harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{{\sigma_{n}}^{2}}{n^{3}}\mathbb{E}\left[ \int_{\tau_{n}}^{\tau_{n}+\theta}\sum_{m=1}^{n}(f^{\prime}(\lambda_{m}^{(n)}(X_{s-}^{(n)})))^{2}ds\right] \leq\frac{{(\sigma+1)}^{2}}{n^{2}}\Vert f^{\prime}\Vert_{\infty}^{2}\delta.
\mathcal{A}(\varepsilon)= \left(\begin{array}{l}{\mathcal{A}}_{1}(\varepsilon)\ \ \ \ \ \ \ \ \ \ \ \ \{\mathcal{A}}_{2}(\varepsilon) (1/\varepsilon){\mathcal{A}}_{3}(\varepsilon)\ \ \ \ (1/\varepsilon){\mathcal{A}}_{4}(\varepsilon)\end{array}\right),
\langle z|(a^{\dagger}a)_{k,\lambda}|z\rangle =\sum_{l=0}^{k}S_{2,\lambda}(k,l)\langle z|(a^{\dagger})^{l}a^{l}|z\rangle=\sum_{l=0}^{k}S_{2,\lambda}(k,l)(\overline{z})^{l}z^{l}\langle z|z\rangle &=\sum_{l=0}^{k}S_{2,\lambda}(k,l)|z|^{2l}=\phi_{k,\lambda}(|z|^{2}) .
\begin{aligned}\inf_{\varphi\in C^{\infty}_{cs}((0,\infty))}\frac{\int_{0}^{\infty}(\varphi')^2\sinh^ndr}{\int_{0}^{\infty}\varphi^2\sinh^{n-2}dr}=\inf_{\varphi\in C^{\infty}_{cs}((0,\infty))}\frac{\int_{0}^{\infty}(\varphi')^2\sinh^ndr}{\int_{0}^{\infty}\varphi^2\cosh^2\sinh^{n-2}dr}=\frac{(n-1)^2}{4}.\end{aligned}
u_1' = 0 \Leftrightarrow \frac{v_1v_0}{1-v_1}+ (2+r)v_0 +v_1 -\frac{r+2}{r+1}=0 & u_2' = 0 \Leftrightarrow (r+2)v_0 + v_1 - \frac{r+2}{r+1}+ \frac{v_0v_1}{1-v_1}& u_3'= 0 \Leftrightarrow (1-v_1)^2 +(1-v_1)(\frac{1}{r+1}-(r+1)v_0) -v_0 = 0
y_{\mathfrak{r}_1, \dots, \mathfrak{r}_k}=\bigg( \prod_{i=1}^k \mu(\mathfrak{r}_i) \varphi(\mathfrak{r}_i)\bigg)\sum_{\substack{\mathfrak{a}_1, \dots, \mathfrak{a}_k \mathfrak{r}_i \mid \mathfrak{a}_i\,\forall i}}\frac{\lambda_{\mathfrak{a}_1, \dots, \mathfrak{a}_k}}{\prod_{i=1}^k |\mathfrak{a}_i|}
\left|\underset{\Gamma_{2p-1}}{\int\sum}\overline{\widehat{\eta}}_k\widehat{(W_tu_0)}_{k_1}\prod_{\substack{j=3 odd}}^{2p-1}\widehat{v}_{k_j}\prod_{\substack{l=2 even}}^{2p-1}\overline{\widehat{v}}_{k_l}\,d\Gamma\right|\lesssim \|\eta\|_{X^{-s-\varepsilon,1/2-}}\|u_0\|_{H^s_x}\|v\|^{2p-2}_{X^{s,1/2+}_T}.
\varrho^4|R_1 - R_2|^2 + \Vert (R_1 - R_2)\,\cdot + c_1 - c_2\Vert^2_{L^2( Q_1 \cup Q_2)}&\le C\sum\nolimits_{t=1,2}\Vert (\bar{R}_{j}\, \cdot + \bar{c}_j) - (R_{t}\, \cdot + c_t)\Vert^2_{L^2(Q_1 \cup Q_2)}.
\|\lceil u, \phi_\ell*\rceil f\|=\|u f_\ell-(uf)_\ell\| \lesssim[u]_\alpha[f]_\beta\int_0^1\big((t^\frac{1}{3}\ell)^{\alpha+\beta}+\ell^\alpha (t^\frac{1}{3}\ell)^\beta\big)dt, \forall \ell\in (0,1].
\begin{aligned}I_3 = \int \left(\nabla_{x}h, [\nabla_{\xi},\mathcal{L}]h\right)\varrho\,m_0 & = -\int \left|\nabla_{x}h\right|^2 \varrho m_0 +\underbrace{\int (\partial_{x_{k}}h) \partial_{i}[(\partial_{k}\sigma^{ij})\partial_{j}h ]\varrho\,m_0}_{=:T_1}\underbrace{-\int h (\nabla_{\xi}\psi,\nabla_{x}h)\varrho\,m_0}_{=:T_2}.\end{aligned}
\lim_{n\rightarrow\infty}F_{\ell,n}(p) =-\int_{\mathbb{T}^{N}}x_{\ell}\partial_{\ell}\left\{p(x) \prod\limits_{j\neq\ell}\left( 1-\frac{x_{j}}{x_{\ell}}\right) \right\}\mathrm{d}\mu_{S}(x)
\det \left[ \left( \begin{matrix}{{A^{( 1 )}}}&{{A^{( 2 )}}}\\{{A^{( 2 )}}}&{{A^{( 3 )}}}\\ \end{matrix}\right) - \mu \right] = ( \alpha - \mu )^{N - 1}[ \mu^{2}- ( \alpha + N + \gamma + \eta^{2}) \mu + ( \alpha + N ) \gamma + \eta^{2}\alpha ] = 0 ~ .
