harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{H}^{(k)}: W^{(k)}\to V_{h,e}^{(k)}: &\begin{cases}\mathcal{H}^{(k)}{u_{B_e}}\in V_{h,e}^{(k)}: a^{(k)}(\mathcal{H}^{(k)}{u_{B_e}},u^{(k)})=0 \forall u^{(k)}\in V_{I,h}^{(k)}, \mathcal{H}^{(k)}{u_{B_e}}_{|\Gamma^{(k)}}={u_{B_e}}_{|\Gamma^{(k)}}, \end{cases}
\begin{array}{r l}{\bar{\phi}_{a b}( z ) : =}{( \bar{\phi}_{a b}^{( 0 )}( z ) , \cdots , \bar{\phi}_{a b}^{( n - 1 )}( z ) ) ,}{\bar{\phi}_{a a + \bar{\epsilon}_{j}}^{( i )}( z ) : =}{( \tilde{\Phi}( z ) )_{j}^{i}/ C h ( z - ( n - 1 ) / 2 ) ,}\end{array}
S_{R^{2}b u l k}^{( 1 )}= \frac{1}{2 k^{2}}\int d^{D}x \frac{\alpha^{\prime}}{4}e^{\gamma \Phi}\bigl ( R^{\mu \nu \rho \sigma}R_{\mu \nu \rho \sigma}- 4 R^{\mu \nu}R_{\mu \nu}+ R^{2}\bigr )
|K(x,y)-K(x,y')| \lesssim \int_0^\infty |S_t(x,y)-S_t(x,y')|\frac{dt}{t}&=\int_0^{\|y-y'\|}|S_t(x,y)-S_t(x,y')|\frac{dt}{t}+\int_{\|y-y'\|}^{\|x-y\|}|S_t(x,y)-S_t(x,y')|\frac{dt}{t}+\int_{\|x-y\|}^\infty |S_t(x,y)-S_t(x,y')|\frac{dt}{t}&=:I\!I_1+I\!I_2+I\!I_3.
h_{\overline{D}_0,\dots,\overline{D}_{n-1}}(Z;s_0,\dots,s_{n-1})=h_{\overline{D}_0,\dots,\overline{D}_{n-1},\overline{D}_f}\big(X_\Sigma;s_0,\dots,s_{n-1},\tilde{f}s_f\big),
\mathbf{X}^{(g)}\left[\mathbf{I}_{K_g}\otimes \mathbf{E}_{L_g,l}\right]=\sum_{\left\{m \; \vert \beta_m^l > 0\right\}}\left(\beta_m^l\right)^{1/2}\boldsymbol{\psi}_m^l \left[\boldsymbol{\phi}_m^l\right]^H
u_{g_1,1}(\lambda,x_{g_1,1})=\frac{\Delta_{B_{g_1-1,1}}^N(\lambda)}{\Delta_{1}(\lambda)}c_{g_1,1}(\lambda,x_{g_1,1})+\sum\limits_{k=g-d(w)+1}^{g-1}\frac{\Delta_{A_{k,1}}^N(\lambda)}{\Delta_{1}(\lambda)}s_{g_1,1}(\lambda,x_{g_1,1}).
B_{\mu\nu}^{c}= ie^{2}\{\epsilon_{\mu\nu\lambda}\partial_{\lambda}\phi -2\pi\int_{\Sigma_{C}}\delta (x-y(\sigma))d\sigma_{\mu\nu}+2\pi\oint \psi_{\mu}(\tau) \psi_{\nu}(\tau) \delta (x-y(\tau))d\tau \}.
(\ref{L_2_estimate}) = \tau \int_{1^-}^{\infty}s\exp\left(-2\tau s^2 \right) \left|\Lambda^\vee \cap B_2(0,s\|\lambda^*\|) \right|ds & \leq \tau\int_{1^-}^{\infty}s\exp\left(-2\tau s^2 + (1 + \varepsilon(k))kF(s)\right)ds
S = \int d^{4}x [ - \frac{1}{4}F_{\mu \nu}^{a}F_{\mu \nu}^{a}+ \frac{1}{2 \alpha}( \partial_{\mu}A_{\mu})^{2}+ + \bar{c}^{a}\partial_{\mu}( \delta^{a b}\partial_{\mu}- g t^{a b c}A_{\mu}^{c}) c^{b}]
\begin{array}{ll}\mbox{minimize}f(R) + n \sum\limits_{k=1}^r |c_k|^2 \mbox{subject to}R = \sigma^2 I + \sum\limits_{k=1}^r |c_k|^2 \left[\begin{array}{c}1 e^{\j\omega_k}\vdots e^{\j (n-1)\omega_k}\end{array}\right] \left[\begin{array}{c}1 e^{\j\omega_k}\vdots e^{\j (n-1)\omega_k}\end{array}\right]^H, \end{array}
\int_{\rho_2}^R f(\rho,R) \, d\rho < \frac{1}{\sqrt{(R + \rho_2)(\rho_2^2 - 1)}}\, \int_{\rho_2}^R \frac{1}{\sqrt{R-\rho}}\,d\rho= \frac{2 \sqrt{R - \rho_2}}{\sqrt{(R + \rho_2)(\rho_2^2 - 1)}}.
