^{p}=( \Gamma(a)-\Gamma(b))^{p}.
\widetilde{C}_{\mu \nu A}{}^{B}=2\partial_{\lbrack \mu}\widetilde{\vartheta}_{\nu ]A}{}^{B}-(\widetilde{\vartheta}_{\mu A}{}^{C}\widetilde{\vartheta}_{\nu C}{}^{B}-\widetilde{\vartheta}_{\nu A}{}^{C}\widetilde{\vartheta}_{\mu C}{}^{B}),
\int d{\bf k}\frac{|{\bf k}|}{( k_{0}^{\mathrm{\scriptsize ~ e f f}})}\Psi ({\bf k},{\bf q}; \omega ) = \frac{\alpha}{\left\{1 - \left( g^{2}/ 4 \pi \right) ( \xi + 1 ) \int \frac{d{\bf k}^{\prime}}{( k_{0}^{\mathrm{\scriptsize ~ e f f}})^{\prime}}\frac{({\bf k}^{\prime})^{2}}{[ \{( k_{0}^{\mathrm{\scriptsize ~ e f f}})^{\prime}\}^{2}- \omega^{2}/ 4 + i \epsilon ]}\right\}}
\begin{array}{lll}X_{37_{1}}=\gamma_{\omega_{2}}^{\frac{1}{2n}}, X_{37_{2}}=\gamma_{\omega_{1}}^{-\frac{1}{2n}}\gamma_{\omega_{2}}^{-\frac{1}{2n}}, X_{37_{3}}=\gamma_{\omega_{1}}^{\frac{1}{2n}}. \end{array}
\mathcal{F}und :=\left\{\coprod_{(r_{1},\ldots,r_{s})\in\Lambda}^{\cdot}\Phi(r_{1},\ldots,r_{s})\right\}& =\left\{(r_{1},\ldots,r_{s},b_{1},\ldots,b_{s+1})\left\vert\begin{array}[c]{l}r_{1},\ldots,r_{s}\in\Lambda b_{1},\ldots,b_{s+1}\in\Phi(r_{1},\ldots,r_{s})\end{array}\right. \right\}
\sigma_1 = \frac{1}{4}+ \frac{3}{2}\cdot \frac{1}{p+1}\in \left(\frac{1}{4}, \frac{1}{2}\right]; \left(\frac{1}{p_2}, \frac{1}{q_2}\right) = \left(\frac{1}{4}, \frac{1}{p+1}\right).
N_{D_5}\big(D^2\big)+N_{D_5}\big(5D^2 \big)=N_{F_5}\big((-1)^0 5^3 |D|^2\big)+ N_{F_5}\big((-1)^0 5^5 |D|^2\big)+N_{F_5}\big((-1)^0 5^7 |D|^2\big)\;.
\frac{\phi_{i}(\lambda_{1}y,\lambda_{2}y^{2})}{y}=\sqrt{\frac{m_{i,i+1}\lambda_{2}}{b_{i}}}\frac{b_{i}\lambda_{1}+\sqrt{b_{i}m_{i,i+1}\lambda_{2}}\tanh (y\sqrt{b_{i}m_{i,i+1}\lambda_{2}})}{b_{i}\lambda_{1}\tanh (y\sqrt{b_{i}m_{i,i+1}\lambda_{2}})+\sqrt{b_{i}m_{i,i+1}\lambda_{2}}}.
\sum_{r \ge 1}|\lambda_f(r)|^2 \cdot g \Big ( \frac{Y(k-1)}{4\pi r}\Big )= \frac{Y(k-1)}{4\pi}\cdot \frac{6}{\pi^2}G(-1) \cdot L(1,^2 f) + O \Big ( \frac{\sqrt{Y}\cdot k}{(\log k)^{1 - \varepsilon}}\Big ).
E_N^{(1)}\le\; 4\pi \mathfrak{a}(N-1)+e_\Lambda \mathfrak{a}^2 &-\frac{1}{2}\sum_{p\in \Lambda_+^*}\left[ p^2+8\pi\mathfrak{a}-\sqrt{|p|^4+ 16\pi \mathfrak{a}p^2}-\frac{(8\pi \mathfrak{a})^2}{2p^2}\right]+ CN^{-1/4},
i\theta_{i}\epsilon^{ijk}L_{jk}=(\begin{matrix}0&\theta_{3}&-\theta_{2}&0&0\\ -\theta_{3}&0&\theta_{1}&0&0\\ \theta_{2}&-\theta_{1}&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \end{matrix}).