\bar\partial_B \bar\partial_B^*\phi= -\sum_{a,b}\bar\omega^a \wedge\bar V_b\,\lrcorner\, \nabla_{\bar V_a}\nabla_{V_b}\phi +\sum_a\bar\omega^a\wedge (\nabla_{\bar V_a}H^{0,1})\,\lrcorner\, \phi &+\sum_a\bar\omega^a\wedge H^{0,1}\,\lrcorner\,\nabla_{\bar V_a}\phi.
\int_{\Omega}\left|\frac{f(\psi_{\epsilon}^{-1})}{f_{\epsilon}(\psi_{\epsilon}^{-1})}\chi_{A_{\epsilon}}(\psi_{\epsilon}^{-1})\right|^p~dx =\int_{A_{\epsilon}}\frac{|f|^p}{|f_{\epsilon}|^{p-1}}~dx &\leq \frac{1}{\delta^{p-1}}\|f\|_{L^p(A_{\epsilon})}^p\rightarrow 0
|u(x,t)|\leq C\left( 1+D+\left( \frac{|\sigma |(\Omega )+|\mu |(\Omega_{T})}{D^{N}}\right)^{m_3}+\mathbb{I}_{2}^{2D}\left[ |\sigma |\otimes \delta_{\{t=0\}}+|\mu |\right] (x,t)\right) ,
\left[ \begin{array}{c}{{\psi_{\alpha}}}\\{{\bar{\psi}^{\dot{\alpha}}}}\\ \end{array}\right]_{\mathrm{W e y l}}= \left[ \begin{array}{c}{{\chi_{1}}}\\{{\xi_{2}}}\\{{i \xi_{1}}}\\{{- i \chi_{2}}}\\ \end{array}\right] .
\begin{aligned}0 \leq -\langle x_n-\bar{x}, (z_{n-1}-x_{n-1}) - (\bar{z}_{n-1}-\bar{x})\rangle & - \lambda \langle x_n-\bar{x}, B_{n-2}(x_{n-1})-B_{n-2}(x_{n-2})\rangle & + \frac{1}{2}\left( \|x_1-\bar{x}\|^2 - \|x_n-\bar{x}\|^2 + \|x_1-x_n\|^2\right).\end{aligned}
\phi-1=(\sum_{i}b_{i})^{-1}[If^{\prime}-f+\sum_{i}\sum_{j}(\lambda_{ij}+I\lambda_{ij}^{\prime})b_{i}b_{j}+2\sum_{i}\sum_{j}\sum_{k}\mu_{ijk}b_{i}b_{j}b_{k}+\cdot\cdot\cdot]
J_{T_1}X = 4(X_2S_1-X_1S_2+X_4S_3-X_3S_4) J_{T_2}X = 4(X_3S_1-X_4S_2-X_1S_3+X_2S_4) J_{T_3}X = 4(X_4S_1+X_3S_2-X_2S_3-X_1S_4).
f_{\mu}([g])= \left\{X\in \mathfrak{sp}(2) \,|\,(g^{-1})X \in S^2\mathbf E \oplus (\mathbf H \otimes \mathbf E)\right\}= (g)\left(S^2\mathbf E \oplus (\mathbf H \otimes \mathbf E) \right)\subset \mathfrak{sp}(2),
z_j = \min\left\{t\ge0\Big|\,(a_l\ldots a_1)_3<3^l-k,\,\, a_i=\vec 1_{x_j^{(i)}>t}+2\vec 1_{-x_j^{(i)}>t}\right\}, z_{ij}^{(a_l\ldots a_1)_3}=\min\left\{-(-1)^{a_\alpha}x_{i'}^{(\alpha)}-| x_{j'}^{(\beta)}|\Big| a_\alpha >0, a_\beta = 0, i',j'\in\{i,j\}\right\}_+, z_+ = \max\{z,0\}.
\frac{p_n}{q_n}=\frac{1}{a_1+[a_2,\ldots,a_n]}=\frac{1}{a_1+\frac{\hat{p}_{n-1}}{\hat{q}_{n-1}}}=\frac{\hat{q}_{n-1}}{a_1\hat{q}_{n-1}+\hat{p}_{n-1}}
\mathbb{Z}_n(\theta_0) = \sum_{i\in\mathcal{D}_2}R_{T,i}(R_{Y,i}- D_i\theta_0) + \sum_{i\in\mathcal{D}_2}R_{T,i}(\widehat{R}_{Y,i}- R_{Y,i}) &\quad+ \sum_{i\in\mathcal{D}_2}(\widehat{R}_{T,i}- R_{T,i})(R_{Y,i}- D_i\theta_0) + \sum_{i\in\mathcal{D}_2}(\widehat{R}_{T,i}- R_{T,i})(\widehat{R}_{Y,i}- R_{Y,i}).
\Delta\phi(x)= \frac{\Delta(|\nabla u(x)|^2)}{(b^2-\rho^2(u(x)))^2}+4\frac{\nabla(|\nabla u(x)|^2)\cdot\nabla(\rho^2(u(x)))}{(b^2-\rho^2(u(x)))^3}+2\frac{|\nabla u(x)|^2\Delta(\rho^2(u(x)))}{(b^2-\rho^2(u(x)))^3}+6\frac{|\nabla u(x)|^2|\nabla(\rho^2(u(x)))|^2}{(b^2-\rho^2(u(x)))^4}.
\mathrm{c^{sm}}(\Omega_{m+1})\mathrm{c^{sm}}(\Omega_{m+1}) =(h+H)^{2m}&=\sum_{l=0}^{2m}h^{2m-l}H^{l}\binom{2m}{l}&=\sum_{l=0}^{m}h^{2m-l}H^{l}\binom{2m}{l}&=\sum_{l=0}^{m}h^{m+l}H^{m-l}\binom{2m}{m-l}&=\sum_{l=0}^{m}h^{m+l}H^{m-l}\binom{2m}{m+l}.