w ( \epsilon ) = \frac{1}{2}\sum_{n = 1}^{\infty}\sqrt{n^{2}+ \epsilon^{- 1}}= \frac{1}{4}\left( \sum_{n = - \infty}^{+ \infty}\sqrt{n^{2}+ \epsilon^{- 1}}- \frac{1}{\sqrt{\epsilon}}\right){,}
U\tilde{R}(-e_{n+1})U^{-1}= \frac12 \tilde{R}(e_{n+1})\tilde{R}(e_n+e_{n+1})\tilde{R}(-e_{n+1})\tilde{R}(e_n+e_{n+1})\tilde{R}(e_{n+1}) &=\frac12 (\tilde{R}(e_{n+1})+\tilde{R}(e_n)+\tilde{R}(e_n)-\tilde{R}(e_{n+1})) =\tilde{R}(e_n).
\sum\limits_{\substack{q\in \mathbb{Z}[i]\setminus\{0\}\mathcal{N}(q)\le Q}}\sum\limits_{\substack{r \bmod{q}(r,q)=1}}\left|\sum\limits_{\substack{n\in \mathbb{Z}[i] \mathcal{N}(n)\le N}}a_n \cdot e\left(\mbox{\rm Tr}\left(\frac{nr}{2q}\right)\right)\right|^2 \ll \left(Q^2+N\log(2Q)\right)Z.
\int_{A_\varepsilon}\left( K_H(x)\nabla \left( u_\varepsilon-q\ln\frac{1}{\varepsilon}\right) | \nabla \left( u_\varepsilon-q\ln\frac{1}{\varepsilon}\right) \right) dx=\frac{1}{\varepsilon^2}\int_{A_\varepsilon}\left( u_\varepsilon-q\ln\frac{1}{\varepsilon}\right)^{p+1}dx,
\sum_{\substack{c>C, d \geq 1 cd=q}}\mu(c) = \sum_{c|q}\mu(c) - \sum_{\substack{c\le C, d \geq 1 cd=q}}\mu(c) = \left\lfloor \frac{1}{q}\right\rfloor - \sum_{\substack{c\le C, d \geq 1 cd=q}}\mu(c).
\mathcal{C}_n=\begin{pmatrix}m_0 0 \ldots 0 0 \ell_1 m_1 \ldots 0 0 \vdots \vdots \ddots \vdots \vdots 0 0 \ldots m_{n-2}0 0 0 \ldots \ell_{n-1}m_{n-1}\end{pmatrix},
\begin{array}{llllllllll}(A_1 \oplus K_2)^1 = \langle x_1, y_1, h_1 \rangle \oplus \langle x_2, h_2 \rangle, (A_1 \oplus K_2)^2 = \langle x_2, y_2, h_2 \rangle \oplus \langle x_1, h_1 \rangle.\end{array}
d_{2p-3}(\tau_1) \dot{=}v_1, d_{2p-3}(\bar{\xi}_1)\dot{=}\alpha_1, d_{2p-2}(\bar{\tau}_1v_1^k) \dot{=}v_1^{k+1}, d_{2p-2}(\bar{\xi}_1v_1^k)\dot{=}\alpha_1v_1^k
J_{15}= \int_b^{b'}\arg{\left( 1 - \frac{1}{\frac{F'}{F}(\sigma+it_q,\chi)}\frac{L'}{L}(1-\sigma-it_q,\overline{\chi}) \right)}d\sigma &\quad- \int_b^{b'}\arg{\left( 1 - \frac{1}{\frac{F'}{F}(\sigma+iT,\chi)}\frac{L'}{L}(1-\sigma-iT,\overline{\chi}) \right)}d\sigma &= O(\delta)
n_1(\xi,\eta,\rho) = O \big( (|\xi-\rho|^{3/2}+ |\rho-\eta|^{3/2}) \mathbf{1}_{[2^2,\infty)}(|\eta|/|\rho-\eta|) \mathbf{1}_{[2^2,\infty)}(|\xi|/|\xi-\rho|) \big) .