\tilde{\phi}_{0}(\textbf{x}) = I \left\lbrace \textbf{x}\in \mathcal{C}^{'}_{1}\cap \mathcal{C}^{'}_{0}, (1-q(\textbf{x},1))P\{Y=1\vert \textbf{X}= \textbf{x}\}> (1-q(\textbf{x},0)) P\{Y=0 \vert \textbf{X}= \textbf{x}\}\right\rbrace &+ I \left\lbrace \textbf{x}\in \mathcal{C}^{'}_{1}- \mathcal{C}^{'}_{0}, q(\textbf{x},1) < 1\right\rbrace.
\left( \partial_{\bar{x}}^{3}+\frac{( g^{\dagger}_{0}g^{\dagger}_{1}+ g^{\dagger}_{0}g^{\dagger}_{2}+ g^{\dagger}_{1}g^{\dagger}_{2}-2 )}{\bar{x}^{2}}\, \partial_{\bar{x}}-\frac{g^{\dagger}_{0}g^{\dagger}_{1}g^{\dagger}_{2}}{\bar{x}^{3}}+p(\bar{x}, \bar{E}) \right) \, \bar{\psi}=0\,,
I_2 \ge t_0 (t-s)^{2H}\int_{|y|\ge 1}\frac{1-\cos(|y|)}{|y|^{2H+1}}dy-\frac{(t-s)^{2H+1}}2 \int_{|y|\ge 1}\frac1{|y|^{2H+1}}dy &-\frac{3(t-s)^{2H+1}}4\int_{|y|\ge 1}\frac1{|y|^{2+2H}}dy &=C_1(t-s)^{2H}-\frac{5H+1}{2H(2H+1)}(t-s)^{2H+1},
V^2(r) \equiv{\Delta (r) \over R^2 (r)}\left\{\frac{\lambda}{R^{2}(r)}+ \frac{1}{R(r)}\frac{\mbox{d}}{\mbox{d}r}\left[ \frac{\Delta (r)}{R^2(r)}\frac{\mbox{d}R(r)}{\mbox{d}r}\right]-{m^2 a^2 \over R^6 (r)}(r^2 + a^2 - Q^2 + 2 M r)\right\},
\big\Vert g_L\big\Vert_{L^p}\lesssim \Big\Vert \Big(\sum_{n=M}^{L}{b_n^22^{-2t_nd(1-1/p)}|\phi_{t_n}|^2}\Big)^{1/2}\Big\Vert_{L^p}&\leq \Big( \sum_{n=M}^{L}{b_n^p 2^{-{t_n}d(p-1)}\big\Vert \phi_{{t_n}}\big\Vert_{L^p}^p}\Big)^{{1}/{p}}\approx\Big(\sum_{n=M}^{L}{b_n^p}\Big)^{{1}/{p}}.
I_{n,B}=\frac1{2(4\pi^2)^2}\int_0^\Lambda\, |\varphi_k\dot\varphi_{k'}-\dot\varphi_k\varphi_{k'}|^2 Q_n(k^2,k'^2)k^2k'^2 \coth\frac{\beta W_k}2 \coth\frac{\beta W_{k'}}2\, dkdk'\ .
\begin{array}{cc}P_x = \Pi_3+J_3 \qquad K_x = \frac{\Pi_3-J_3}{2}\null \null P_t = -\Pi_0+N_2 \qquad K_t = -\frac{\Pi_0+N_2}{2}\null \null P_w= \Pi_1-J_1 \qquad K_w = \frac{\Pi_1+J_1}{2}\end{array}
\rho_j(S) \lambda_i(T)b &= \rho_j(S) \left( E_{i}(T)b \oplus (T-L_{E_i(T)})b \right) &= E_{i}(T)b E_j(S) \oplus (S-R_{E_j(S)})E_{i}(T)b \oplus R_{E_j(S)}(T-L_{E_i(T)})b \oplus \left([(T-L_{E_{i}(T)})b] \otimes [(S-R_{E_j(S)}) 1_B] \right).