\mathrm{Re}\bigg \langle \frac{f}{|x|^2}, \partial_r^2 f \bigg\rangle = \mathrm{Re}\bigg\langle \frac{f}{|x|^2}, \Delta_r f \bigg\rangle + 2(\mathbf{H}_{3}-1) \bigg\| \frac{f}{|x|^2}\bigg\|_{2}^2.
{}_{r+5}V_{r+4}\Big[a_0; a_1,\,\cdots,a_r\, \Big|\,z\Big]={}_{r+2}F_{r+1}\left[\begin{matrix}a_0, \tfrac{1}{2}a_0+1, a_1, \ldots a_r begin{align*}2pt]\tfrac{1}{2}a_0, \b_1,\ \ldots, \b_r\end{matrix};\,z \right]
H_{0}= \int d^{3}x \biggl ( \psi^{\dagger}( i \mathrm{\boldmath ~ \a l p h a ~}\cdot{\bf D}+ \gamma_{0}m ) \psi +{\frac{1}{2}}\mathrm{\boldmath ~ \p i ~}^{2}+{\frac{1}{4}}F^{i j}F_{i j}\biggr ) \, .
u_t = D\Delta u+H_r(u) x\in \Omega, t>0 v_t = \tilde D\Delta_{M}v+F_r(u,v) x\in M, t>0 D\frac{\partial u}{\partial \eta}=G_r(u,v) x\in M, t>0 u =u_0 x\in\Omega , t=0 v =v_0 x\in M , t=0
\quad\;\int\Lambda(\varepsilon+Q_b)\psi_B\big{[}(\varepsilon+Q_b)|\varepsilon+Q_b|^{p-1}-Q_b^{p}\big{]}&=\frac{1}{p+1}\int\bigg{(}\frac{p+3}{p-1}\psi_B-y\psi'_B\bigg{)}\big{[}|Q_b+\varepsilon|^{p+1}-Q_b^{p+1}-(p+1)\varepsilon Q_b^p\big{]}&\quad+p\int \psi_B\varepsilon Q_b^{p-1}(\Lambda Q_b),
(a;q)_k =\begin{cases}1;~ k=0, (1-a)(1-aq)(1-aq^2)\cdots(1-aq^{k-1}) ; k\in \mathbb{N}:=\{1,2,3,\cdots\}\end{cases}\ (a;q)_{\infty}=\prod_{k=0}^{\infty}(1-aq^k).
\lim_{h\rightarrow 0}V_{\alpha}= \lim_{h\rightarrow 0}\frac{\mu(h)}{2}\sum_{p=0}^{N-1}\sum_{q=0}^{N-1}u_pu_q^*f^{(\alpha)}(|p-q|)=\frac{\rho_0A_{\alpha}}{2}\int_0^L\int_0^Lu(x)u(x'){\cal K}^{(\alpha)}(|x-x'|){\rm dx}{\rm d x}'
\sum_{1 \leq i<j \leq r}a_{i,j}= \sum_{j \in [r] \setminus \{1 \}}a_{1,j}+ \sum_{l,t \in [r] \setminus \{1 \}, l<t}a_{l,t}\geq r-1 + \frac{(r-1)(r-2)}{2}= \binom{r}{2}.
\frac{1}{A+B}=\frac{1}{A}-\frac{1}{A}B\frac{1}{A}+\frac{1}{A}B\frac{1}{A}B\frac{1}{A}-\left(\frac{1}{A}B\right)^{3}\frac{1}{A}\cdots\pm\left(\frac{1}{A}B\right)^{k}\frac{1}{A+B}.
\Phi = \left( \begin{array}{c c c c}{{0}}&{{\phi_{1}\otimes M_{e}+ \phi_{2}\otimes M_{\nu}}}&{{0}}&{{0}}\\{{\phi_{1}^{*}\otimes M_{e}^{*}+ \phi_{2}^{*}\otimes M_{\nu}^{*}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right)
\mathcal{I}(\delta) \leq \mathbb{E}[Y_2] - \mathbb{E}[Y_1] + \mathrm{R}(\lambda_{\mathrm{D}}^\star, \delta) \leq \mathbb{E}[Y_2] - \mathbb{E}[Y_1] + \mathrm{R}_{\mathrm{D}}(\lambda_{\mathrm{D}}^{\star}, \delta) = \mathcal{I}_{\mathrm{D}}(\delta).
R^s_{,\ell}(\mathbf{H}_{\ell,\mathcal{K}},d_\ell) = \frac{1}{2}\log \left( 1 + \frac{P \|\mathbf{H}_{\ell,\mathcal{K}}\|^2 +1}{d_\ell}\right), \ \ \forall \ell \in \mathcal{L}
ev_1\circ h_{n+1}(v) = \sum_{i\geq 0}\alpha_i(v) & = f_{n+1}(v) - \sum_{i\geq 0}\frac{d\beta_i(v)+\beta_i(dv)}{i+1}= f_{n+1}(v)-d\beta_0(v) -\sum_{i\geq 0}\frac{\beta_id(v)}{i+1}= f_{n+1}(v) +\left(g_{n+1}(v)-f_{n+1}(v)+\sum_{i\geq 0}\frac{\beta_id(v)}{i+1}\right)-\sum_{i\geq 0}\frac{\beta_id(v)}{i+1}=g_{n+1}(v)
\mathfrak{G}_{\pi_{1}^{(5)}}^{(\beta)}\big(1,q,q^2,q^3\big)=q^4(1,3,3,3,2,1,1)+q^5(1,3,5,6,3,3) \beta+q^7(1,2,3,3) \beta^2+q^{10}\beta^3.