P'_{k}(x_0)/P_k(x_0)-P'_{k-1}(x_0)/P_{k-1}(x_0) = \sum_{j=1}^k \frac{1}{x_0-r_j}- \sum_{j=1}^{k-1}\frac{1}{x_0-s_j}&= \frac{1}{x_0-r_k}+ \sum_{j=2}^{k-1}\frac{r_j-s_j}{(x_0-r_j)(x_0-s_j)},
U_{1,-\frac{2e_n}{\delta}}(y) =\frac{\alpha_n}{\left(1+\left|y-\frac{2e_n}{\delta}\right|^2\right)^\frac{n-2}{2}}=O(\delta^{n-2}), \delta \varphi_0(y)-\delta \varphi_0\Big(\frac{2e_n}{\delta}-y\Big) =O\left(\frac{\delta}{\left(1+\left|y-\frac{2e_n}{\delta}\right|\right)^{n-3}}\right)=O(\delta^{n-2}),
\begin{array}{rclcl}Q^T \bar{X}Q = \begin{bmatrix}U^T X_L U 0 0 I \end{bmatrix}= \begin{bmatrix}0 0 0 0 \Lambda 0 0 0 I \end{bmatrix}, Q^T X Q = \begin{bmatrix}U^T A U U^T B B^T U C \end{bmatrix}= \begin{bmatrix}V_{11}V_{12}V_{13}V_{12}^T V_{22}V_{23}V_{13}^T V_{23}^T V_{33}\end{bmatrix},\end{array}
V_i(x_i) = e^{L_i \tau_i}W_i(z_i) \geq W_i(z_i) \geq \max_{j=1}^{n}\tilde\chi_{ij}(W_j(z_j)) = \max_{j=1}^{n}\chi_{ij}(e^{L_j N_{0j}^* \delta_j^*}W_j(z_j)) \geq \max_{j=1}^{n}\chi_{ij}(V_j(x_j)),
\begin{pmatrix}x_0\,g_{00}+x_1g_{10}x_0\,g_{01}+x_1g_{11}x_0\,h_{00}+x_1h_{10}x_0\,h_{01}+x_1h_{11}\end{pmatrix}\begin{pmatrix}y_0 y_1\end{pmatrix}\xmapsto{t_j \mapsto f_j}\begin{pmatrix}0 0\end{pmatrix}
\Sigma \leqslant 2\langle\nabla G(\xi(-\tau^*)),\xi(-\tau^*)-\xi(-T^*)\rangle+C|\xi(-\tau^*)-\xi(-T^*)|^2 &=2\Big\langle\nabla G(\xi(-\tau^*)),(\frac{\xi(-\tau^*)-y^-}{2}+\frac{\xi(-\tau^*)-y^+}{2})\Big\rangle+C|\xi(-\tau^*)-\xi(-T^*)|^2 &\leqslant C|\xi(-\tau^*)-y^-|^2+C|\xi(-\tau^*)-y^+|^2+C|\xi(-\tau^*)-\xi(-T^*)|^2 &=:a_1+a_2+a_3,
\Phi(z_{1},...,z_{n-1})=\sum_{\lambda_{1}=0}^{L}\cdot\cdot\cdot\sum_{\lambda_{n}=0}^{L}p(\lambda_{1},...,\lambda_{n})\alpha_{1}^{(\lambda_{1}+\lambda_{n}\beta_{1})z_{1}}\cdot\cdot\cdot\alpha_{n-1}^{(\lambda_{n-1}+\lambda_{n}\beta_{n-1})z_{n-1}}
\phi_{i}( \tau ) = \big ( \pi_{i}^{\mu}( \tau ) + \frac i 2 \xi_{i}^{\mu}( \tau ) \big ) \int{d^{3}\sigma \rho_{\mu}( \tau , \vec{\sigma}) \equiv \big ( \pi_{i}^{\mu}( \tau ) + \frac i 2 \xi_{i}^{\mu}( \tau ) \big ){p_{s}}_{\mu}}\approx 0 ,
u(t_n,x;s,u_n,a,b) =u(t_n,x;t_n-1,u(t_n-1,\cdot;s_n,u_n,a,b),a,b) &=u(1,x;0,u(t_n-1,\cdot;s_n,u_n,a,b),a_n,b_n).
\deg (\pi_{2,3}^{*}\left[T_{\eta^{+}}\right] \cdot f^{*}[\eta^{+}]) =\deg (f_{*}(\pi_{2,3}^{*}F_1 + \pi_{2,3}^{*}F_2 + \pi_{2,3}^{*}\Delta_{C\times C})\cdot [\eta^{+}]).
|b_{14}^{(k)}-b_{23}^{(k)}| =|b_{14}^{(k)}|+|b_{23}^{(k)}| \geq \sqrt{|b_{14}^{(k)}|^{2}+|b_{23}^{(k)}|^{2}}=S(B^{(k+1)}) > (1-\epsilon)S(B)>2\sqrt{\epsilon}S(B).