\frac{1}{\beta(n)}=\frac{1}{\beta((l_n-1)d+\overline{n})}+\frac{1}{\beta(d+\overline{n})}-\frac{1}{\beta(\overline{n})}\cr =\frac{1}{\beta((l_n-2)d+\overline{n})}+\frac{2}{\beta(d+\overline{n})}-\frac{2}{\beta(\overline{n})}\cr =\cdots\ \cdots\cr =\frac{l_n}{\beta(d+\overline{n})}-\frac{l_n-1}{\beta(\overline{n})}.
\varphi(x)=\pm\sqrt{\frac{2\mu^{4}}{\mu_{0}^{2}+\sqrt{\mu_{0}^{4}+2\lambda\mu^{4}}}}sn(p\cdot x+\theta,\sqrt{\frac{-\mu_{0}^{2}+\sqrt{\mu_{0}^{4}+2\lambda\mu^{4}}}{-\mu_{0}^{2}-\sqrt{\mu_{0}^{4}+2\lambda\mu^{4}}}})
A^*(Z)= \pi^k c_k \sum_{r,s}\sum_{D_1,D_2,t_1,t_2}\sum_{d_1|c(T_1)}\sum_{d_2|c(T_2)}(d_1d_2)^{k-1}S(D_1d_1^{-2},D_2d_2^{-2})e(\cdots) =\pi^k c_k \sum_{T_1,T_2>0}\sum_{d_1|c(T_1)}\sum_{d_2|c(T_2)}(d_1d_2)^{k-1}S(D_1d_1^{-2},D_2d_2^{-2}) e(Tr(T_1Z+T_2Z)).
\frac{\delta \Gamma}{{\delta}A_{\mu}}= \langle \frac{{\delta}S}{\delta \hat{A}_{\mu}}\rangle , \qquad \frac{\delta \Gamma}{\delta{\bar{\psi}}^{\alpha}}= \langle \frac{{\delta}S}{\delta \hat{\bar{\psi}}^{\alpha}}\rangle , \qquad \frac{\delta \Gamma}{\delta{\psi}^{\alpha}}= \langle \frac{{\delta}S}{\delta \hat{\psi}^{\alpha}}\rangle .
\big\|\big(\frac1\tau \sum_{j=0}^k \delta_j v_{n-j}\big)_{n=k}^N\big\|_{L^p(D)}\le C\big (\|(f_n)_{n=k}^N\|_{L^p(D)}+ \|(v_n)_{n=k}^N\|_{L^p(D)}\big ) .
\frac{m(t,k,\eta,l)}{m(t,k,\eta',l')}= \frac{k^2 + l^2}{k^2 + (l')^2}\frac{k^2 + (1000 \nu^{-1/3}k)^2 + (l')^2}{k^2 + (1000 \nu^{-1/3}k)^2 + l^2}\lesssim \frac{1 + L^2}{1+ (L')^2}\frac{\nu^{-2/3}+ (L')^2}{\nu^{-2/3}+ L^2},
\frac{d}{dt}g_{ij}= X \cdot g(e_i, e_j) = g(\bar \nabla_X e_i,e_j) + g(e_i, \bar \nabla_X e_j) = g(\bar \nabla_{e_i}X, e_j) + g(e_i, \bar \nabla_{e_j}X) &= -2 g(h(e_i, e_j), X) = -2g(A_X e_i, e_j),
\mathrm{d}\mathcal{E}_t = \frac{\partial \mathcal{E}_t}{\partial t}\mathrm{d}t + \left\langle\frac{\partial \mathcal{E}_t}{\partial X_t}, \mathrm{d}X_t \right\rangle + \left\langle \frac{\partial \mathcal{E}_t}{\partial Z_t}, \mathrm{d}Z_t \right\rangle + \frac{\dot \beta_t^2 e^{\beta_t}}{2 \mu}\left( \sigma(X_t, t)^\top \sigma(X_t, t) \right) \mathrm{d}t.