\begin{aligned}b_{2j+1}=\sharp D_{2j+1}=\prod_{i=1}^{2j+1}\sharp G_i \geq c(\overline{\beta})\overline{\beta}^{\frac{p-1}{p}n_{2j+1}- \frac{1}{p}\sum\limits_{i=1}^j n_{2i-1}-\sum\limits_{i=1}^{2j}d_i}.\end{aligned}
\int_{\mathfrak{g}^{(l)}}f(u)\rho_{t}(u)du =\int_{\mathfrak{g}^{(l)}}f(u)Q_{t}(du) =\int_{\mathfrak{g}^{(l)}}f(\delta_{t^{H}}u)Q_{1}(du) =\int_{\mathfrak{g}^{(l)}}f(\delta_{t^{H}}u)\rho_{1}(u)du =\int_{\mathfrak{g}^{(l)}}t^{-H\nu}f(u)\rho_{1}(\delta_{t^{-H}}u)du,
D_{\mu}^{i j}[ A ] \frac{\delta W}{\delta A_{\mu}^{j}}+ g f^{i j k}( J_{\mu}^{j}\frac{\delta W}{\delta J_{\mu}^{k}}+ \xi^{j}\frac{\delta W}{\delta \xi^{k}}+ \overline{{\xi}}^{j}\frac{\delta W}{\delta \overline{{\xi}}^{k}}) = 0 .
\frac{\sum_{|I|=n}(\det\U_I)^2}{\sum_{|I|=n-1}(\det\U'_I)^2}\ge\frac{\sum_{|I|=n}(\det\V_I)^2}{\sum_{|I|=n-1}(\det\V'_I)^2}+\frac{\sum_{|I|=n}(\det\W_I)^2}{\sum_{|I|=n-1}(\det\W'_I)^2}.
Z_{\; \; \alpha_{L}}^{\alpha_{L - 1}}Z_{\; \; \alpha_{L + 1}}^{\alpha_{L}}= C_{\alpha_{L + 1}}^{\alpha_{L - 1}\beta_{0}}\frac{\delta S_{0}}{\delta \Phi^{\beta_{0}}}, \; \alpha_{L + 1}= 1 , \ldots , M_{L + 1},
\mathsf{E}^{\mathsf{Q}}\left[ X{\bf 1}_{[ Y_1 > k ]}{\bf 1}_{[ Y_2 > h ]}\right] = \sigma_1 T \bar{\Phi}(\hat{x}, \hat{y}, \rho) + \sqrt{T}\int_{\hat{x}}^\infty \int_{\hat{y}}^\infty y_1 \phi_{\rho}(y_1, y_2) dy_1 dy_2
a_{L}( \vec{k}) = \bar{k}^{\mu}a_{\mu}( \vec{k}) , \; \; a_{T}( \vec{k}) = k^{\mu}a_{\mu}( \vec{k}) , \; \; a_{\lambda}( \vec{k}) = - e_{\lambda}^{\mu}( \vec{k}) a_{\mu}( \vec{k}) .
\frac{\langle \theta | \bar{\psi}( x ) \psi ( x ) | \theta \rangle}{\langle \theta | \theta \rangle}= \frac{1}{\langle \theta | \theta \rangle}\Bigl [ \langle \theta | \psi_{1}^{+}( x ) \psi_{2}( x ) | \theta \rangle + \langle \theta | \psi_{2}^{+}( x ) \psi_{1}( x ) | \theta \rangle \Bigr ] \ .
\delta\cong[\frac{d^{3}\delta}{dL^{3}}]_{L=0}\frac{L^{3}}{3!}=-\frac{1}{4a^{2}}\frac{L^{3}}{6}=\frac{-L^{3}}{24a^{2}}=\frac{-L^{3}\omega^{2}g^{2}}{24P^{2}}
N_{p\alpha +k}(\xi,t) = w^{(i)}_{p\alpha +k}(0,t) + \Psi^{(i)}_{p\alpha +k}(\xi_i,t) + \mathcal{O}(\exp(-\beta_0\, \xi_i))\mbox{as}\ \ \xi_i\to+\infty, \ \ \xi \in \Xi^{(i)},
\begin{array}{l l}{\}&{e^{- 2 \Phi}={\frac{M}{\lambda}}- \lambda^{2}x^{+}x^{-}={\frac{M}{\lambda}}+ e^{2 \lambda \sigma}}\\{\Rightarrow}&{e^{- 2 \left( \Phi + \lambda \sigma \right)}= 1 +{\frac{M}{\lambda}}e^{- 2 \lambda \sigma}}\\{\Rightarrow}&{\Phi \sim - \lambda \sigma -{\frac{M}{2 \lambda}}e^{- 2 \lambda \sigma}.}\\ \end{array}
H(f) = \begin{bmatrix}f_0 f_1 f_2 \hdots f_{t-1}f_1 f_2 f_3 \hdots f_t f_2 f_3 f_4 \hdots f_{y+1}\vdots \vdots \vdots \cdots \vdots f_{t-1}f_t f_{t+1}\hdots f_{2t-2}\end{bmatrix}
{\tilde{\hat{\cal Y}}}_{J_1}^{R_1}{\tilde{\hat{\cal Y}}}_{J_2}^{R_2}= \sum_{R, J}\left \langle \begin{array}{l}R J \end{array}\right. \left | \begin{array}{ll}R_1 R_2 J_1 J_2 \end{array}\right \rangle \left \{\begin{array}{lll}R R_1 R_2 \frac{P}{2}\frac{P}{2}\frac{P}{2}\end{array}\right \}[D(\frac{P}{2})]^{-1}{\tilde{\hat{\cal Y}}}_J^R,
\textbf{err}^{(4)}= \mathcal{O}\Bigg(\varepsilon^{\frac{23}{4}}\sqrt[4]{\mathcal{K}^{(\mathrm{T_0})}_{\mathrm{r}}\mathcal{S}^{(\mathrm{T_0})}_{\mathrm{r}}}\frac{1}{|\xi-z|^{\frac{7}{2}-2\mathrm{r}}}\Big\Vert|\mathrm{E}|^{2}\Big\Vert_{\mathrm{L}^2(\Omega)}\Bigg).