\limsup_{N \to \infty}\left| \frac{1}{N}\sum_{n=1}^{N}F_{1, h}(T^{an}x) F_{2, h}(T^{bn}x) \right|^2 &\leq \frac{C}{K}+ \frac{C}{(K+1)^2}\sum_{k=1}^K (K + 1 - k) \left( \limsup_{N \to \infty}\frac{1}{N}\sum_{n=1}^{N}\left( F_{1, h}\cdot F_{1, h}\circ T^{ak}\right)(T^{an}x) \left( F_{2, h}\cdot F_{2, h}\circ T^{bk}\right) (T^{bn}x) \right).
\left(\mathcal{A}_s^N\vec{f}\right)_3(u)=\frac{1}{N}\sum_{j=1}^N\mathbb{E}S_s^N(j)\lambda\left(\frac{j}{N}, u\right)\left(-f_1\left(\frac{j}{N}\right)+f_2\left(\frac{j}{N}\right)\right)-\phi(u)f_3(u),
S\;=\; \int dt d^2x \, \left[ -h b_0{\tilde J}_0 - h b_k{\tilde J}_k+ \frac{m}{2 \rho_b}{\vec J}^2_b - \frac{\hbar^2}{8 m \rho_b}\partial_j\rho_b \partial_j \rho_b \,-\, V(\rho_b) \right] \;,
(s_*d_C - d_{Sd_r\,C}s_*)_{\widetilde{\rho},\tau,n}= (s_*)_{\widetilde{\rho},\widetilde{\rho},n-1}(d_C)_{\widetilde{\rho},\tau,n}- \sum_{\widetilde{\rho}\leqslant \rho \leqslant \tau}(d_{Sd_r\,C})_{\widetilde{\rho},\rho,n}(s_*)_{\rho,\tau,n}&= (d_C)_{\widetilde{\rho},\tau,n}- (d_C)_{\widetilde{\rho},\tau,n}= 0.
\langle 0|\varphi^{2}(x)|0\rangle =\langle 0^{(\alpha )}| \varphi^{2}(x)|0^{(\alpha )}\rangle -\frac{\sigma^{1-D}}{\pi nS_{D}r^{n}}\sum_{l=0}^{\infty}D_{l}\int_{m}^{\infty}dz\frac{z\Omega_{\alpha \nu_l}(a z, b z)}{\sqrt{z^{2}-m^{2}}}G_{\nu_l}^{(\alpha )2}(\alpha z,r z),
GP(G) = d(1,\alpha_1(1)) + d(6,\alpha_1(6)) + d(4,\alpha_2(4)) + d(7,\alpha_2(7)) + d(5,\alpha_2(5)) + d(8,\alpha_2(8)) & + d(1,\alpha_3(1)) + d(6,\alpha_3(6)) + d(4,\alpha_3(4)) + d(7,\alpha_3(7)) + d(5,\alpha_3(5)) + d(8,\alpha_3(8)) & = 2+2+2+2+4+4+2+2+2+2+4+4 & = 32.
\stackrel{\wedge}{\Sigma}_{( 2 )}( p , m , \lambda , \Lambda_{0}) = \frac{i \lambda^{2}}{1 9 2 \pi^{6}}\int_{0}^{1}d x \int_{0}^{1}d y \frac y{\left[ y C \right]^{3 / 2}}\int d^{3}k d^{3}l \frac 1{\left[ k^{2}+ l^{2}+ D p^{2}- m^{2}+ i \epsilon \right]^{3}}
\eta ( \alpha ) u^{\beta ( \alpha )}( - \vec{p}) = - ( - 1 )^{\alpha}\gamma^{2}\gamma^{4}\gamma^{5}u^{\alpha *}( \vec{p}) , \qquad e_{\mu}^{- \ell *}( - \vec{q}) = - R_{\mu \nu}e_{\nu}^{\ell}( \vec{q}) ,
u_{3,j}v_j^* - (I-bT_j) \big[ H_{3,j}T_j^* + H_{1,j}(I-bT_j^*) \big] =0, u_{4,j}v_j^* - (I-aT_j) \big[ H_{4,j}T_j^*-H_{1,j}(I-aT_j^*) \big] =0.
\left\vert\tilde{\mu}\left( v_{\Sigma}- \tilde{v}_\Sigma \right)\right\vert^2 = \left\vert \left(\mathbf{0}, \tilde{\mu}_2\right)(v_\Sigma - \tilde{v}_\Sigma)\right\vert^2= O\big(e^{-cn^{1/2}}\big)
H_\mu(\alpha_{F_i}) \, \leq \, H_\mu(\beta)+H_\mu(\alpha_{F_i}|\beta)\, \leq \, H_\mu(\beta)+H_\mu(\alpha_g|\beta) &= \, H_\mu(\beta)+H_\mu(\alpha|\beta_{g^{-1}})\, \leq \, H_\mu(\beta)+|F_i|\epsilon.