\begin{aligned}\int_{0}^{t}\left[\ddot{u}_{i}(t+\tau)u_{i}(t-\tau)-\ddot{u}_{i}(t-\tau)u_{i}(t+\tau)\right]d\tau= \qquad=\dot{u}_{i}(2t)u_{i}^{0}+u_{i}(2t)v_{i}^{0}-2u_{i}(t)\dot{u}_{i}(t),\end{aligned}
\hat{m}= \left( \begin{array}{c c c}{l_{1}}&{l_{2}}&{l_{3}}\\{m_{1 1}}&{m_{1 2}}&{m_{1 3}}\\{m_{1 2}}&{m_{2 2}}&{m_{2 3}}\\{m_{1 3}}&{m_{2 3}}&{m_{3 3}}\\ \end{array}\right)
\exp [\pm \sum_{n=1}^{\infty}\sum_{j=1}^{M}{\alpha_{-n}\over \sqrt{\beta}}{(y_j)^n \over n}]= \sum_{\{\lambda \}}(-1)^{|\lambda|}J_{\{\lambda'\}}^{(1/\beta)}(y_j)J_{\{\lambda \}}^{(\beta)}(\{\mp{\alpha_{-n}\over \sqrt{\beta}}\}).
\begin{array}{c}\left\{Q_\alpha ,Q_\beta \right\}=\left( S_A\right)_{\alpha \beta}\left( S_A\right)_{\alpha \beta}=\left( C\gamma^m\right)_{\alpha \beta}\,\,P_m+\left( C\gamma^{m_1m_2}\right)_{\alpha \beta}\,\,\,Z_{m_1m_2}+\left( C\gamma^{m_1\cdots m_5}\right)_{\alpha \beta}\,\,\,X_{m_1\cdots m_5}.\end{array}
|u|_\vartheta\leq \|u\|_{\sigma,\alpha}\int_{\Gamma_0}\exp\left((\alpha + \vartheta) |\eta|\right) e(\phi_\sigma, \eta)\lambda(d\eta) = \|u\|_{\sigma,\alpha}\exp\left( (\alpha + \vartheta)\langle \phi \rangle\right).
u(t,x) =\Pi_{x}\left[e_{a}(t)h(\xi_{t})f(\xi_{t})\right]+\Pi_{x}\left[\int_{0}^{t}e_{a}(s)\gamma(\xi_{s},u^{t-s}){\rm d}s\right] &=\Pi_{x}\left[h(\xi_{t})f(\xi_{t})\right]-\Pi_{x}\left[\int_{0}^{t}a(\xi_{s})u(t-s,\xi_{s}){\rm d}s\right]+\Pi_{x}\left[\int_{0}^{t}\gamma(\xi_{s},u^{t-s}){\rm d}s\right].
\Bigg\|Ue_i - \sum_{k=1}^{\lceil \theta^{-1}\rceil - 1}Ue_{j_k}\Bigg\| = \bigg\| \varepsilon_i e_{\lceil \theta^{-1}\rceil}- \sum_{k=1}^{\lceil \theta^{-1}\rceil - 1}\varepsilon_{j_k}e_{j_k}\bigg\| = \theta \cdot \lceil \theta^{-1}\rceil > 1.
+ \frac{1}{2}h^{- 5 / 3}v^{\hat{I}}\partial_{\hat{J}}h \bar{\theta}\Gamma_{~ \tilde{I}}^{0 ~ \tilde{J}}\theta + h^{- 1 / 6}v^{\hat{I}}\bar{\theta}( \Gamma_{\tilde{I}\tilde{1}}\partial_{2}- \Gamma_{\tilde{I}\tilde{2}}\partial_{1}) \theta + \cdots \Big ] d^{3}\zeta
\sum_{i=1}^n \mathbb{P}(|X_i|>\varepsilon n^{1/p}) \ge \sum_{i=\lfloor n/2 \rfloor}^n \mathbb{P}(|X_i|>\varepsilon n^{1/p}) &\ge \sum_{i=\lfloor n/2 \rfloor}^n \frac{1}{n}\ge \frac{1}{2}.