\langle \tau_t( \sigma_\xi^w) \, \sigma_\eta^{w'}\rangle = (-1)^{w+w'}\langle \left( \prod_{\ell=1}^{\xi-1}i a_\ell^+(t) a_\ell^-(t) \right) a^{\#_w}_\xi(t) a^{\#_{w'}}_\eta \left( \prod_{m=1}^{\eta-1}i a_\ell^+ a_\ell^- \right) \rangle \, ,
{\cal A}=dy^{0}\wedge dy^{1}\wedge ...\wedge dy^{q-1}\wedge[dz^{1}_{1}\wedge...\wedge dz^{r_{1}}_{1}h_{1}e^{C_{1}}+...+dz^{1}_{n}\wedge...\wedge dz^{r_{n}}_{n}h_{n}e^{C_{n}}],
\Omega_{\gamma}\approx \frac{1}{2}\int_{\gamma}d x_{\mu}^{\prime}\, \varepsilon_{\mu \nu}\left( \frac{x_{\nu}^{\prime}- x_{\nu}}{| x^{\prime}- x |^{2}}- \frac{x_{\nu}^{\prime}- y_{\nu}}{| x^{\prime}- y |^{2}}\right) .
\frac{Q\left(f_{n};\phi\right)}{T_{A}(f_{n})}=\frac{\alpha_{n}+\beta_{n}}{a_{n}\alpha_{n}+\gamma_{n}}=\frac{1}{a_{n}}\left(1+\frac{\beta_{n}-\gamma_{n}/a_{n}}{\alpha_{n}+\gamma_{n}/a_{n}}\right)
s_{\mathfrak{z}_{d}}^{\prime}=im\left( -\mathfrak{d}h\right)=\omega^{\sharp}\circ im\left( d^{G\backslash}\left( -h\right)\right) =\omega^{\sharp}\circ p_{\^{1}T^{\ast}T^{\ast}G}\circ im\left( -^{1}dh\right) .
u = z^{\alpha}x_{I_{1}}^{\xi_{1}}\cdots x_{I_{a}}^{\xi_{a}}y_{\max(J_{1})}^{\nu_{1}}\cdots y_{\max(J_{b})}^{\nu_{b}},\, \, \, \, \, \, \, \, \, \, v = z^{\alpha'}x_{I'_{1}}^{\xi'_{1}}\cdots x_{I'_{a'}}^{\xi'_{a'}}y_{\max(J'_{1})}^{\nu'_{1}}\cdots y_{\max(J'_{b'})}^{\nu'_{b'}},
\delta=\min_{1\le i\le n}\frac{x_{i}-x_{i-1}}{3}\; \textrm{and}\;\;V(x_1,\ldots,x_{n-1})=\{(y_1,\ldots,y_{n-1})\in\Omega_{n-1}: \vert y_i-x_i\vert< \delta,\forall i\}.
\begin{aligned}S(U,f^{n+k}) \geq\frac13 \frac{\log M^{n-1}(R_2,f)}{\log R_2}\frac{\log R_2}{\log r_2}-3\frac{\log^+ M^{n-1}(R_1,f)}{\log R_1}\frac{\log R_1}{\log r_1}.\end{aligned}
\begin{aligned}(\boldsymbol{D}^{-s}\psi, \boldsymbol{D}^{s}\psi) =(\widehat{\boldsymbol{D}^{-s}\psi}, \overline{\widehat{\boldsymbol{D}^{s}\psi}}) =(\mathcal{F}(\boldsymbol{D}^{-s}\psi), \overline{\mathcal{F}(\boldsymbol{D}^{s}\psi)}) =((2\pi i\xi)^{-s}\widehat{\psi}, \overline{(2\pi i \xi)^{s}\widehat{\psi}}). \end{aligned}
S(x) \le \begin{cases}x^\theta \log^6(x+1) \textrm{if}\theta=\theta_0; Z_\alpha(x) + O\left(\frac{1-\theta}{(\alpha-\theta)^3}\log^3\frac{x+1}{\alpha-\theta}~x^\alpha \right) \ll\frac{1-\theta}{(\alpha-\theta)^3}\log^3\frac{x+1}{\alpha-\theta}~x^{\theta_0}\textrm{if}\theta<\alpha<\theta_0.\end{cases}
S_{1}:=\mathrm{span}\left\{P_{0}^{(-2,\beta)},P_{1}^{(-2,\beta)}\right\}=\left\{f\in S\mid f^{\prime\prime}(x)=0\right\}\,,\text{and}S_{2}:=\mathrm{span}\left\{P_{n}^{(-2,\beta)}\right\}_{n=2}^{\infty}=\left\{f\in S\mid f(1)=f^{\prime}(1)=0\right\}\,.
\sum_{I \subset [1,n] \atop |I|=k}\mathfrak{D}_{I}\prod_{i \notin I, j \in J \atop i < j}\big(1+t \beta-t^2 q_{ij}\big){}= \sum_{I \subset [1,n] \atop I=\{1 \le i_1 < \ldots < i_k \le n \}}\prod_{a= 1}^{k}(1+t \beta)^{n-k-i_{a}+a}\left(z_{i_{a}}(1+ t \beta)^{i_{a}-a}+1 + h_{i_{a}}\frac{(1+t \beta)^{i_{a}- a}-1}{\beta}\right).