\alpha_{i,j}= \begin{cases}\pm M((q_i^1)_{\pm}) \; j = j_{\pm}(i) \; \; j_{-}(i) \not= j_{+}(i); M((q_i^1)_{+1}) - M((q_i^1)_{-1}) \; j = j_{-}(i) = j_{+}(i); 0 \; j \notin \{j_{-}(i), j_{+}(i)\}. \end{cases}
\forall i\in\{1,\ldots,n\}:\,\Bigl{(}\frac{\partial u^i}{\partial \nu}+\alpha^i u^i\frac{\partial\mathcal{V}}{\partial \nu}\Bigr{)}(t, x)=\sigma^i(t, x,u^i_{|_{\partial\Omega}}, \mathcal{V}_{|_{\partial\Omega}}),(t, x)\in[0, T]\times\partial\Omega,
\phi_{l m}= \left( \frac{2 j + 1}{2 j}\right)^{1 / 2}\phi_{l^{\prime}m^{\prime}}^{\prime}\delta_{l l^{\prime}}\delta_{m m^{\prime}}= \left( \frac{2 j + 1}{2 j}\right)^{1 / 2}\phi_{l m}^{\prime}
\mathrm{d e t}_{r e n}[ 1 - K ( \not{\! \! B}) ] = \mathrm{d e t}_{3}[ 1 - K ( \not{\! \! B}) ] \; \exp \Big ( - \frac{1}{2}\mathrm{T r}_{r e n}\big [ K^{2}( \not{\! \! B}) \big ] \Big ) \; .
\mathsf{d}_{cc}\Bigg(\prod_{l=1}^jy\big(\bigl\{X_{ni}^{(l)}\bigr\}\big)\Bigg)\leq 2^{k-1}\mathsf{d}_{\mathrm{com}}^{(j)}\big(\bigl\{X_{ni}^{(1)}\bigr\}, \dots,\bigl\{X_{ni}^{(j)}\bigr\}\big).
h^{n - 1}= \partial_{t}^{n - 1}\bar{f}+\sum_{m = 1}^{n - 1}\binom{n - 1}{m}\big(\partial_{t}^{m}\bar{a}_{ij}\big)\partial_{x_{i}}\partial_{x_{j}}\partial_{t}^{n - 1 - m}z.
N_{lm}(\delta_1,\delta_2) = \frac{(-1)^{\frac{m-|m|}{2}}}{2^{|m|}}\sqrt{\frac{(2l+\delta_1+\delta_2+1)(l-|m|)!\Gamma(l+|m|+\delta_1+\delta_2+1)}{2^{\delta_1+\delta_2+2}\>{\pi}\>\Gamma(l+\delta_1+1) \>\Gamma(l+\delta_2+1)}}.
S_{i n t}= \frac{i}{2 \pi \alpha^{\prime}}\int_{z^{2}= 0}d z^{1}\left( \frac{1}{2}\partial_{\nu}F_{\mu \lambda}\xi^{\nu}\xi^{\lambda}\partial_{1}\bar{X}^{\mu}+ \frac{1}{3}\partial_{\nu}F_{\mu \lambda}\xi^{\nu}\xi^{\lambda}\partial_{1}\xi^{\mu}\right) .
a(t)\geq0, \; t\in I^{+}_{i}, a(t)\not\equiv0 \; I^{+}_{i}, i=1,\ldots,m; & a(t)\leq0, \; t\in(\mathbb{R}/T\mathbb{Z})\setminus\bigcup_{i=1}^{m}I^{+}_{i}.
F_{_{( 1 )}}= \left[ \begin{array}{c c c c}{{0}}&{{- 1}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{3 \alpha^{2}- k}}\\{{0}}&{{0}}&{{k - 3 \alpha^{2}}}&{{0}}\\ \end{array}\right] \; .
\Psi ( p = 0 ) = \frac{c_{1}}{\sqrt 2}\left( \begin{array}{c}{{\omega + 1}}\\{{\omega - 1}}\\{{0}}\\{{0}}\\ \end{array}\right) + \frac{c_{2}}{\sqrt 2}\left( \begin{array}{c}{{0}}\\{{0}}\\{{\omega - 1}}\\{{\omega + 1}}\\ \end{array}\right) + \frac{c_{3}}{\sqrt 8}\left( \begin{array}{c}{{\omega - 1}}\\{{\omega + 1}}\\{{\lambda ( \omega + 1 )}}\\{{\lambda ( \omega - 1 )}}\\ \end{array}\right) ,
\sum_{\ell=1}^{N}\bigg(\frac{2\rho_{\ell}\mu_{\ell}}{p_\ell}+\frac{c_\ell p_\ell}{\mu_\ell}\bigg) \le \sum_{\ell=1}^{N}\bigg(\frac{2\rho_{\ell}\mu_{\ell}}{f_\ell^\star}+\frac{c_\ell f_\ell^\star}{\mu_\ell}\bigg).