g_{\mathrm{m j}}\ddot{x}^{\mathrm{j}}+ \biggl ({\frac{\partial g_{\mathrm{m l}}}{\partial x^{\mathrm{k}}}}-{\frac{1}{2}}{\frac{\partial g_{\mathrm{k l}}}{\partial x^{\mathrm{m}}}}\biggr ) \dot{x}^{\mathrm{k}}\dot{x}^{\mathrm{l}}= -{\frac{\partial V}{\partial x^{\mathrm{m}}}}
\begin{array}{r l}{{\tan \left( \frac{\pi}{4}+ \frac{\Theta}{2}\right) =}}&{{( - 1 )^{n}\left( \frac{E + m}{E - m}\right)^{1 / 2}\left( \frac{k}{2 m}\right)^{2 \nu + 1}}}\\{{}}&{{\times \frac{\Gamma ( - \nu )}{\Gamma ( \nu + 1 )}\tan \mu .}}\\ \end{array}
\begin{array}{lcl}\phi_0=\{T,u\}_{a,E}\qquad \psi_0=1 \phi_1=\{T,M\}_{a,E}u_x \qquad \psi_1=\int_0^x\phi_2(s)ds \phi_2=\{u,T,M\}_{a,E,c}\qquad \psi_2=\{T,M\}_{a,E}+\{T,M\}_{a,E}u\end{array}
\left| \frac{\partial(a_1^1, a_1^2, a_1^3, a_2^1, a_2^2, a_2^3)}{\partial (r,\Theta,\Phi,\varphi^1,\varphi^2,\varphi^3)}\right| = \frac{1}{8}\: r^5 \sin(4\Theta) \;.
\mathcal{C}_{q}=3^{-\frac{7}{4}}q^{\frac{3}{q}}\left(q-2\right)^{\frac{3}{2q}}\left(\frac{q}{q-2}\right)^{\frac{3}{4}}(4\pi(1-\tau))^{-\frac{3}{2}(\frac{1}{3}-\frac{1}{q})}e^{-6(\frac{1}{3}-\frac{1}{q})}.
\int_{B_{R}}\left\langle a(x) \left\lvert du \right\rvert^{p-2}du; d\phi \right\rangle = \int_{B_{R}}\left\langle f; \phi \right\rangle \phi \in W_{\delta, T}^{1,p}\left( B_{R}; \Lambda^{k}\mathbb{R}^{n}\otimes \mathbb{R}^{N}\right).
dX^{\epsilon,\gamma}(t) = \Bigl (A + \sum\nolimits_{i=1}^n B_i \gamma_i\Bigr)X^{\epsilon,\gamma}(t) dt + \sqrt{\epsilon}\sigma\bigl(X^{\epsilon,\gamma}(t)\bigr) dW(t), \,\, X^{\epsilon,\gamma}(0)=x_0,
{{\mathrm{\boldmath ~ \s i g m a ~}}\cdot{\bf p}}{{\mathrm{\boldmath ~ \s i g m a ~}}\cdot{\bf n}}{\frac{f ( r )}{r}}{\phi^{l}}_{j m}= -{\frac{i}{r}}({\frac{d f}{d r}}-{\frac{\kappa f}{r}}){\phi^{l}}_{j m},
\begin{array}{l}{{n_{+}= ( \bar{n}_{+}\otimes 1 ) \oplus ( \bar{{\cal G}}\otimes \lambda{\bf C}[ \lambda ] ) ,}}\\{{n_{-}= ( \bar{n}_{-}\otimes 1 ) \oplus ( \bar{{\cal G}}\otimes \lambda^{- 1}{\bf C}[ \lambda^{- 1}] ) ,}}\\ \end{array}
e(\vec{H}^\ell[W]) \geq \frac{p^{1/\ell}\cdot\ell\alpha^2}{2}\geq \frac{\ell p^{1/\ell}}{(\ell+1)^2}\cdot \binom{|W|}{2}\geq \frac{p^{1/\ell}}{2(\ell+1)}\cdot \binom{|W|}{2}.
\psi(h_5) =\psi(h_6)=\psi(h_7)=\psi(h_8)=\frac{n+1}{4}e \, n\equiv 3 \pmod{4}\,, \psi(h_5) =\psi(h_6)=\psi(h_7)=\psi(h_8)=\frac{3n+1}{4}e n\equiv 1 \pmod{4}\,.
p\lim_{\theta}\left(\frac{\hat{\nu}^{11}}{\hat{L}^{11}}\right) = \frac{\left\{\textrm{asy. var. of}\sqrt{n}\left(\hat{\theta}^{M}_1\right)\right\}}{\left\{\textrm{asy. var. of}\sqrt{n}\left(\hat{\theta}^{ML}_1\right)\right\}}= \textrm{asy. eff.}\left( \hat{\theta}^M_1, F \right)^{-1}\,.