\bar{\Delta}V_{n}=\sum_{v=1}^{n-1}\Delta_{v}\left( \frac{\hat{a}_{nv}P_{v}\lambda_{v}}{vp_{v}}\right)\sum_{r=1}^{v}a_{r}+\frac{\hat{a}_{nn}P_{n}\lambda_{n}}{np_{n}}\sum_{r=1}^{n}a_{r}\bar{\Delta}V_{n}=\sum_{v=1}^{n-1}\Delta_{v}\left( \frac{\hat{a}_{nv}P_{v}\lambda_{v}}{vp_{v}}\right)s_{v}+\frac{\hat{a}_{nn}P_{n}\lambda_{n}}{np_{n}}s_{n},
E_{m}=\frac{RT}{F}ln(\frac{P_{Na^{+}}[Na^{+}]_{o}+P_{K^{+}}[K^{+}]_{o}+P_{Cl^{-}}[Cl^{-}]_{i}}{P_{Na^{+}}[Na^{+}]_{i}+P_{K^{+}}[K^{+}]_{i}+P_{Cl^{-}}[Cl^{-}]_{o}})
T_{\nu}^{\mu}= ( \stackrel{0}{\rho}+ \stackrel{0}{p}) \stackrel{0}{{u}^{\mu}}{\stackrel{0}{u}}_{\nu}+ \stackrel{0}{p}\delta_{\nu}^{\mu}\equiv T_{( r a d . ) \nu}^{\mu}+ T_{( c o s . ) \nu}^{\mu}
\begin{array}{r c l}{}&{}&{m ( S \otimes \mathrm{i d}) \Delta ( H ) = S ( H ) + H = 0 ,}\\{}&{}&{m ( S \otimes \mathrm{i d}) \Delta ( X^{+}) = S ( X^{+}) q^{- H}+ S ( q^{H}) X^{+}= 0 .}\\ \end{array}
\sum\limits_{i = 1}^{k}{(\alpha_i - \beta_i )v_i}+ \sum\limits_{i = k+1}^m{\alpha_i v_i}- \sum\limits_{i = k+1}^{2d - m}{\beta_i w_i}+ \left({\sum\limits_{i = 1}^m{\alpha_i K_i}}\right)u_0 = 0
T_{l-1}=\frac{f^{(l)}(\tau)}{p^{l}}\sum_{\substack{0\leq a_1+a_2\leq l a_1=a_2}}(-2)^{l-a_1-a_2}\frac{1}{a_1!a_2!(l-a_1-a_2)!}\|\mathcal{T}_{1,l}^{(a_1,a_2)}-\mathcal{T}_{2,l}^{(a_1,a_2)}\|_{\rm F}^2.
\Phi =dr\wedge\left[PQ\left(\theta^{125}+\theta^{136}+\theta^{147}\right)+\frac{rR_{+}}{2}\left(\theta^{237}+\theta^{264}+\theta^{534}\right)-\frac{rR_{-}^{2}}{2R_{+}}\theta^{567}\right] &+R_{+}^{3}\theta^{1234}-R_{+}R_{-}^{2}\left(\theta^{1267}+\theta^{1537}+\theta^{1564}\right)+Q^{2}\left(\theta^{2536}+\theta^{2547}+\theta^{3647}\right).
{}_pF_q(a_1,\dotsc,a_p;b_1,\dotsc,b_q;z)=\sum_{\ell=0}^\infty\frac{(a_1)_\ell\dotsm(a_p)_\ell}{(b_1)_\ell\dotsm(b_q)_\ell}\frac{z^\ell}{\ell!}
\big\|e^{-\beta t}M^{\phi}(t)-\Delta\big\|_k \le \Big\|e^{-\beta t}M^{\phi}(t)-\max_{i\le n}e^{-\beta t}D_i^{\phi}(t-\tau_i)\Big\|_k &+\Big\|\max_{i\le n}e^{-\beta t}D_i^{\phi}(t-\tau_i)-\max_{i\le n}e^{-\beta \tau_i}Y_i\Big\|_k &+\Big\|\Delta-\max_{i\le n}e^{-\beta \tau_i}Y_i\Big\|_k &=: Q_1+Q_2+Q_3.
\beta_{i}^2 = \min\left\{\max\left\{0,\frac{z_i-\frac{\epsilon_{1,i}}{\rho}}{z_i(1+\epsilon_{1,i})}\right\},\max\left\{0,\frac{x_i-\frac{\epsilon_{1,i}}{\rho}}{x_i(1+\epsilon_{1,i})}, \right\}\right\}.
f^0(\lambda)\big|A_N\bigl(e^{i\lambda}\bigr)\bigl(1-e^{i\lambda\mu}\bigr)^{n}\bigl[p^0(\lambda)-\beta^2f^0(\lambda)\bigr]_++\lambda^{2n}C_{\mu,N}^{\beta,0}\bigl(e^{i\lambda}\bigr)\big| &=|\lambda|^{n}\big|1-e^{i\lambda\mu}\big|^{n}\bigl(\alpha_1p^0(\lambda)+\alpha_2|\beta|f^0(\lambda)\bigr),
\int_{\mathbb{R}^n}e^{-\psi}\, dx \int_{\mathbb{R}^n}e^{-\psi^*}\, dx = (2\pi)^n \int_{\mathbb{R}^n}e^{-\phi}\, d\gamma \int_{\mathbb{R}^n}e^{Q_1 \phi}\, d\gamma,\;\;\;{\rm for}\;\;\; \psi = \phi + \frac12|\cdot|^2.