\left. \int \, e^{i q A_{b}}\, G ( A ) \, d \mu_{I_{\beta}}\, ( A ) = e^{- \tilde{E}_{\beta}( a , q )}\, \int \, e^{i q A_{b}}\, \left[ \prod_{p : b \in \partial p}\, i_{\beta}\, ( \epsilon ( b , p ) a , d A ) \right] \, G ( A ) \, d \mu_{I_{\beta}}\, ( A ) \ , \right.
\frac 12\frac{d}{dt}\int_{\mathbb{R}^2}(|V^{(\alpha,a)}|^2+|H^{(\alpha,a)}|^2)dx &-\int_{\mathbb{R}^2}\mu\Delta\sum\limits_{l=0}^{\alpha}C_\alpha^l (-1)^{\alpha-l}V^{(l,a)}\cdot V^{(\alpha,a)}dx\lesssim \langle t\rangle^{-\frac 32}E_{\kappa-3}E_{\kappa-1}^{\frac 12}.
\overline{v}=\frac{R_{0}}{\int_{0}^{R_{0}}\frac{dR}{v}}=\frac{R_{0}}{\int_{0}^{R_{0}}\frac{dR}{\sqrt{\frac{P}{\rho}}\sqrt{\frac{2}{3}}\sqrt{\frac{R_{0}^{3}}{R^{3}}-1}}}=\frac{\sqrt{\frac{2}{3}}}{\int_{0}^{1}\frac{dx}{\sqrt{\frac{1}{x^{3}}-1}}}\sqrt{\frac{P}{\rho}}=1.09328\sqrt{\frac{P}{\rho}}
\frac{1}{2\pi i}\int_{\sigma_0 -iT}^{\sigma_0 + iT}\alpha(s) \frac{x^s}{s}ds=\sum_{n=1}^{\infty}a_n\frac{1}{2\pi i}\int_{\sigma_0 -iT}^{\sigma_0 + iT}\frac{(x/\l_n)^s}{s}ds.
Z_{\tau_t}= X_{\tau_t}+ \frac{1}{\dot \beta_{\tau_t}}\dot X_{\tau_t}= Y_t + \frac{1}{\dot \beta_{\tau_t}\dot \tau_t}\dot Y_{t}= Y_t + \frac{1}{\dot{\tilde \beta}_{t}}\dot Y_{t}
||\dot{{\mathbf U}}||_{s,\ast,T}+||\varphi||_{H^s(\Gamma_T)}\leq C(K_0)\Big(||\mathbf{F}||_{s,\ast,T}+||\mathbf{F}||_{6,\ast,T}||\hat{W}||_{s+4,\ast,T}\Big)\,,
C_{\theta}( \tau ) : = \lim_{\tau \rightarrow \infty}\Big [ \langle{\cal A}( \tau ){\cal B}( 0 ) \rangle_{0}^{\theta}- \langle{\cal A}( 0 ) \rangle_{0}^{\theta}\langle{\cal B}( 0 ) \rangle_{0}^{\theta}\Big ] .
4 . ( \frac{1}{2}+ \frac{1}{3}+ \cdots + \frac{1}{2 0 0 6}) ( 1 + \frac{1}{2}+ \cdots + \frac{1}{2 0 0 5}) - ( 1 + \frac{1}{2}+ \cdots + \frac{1}{2 0 0 6}) ( \frac{1}{2}+ \frac{1}{3}+ \cdots + \frac{1}{2 0 0 5})
g(z)=\sum_{m=0}^{\infty}a_mz^{q(m+1)}=\sum_{n=1}^{\infty}a_{n-1}z^{qn}=\sum_{n=0}^{\infty}b_nz^{qn},b_n=\begin{cases}a_{n-1}, n\geq 1, 0 , .\end{cases}
\int\limits_{Q_T}\frac{f(\cdot,\cdot,u_\varepsilon,\nabla\varphi_\varepsilon)}{1+\varepsilon f(\cdot,\cdot,u_\varepsilon,\nabla \varphi_\varepsilon)}\,z\,dxdt-\int\limits_{Q_T}f(\cdot,\cdot,u,\nabla\varphi)z\,dxdt=J_{1,\varepsilon}+ J_{2,\varepsilon}+J_{3,\varepsilon}
I^{(2)}(\omega) = \int_{s_1}^{s_2}U\big( \varphi(s) \big) \, \varphi'(s) \, e^{i \omega s}\, ds & = (i \omega)^{-1}\bigg( \Big[ (U \circ \varphi)(s) \, \varphi'(s) \, e^{i \omega s}\Big]_{s_1}^{s_2}- \int_{s_1}^{s_2}\big( (U \circ \varphi) \, \varphi' \big)'(s) \, e^{i \omega s}\, ds \bigg) \; ;
\nabla_{\vec{x}}V(\vec{x}) =\left[\dfrac{\partial f}{\partial \vec{x}}(\vec{z}^{*}(\vec{x}),\vec{x})\right]^{\top}+\left[\dfrac{\partial \vec{g}}{\partial \vec{x}}(\vec{z}^{*}(\vec{x}), \vec{x})\right]^{\top}\boldsymbol{\lambda}^{*}(\vec{x}),
\Lambda[0](\mu,c)(x){=}\frac{1}{2}\mu(x) +W_{q}^\ast[\omega,\mu](x) +\tilde b(x) c \forall x \in \partial \Omega_Q\, , \forall (\mu,c) \in C^{m-1,\alpha}(\partial \Omega_Q,\mathbb{R}^n)_0\times \mathbb{R}^n\, ,
\mathcal{T}\exp \Big(\int_{t}^{T}H(s)\, \mathrm{d}s\Big)=\sum_{k=0}^{\infty}\int_{t}^{T}\int_{t_1}^T\ldots\int_{t_{k-1}}^{T}H(t_{1})\ldots H(t_{k})\, \mathrm{d}t_{k}\ldots\, \mathrm{d}t_{1}.