\prod_{\ell\in\lambda}([1_{\mathcal H}]-\psi_\ell\psi_\ell^*)*k_r = \tfrac{1}{r^2}\left((r^2-1)[1_{\mathcal H}] + [1_{\mathcal H}] - \psi_{r,\ell}\psi_{r,\ell}^*\right)*k_r \geq 0,
\lambda^{a,\alpha \beta}\lambda_{a~\delta}^{~\gamma}- \lambda^{a,\gamma \alpha}\lambda_{a~\delta}^{~\beta}= \frac{2}{\sigma_{0}}\left(\Omega^{\alpha \beta}\delta^{\gamma}_{~\delta}- 2 \Omega^{\beta \gamma}\delta^{\alpha}_{~\delta}+ \Omega^{\gamma \alpha}\delta^{\beta}_{~\delta}\right),
\lVert \overline{\xi}_{\overline{x}}- \overline{\xi}_{\overline{y}}\rVert = \frac{1}{|F|}\sum_{\Lambda z\in \Gamma/\Lambda}\lvert \, \lvert \Lambda z\cap xF\rvert - \lvert \Lambda z\cap yF\rvert \, \rvert &\le \frac{1}{|F|}\sum_{\Lambda z\in \Gamma/\Lambda}|(xF\Delta yF)\cap \Lambda z| &= \frac{| xF\Delta yF|}{|F|}&\le \epsilon.
f^{\ast}(u)\leq\inf\bigg\{\sum_{n\geq1}f_{n}^{\ast}(u_{n})\mid(u_{n})_{n\geq1}\subset\left( \operatorname*{dom}f_{n}^{\ast}\right)_{n\geq1},\u=\sum_{n\geq1}u_{n}\bigg\}<\infty\quad\forall u\in\mathbb{R}_{+}.
\big| \ \partial^m_\alpha|_{\alpha=0}\,(1-2e^\alpha C\varkappa^2)^{-1}\ \big| \le 2^m m!\max_{1\le j\le m}(1-2C\varkappa^2)^{-j}= \big(\frac{2}{1-2C\varkappa^2}\big)^m m!
\sum_{n=0}^{\infty}p(5n+4)q^{n}=5\dfrac{(q^{5};q^{5})_{\infty}^{5}}{(q;q)_{\infty}^{6}}\equiv5\dfrac{(q^{5};q^{5})_{\infty}^{4}}{(q;q)_{\infty}}=5(q^{5};q^{5})_{\infty}^{4}\sum_{n=0}^{\infty}p(n)q^{n},
\nu(x)=\frac{( \nabla \rho,-1 )}{\sqrt{1+|\nabla \rho|^2}}=\frac{(\frac{x'}{\sqrt{1-|x'|^2}},-1 )}{\sqrt{1+\frac{|x'|^2}{1-|x'|^2}}}=(x',-\sqrt{1-|x'|^2}).
G(t,s,\xi):=\begin{cases}-X^{t}(\xi)\left[P_{\xi}^{+}+P_{\xi}^{0}\right]\left[X^{s}(\xi)\right]^{-1}t\le s X^{t}(\xi)P_{\xi}^{-}\left[X^{s}(\xi)\right]^{-1}t>s.\end{cases}
Z_{G}[ \overline{{{\xi}}}, \xi ] = \exp \left\{i \sum_{a , b = 1 , . . 8}\int d^{4}x d^{4}y \overline{{{\xi}}}^{a}\left( x \right) \widetilde{D}_{G}^{a b}( x - y ) \xi^{b}\left( y \right) \right\},
d s^{2}= ( F f_{1}\cdots f_{7}f_{8})^{1 / 2}( a_{1}( r ) \cdots a_{7}( r ) )^{1 / 2}\times \left({\frac{( \sigma_{1})^{2}}{f_{1}^{2}a_{1}^{2}}}+ \cdots{\frac{( \sigma_{7})^{2}}{f_{7}^{2}a_{7}^{2}}}+{\frac{d r^{2}}{f_{8}^{2}}}\right)
\|F_m^l\|_{l,r}\le 1_{m=2}\sum_{n=4}^N 2^n (c_0M^{a_1(l+1)})^{\frac{n-2}{2}}\|J_n^{l+1}\|_{l,r}&+1_{m\ge 4}\Bigg(\|J_m^{l+1}\|_{l,r}+\sum_{n=m+2}^N 2^n (c_0M^{a_1(l+1)})^{\frac{n-m}{2}}\|J_n^{l+1}\|_{l,r}\Bigg).
\mathrm{Ad}(\exp H) S_{\lambda, i}= \cos \langle \lambda, H \rangle S_{\lambda, i}+ \sin \langle \lambda, H \rangle T_{\lambda, i}, \mathrm{Ad}(\exp H) T_{\lambda, i}= -\sin \langle \lambda, H \rangle S_{\lambda, i}+ \cos \langle \lambda, H \rangle T_{\lambda, i}.
\begin{array}{@{}l@{}}{\displaystyle \psi_{h}^{0}=R_h\psi_{0},\quad\mathbf{A}_{h}^{0}={\pi}_h \mathbf{A}_{0},\mathbf{A}_{h}^{-1}=\mathbf{A}_{h}^{0}-\Delta t{\pi}_h\mathbf{A}_{1},}begin{align*}2mm]{\displaystyle \qquad\qquad\phi_{h}^{0}= I_{h}\phi_{0}, \phi_{h}^{-1}= \phi_{h}^{0}- \Delta t I_{h}\psi_{1},}\end{array}
I m \Pi_{2}^{\mathrm{v a c}}= \frac{3 \lambda g^{2}}{1 6 \pi^{3}}\left\{\frac{1}{\epsilon}+ 4 - 2 \gamma_{E}+ 2 \ln \left( \frac{4 \pi \mu^{2}}{P^{2}}\right) \right\}.
\widehat{S}_n=-\frac{n!^{A-15}}{i\pi}\int_L (2t+n-1)\frac{\Gamma(t-\frac{1}{2})^{A+3}\Gamma(2t+4n)^3\Gamma(2n-2t+2)^3}{\Gamma(t+n+\frac12)^{A+3}}\cos(\pi t)^4dt.
\begin{array}{r c l r c l}{{F_{1}}}&{{=}}&{{\bar{U}d V - V d \bar{U},}}&{{F_{3}}}&{{=}}&{{\bar{U}H + V \bar{H},}}\\{{F_{5}}}&{{=}}&{{d A_{4}+ \mathrm{i}( B \bar{H}- \bar{B}H ) ,}}&{{H}}&{{=}}&{{d B ,}}\\ \end{array}
\tilde P(A) = \frac{e^2\sqrt{(A-1+u)(A+2u)}(2A-1+2u)^{u}}{u!e^u\sqrt{2\pi}\sqrt{2A-1+2u}\cdot 2^u}& \leq e^2\sqrt{(A-1+u)(A+2u)}(2A-1+2u)^{u}&\leq e^2 (2A-1+2u)^{u+1}&\leq CA^{u+1}
\mathcal{H}(\lambda)\setminus \mathcal{H}(\lambda^+) =\left( \{|x_i-y_j|:1\leq j \leq m \}\cup \{x_i+y_j-1:1\leq j \leq m \}\right) &\setminus \bigl( \{1,x_i, 2x_i-2 \}\cup \{|x_i-x_j|:1\leq j \leq m, j\neq i \}\qquad\cup \{x_i+x_j-1:1\leq j \leq m, j\neq i \}\bigr),
\exp\left(\sum_{i, k \geq 0}I_{i, k}\hbar^{i - 1}\right) = \sum_{j \geq 0}\frac{(\sum_{i, k \geq 0}I_{i, k}\hbar^{i - 1})^j}{j!}= \sum_{j \geq 0}\sum_{\,|\{n_{i, k}\}| = j \,}\prod_{i, k \geq 0}\frac{\hbar^{(i - 1)n_{i, k}}}{n_{i, k}!}I_{i, k}^{n_{i, k}},
0 = \frac{\nabla_E w(x_0)-\nabla_Ew(y_0)}{\bar w}- \frac{m}{\bar w}(\nabla_E\bar w), 0 \geq \frac{\nabla^2_{E,E}w(x_0) - \nabla^2_{E,E}w(y_0)}{\bar w}-\frac{m}{\bar w}\nabla^2_{E,E}\bar w.
\begin{array}{r c l}{Y}&{=}&{\frac{1}{2}z^{- 2}( x + y + 2 \prod_{i = 1}^{n}z_{i})}\\{X}&{=}&{\frac{1}{2}z^{- 1}( x - y )}\\{Z}&{=}&{z^{2}.}\\ \end{array}
\Lambda(x,\delta) = \max_{j_1,\ldots, j_n\in \{1,\ldots, q\}}\left|\det \left( X^{\delta}_{j_1}(x) | \cdots | X^{\delta}_{j_n}(x) \right)\right| & = \max_{j_1,\ldots, j_n\in \{1,\ldots, q\}}\left|\det \left( (AX^{c\delta})_{j_1}(x) | \cdots | (AX^{c\delta})_{j_n}(x) \right)\right|,
\alpha(N, c)=\begin{cases}N \gcd(c, N/c)>2, \frac{N}{c}\gcd(c, N/c)=1, \frac{N}{2c}\gcd(c, N/c) = 2 \frac{N}{2c}\frac{2N}{c}\gcd(c, N/c) = 2 \frac{N}{2c}\end{cases}
\Big| \frac{1}{2 \varepsilon n^2}\sum_{x}\Phi_n(s, \tfrac{x}{n}) \sum_{r=1}^{\varepsilon n}\int_{\Omega_1(x)}\overleftarrow{\eta}^{\ell}(x) [\eta(x+1)-\eta(x+1+r) ] [ f ( \eta_{2,x,r}) - f( \eta_{3,x,r}) ] d\nu_{b}\Big|,
e ( D e f ( \Sigma )^{m}) = \prod_{i = 1}^{h_{1}}\left({\frac{\lambda_{1}- \lambda_{2}}{m_{i}}}H - \psi_{i}\right) \prod_{j = 1}^{h_{2}}\left({\frac{\lambda_{1}}{n_{j}}}H - \psi_{j}\right)
\delta S_{a b}^{c d}\left( \vartheta \right) = - i g \frac{F_{\bar{c}\bar{d}a b}^{\Psi}\left( i \pi , \, \vartheta + i \pi , \, 0 , \, \vartheta \right)}{m_{a}m_{b}\sinh \vartheta}\quad , \quad \vartheta = \vartheta_{a}- \vartheta_{b}\: .
\left|\frac{dg(z)}{d\bar{z}}\right|^2 - 2\pi\mathrm{Re}\left(\bar{z}g(z)\frac{dg(z)}{d\bar{z}}\right) &= \frac{1}{4m^3 n^3}e^{-\left[ \frac{2}{x(1-x)}+\frac{2}{y(1-y)}\right]}\Bigg{[}\frac{(2x-1)^2}{x^4(1-x)^4}+ \frac{(2y-1)^2}{y^4(1-y)^4}+\frac{4\pi(2x-1)}{x(1-x)^2}+\frac{4\pi(2y-1)}{y(1-y)^2}\Bigg{]}\geq 0.
Q(C_{n(i_1-1) + j_1},C_{n(i_2-1) + j_2}) \ = \Q(B_{i_1},B_{i_2}) \cdot N^2 \i_1 \not= i_2, Q(C_{n(i-1) + j_1},C_{n(i-1) + j_2}) \ = \Q(A_{j_1},A_{j_2}) \cdot K \j_1 \not= j_2.
\sum_{N^-<p<N^+}\frac{H(D_N(p))}{p}=\frac{1}{N}\sum_{N^-<p<N^+}H(D_N(p))+O\left(\frac{\log \log N}{\sqrt{N}\log N}\right)\intertext{Also}\left(\sum_{N^-<p<N^+}\frac{H(D_N(p))}{p}\right)^j =\frac{1}{N^j}\left(\sum_{N^-<p<N^+}H(D_N(p))\right)^j+O\left(\frac{1}{\sqrt{N}}\right)