V^{(m,n)}_{\texttt{a};\lambda}(\boldsymbol{\xi}) := \sum_{1\le j < k \le n}\int_0^{\xi_j-\xi_k}v_q(\vartheta)\vartheta + \sum_{1\leq j\leq n}\left({\textstyle \frac{m}{2}}\xi_j^2-2\pi (\varrho_{\texttt{a};j}+\lambda_j)\xi_j \right) ,
R^{CP}_{m,i}= \log\left(1+\frac{\rho \alpha_i^2\frac{1}{{L\left(||x_m-x_0||\right)}}}{\rho \frac{1}{{L\left(||x_m-x_0||\right)}}\sum^{M_s}_{j=i+1}\alpha_j^2+1}\right).
\begin{array}{l}{{E_{\mathrm{v i b}}= \displaystyle h c \omega_{e}( v +{\frac{1}{2}}) - h c \omega_{e}x_{e}( v +{\frac{1}{2}})^{2}+ h c \omega_{e}y_{e}( v +{\frac{1}{2}})^{3}+ \cdots ~ ,}}\\{{E_{\mathrm{r o t}}= B h c J ( J + 1 ) - D h c J^{2}( J + 1 )^{2}+ H h c J^{3}( J + 1 )^{3}+ \cdots ~ .}}\\ \end{array}
\begin{array}{rcl}\Phi\left(S^2(Z)\right) = e_1 \widetilde C Z +e_0 \widetilde C W \Phi\left(S^2(Z,W)\right) = -e_2\widetilde C Z -e_1\widetilde C W \Phi\left(S^2(W)\right) = e_3\widetilde C Z +e_2\widetilde C W, \end{array}
L_f(s) =\prod_{p|N}(1-\lambda_f(p)p^{-s})^{-1}\prod_{p\nmid N}(1-\lambda_f(p)p^{-s}+p^{-2s})^{-1}&=\prod_{p|N}(1-\lambda_f(p)p^{-s})^{-1}\prod_{p\nmid N}(1-\alpha_f(p)p^{-s})^{-1}(1-\beta_f(p)p^{-s})^{-1}.
\|-\Delta_{g}Z_{\delta_{\varepsilon}(t),\eta}^{k}+hZ_{\delta_{\varepsilon}(t),\eta}^{k}-f_{\varepsilon}'(W_{\delta_{\varepsilon}(t),\eta})Z_{\delta_{\varepsilon}(t),\eta}^{k}\|_{\frac{2m}{m+2}}=O(|\varepsilon|^{\theta}).
0 = \int_\Omega (-\Delta w_0) \frac{\psi_1(0)^2}{w_0}= - \int_\Omega \left\vert \frac{\psi_1(0)}{w_0}\nabla w_0 - \nabla \psi_1(0) \right\vert^2 + \int_\Omega |\nabla \psi_1(0)|^2 - \int_{\partial \Omega}b w_0^{q-2}\psi_1(0)^2,
x_{\Delta}=\sqrt{\frac{A}{4\pi}}\ \{\rm with}\ \A= 2\pi \int_{0}^{\pi}d\theta\sin\theta x_h^2 \frac{\sqrt{l_2(x=x_h,\theta)m_2(x=x_h,\theta)}}{f_2(x=x_h,\theta)}\ .
(f_2\psi_M)^1(\gamma)(m) = f_2^0(\psi_M(\gamma)(m)) + \sum_{i,j=1}^{n}f_2^1(\gamma_j\gamma(p_jp_i))\psi_M(\gamma_i)(m) = f^0(m w_{1,2}\gamma) + \sum_{i,j=1}^{n}f^1(w_2 \gamma_j\gamma(p_jp_i))(m w_{1,2}\gamma_i) = f^0(m w_{1,2}\gamma) + \sum_{i,j=1}^{n}f^1(w_2 \gamma_j)(m w_{1,2}\gamma_i)\gamma(p_jp_i);
\sum_{s=1}^{q}\sum_{j,k=1}^{n}\frac{\partial^{2}h_{A}(z)}{\partial z_{j}\partial\overline{z_{k}}}(t^{s})_{j}\overline{(t^{s})_{k}}\geq A\;, \;t^{1}, \hdots, t^{q}\;\;\mathbb{C}^{n}
\psi^{m_j}_{j-1/2}:=\frac{1}{\sqrt{2j}}\left(\begin{array}{c}\sqrt{j+m_j}\,Y^{m_j-1/2}_{j-1/2}\sqrt{j-m_j}\,Y^{m_j+1/2}_{j-1/2}\end{array}\right), \psi^{m_j}_{j+1/2}:=\frac{1}{\sqrt{2j+2}}\left(\begin{array}{c}\sqrt{j+1-m_j}\,Y^{m_j-1/2}_{j+1/2}-\sqrt{j+1+m_j}\,Y^{m_j+1/2}_{j+1/2}\end{array}\right);
\frac{1}{4\pi r^2}\left\vert \int_{B(z,r)}\Delta_x f\,dx\right\vert = \frac{1}{r^2}\left\vert \int_{B(z,r)}\Big(1-\sum_{\substack{k\in\mathbb{Z}^3 |k|_\infty \leq m-1}}e^{i(2\pi k-\eta)\cdot x}\Big)\,dx\right \vert \leq \frac{4\pi[(2m-1)^3+1]}{3}\,r,
{\mathbf{A}}=\mathbf{Anti}(\vartheta_1,\vartheta_2,\vartheta_3):=\begin{footnotesize}\begin{pmatrix}0 -\vartheta_3 \vartheta_2 \vartheta_3 0 -\vartheta_1 -\vartheta_2 \vartheta_1 0\end{pmatrix}\end{footnotesize}\in \mathfrak{so}(3),
\Phi_G^{+}=\{2\,\omega_1, -3\, \omega_1+\omega_2,-\omega_1+\omega_2,\omega_1+\omega_2,2\,\omega_2,3\, \omega_1+\omega_2\}\Phi_K^{+}=\{2\,\omega_1,2\,\omega_2\}\,.
\left[ J^{\alpha \beta}( x ) , S_{+}^{\alpha^{\prime}\beta^{\prime}}( y ) \right] = \delta ( x - y ) \left( \delta^{\beta \alpha^{\prime}}S_{+}^{\alpha \beta^{\prime}}( x ) + \delta^{\beta \beta^{\prime}}S_{+}^{\alpha \alpha^{\prime}}( x ) \right) ~ ~ ,
0 < \, \mathbf{x}\cdot ( \mathbf{e}_{1}^{+}+ \mathbf{e}_{1}^{-}+ \mathbf{e}_{2}^{+}+ \mathbf{e}_{2}^{-}) = 4 \cos ( \tau ) \cos ( \omega_{-}) \big ( x_{1}\cos ( \omega_{+}) - x_{2}\sin ( \omega_{+}) \big )
\eta(s,t,x_{1},x_{2}) = \eta(D^{2}s,D^{3}t,Dx_{1},Dx_{2}) = \frac{h(X+P)}{h(X)}+\frac{h'(X+P)}{h'(X)}= e_{m}\left(P,\left[\tfrac{n}{m}\right]Q\right)+e_{m}\left(P,-\left[\tfrac{n}{m}\right]Q\right)=\zeta_{m}+\zeta_{m}^{-1}.
\sum_{i=a_{j_{0}}+1}^{a_{j_{0}+1}-1}p_{A,i}\left(C\right)\left\langle Y,B,i\right\rangle +\sum_{i=a_{j_{0}+1}+1}^{a_{j_{0}+2}-1}p_{A,i}\left(C\right)\left\langle Y,B,i\right\rangle +\cdots+\sum_{i=a_{k-1}+1}^{n}p_{A,i}\left(C\right)\left\langle Y,B,i\right\rangle =0,
C^{-1}(\{M^2 /2\})\cap U_{0}=C^{-1}(\{M^2 /2\})\cap\left(\{(x,y,z)\in\mathbb{R}^3 : x=0\}\cup \{(x,y,z)\in\mathbb{R}^3 : y=0\}\right) &=\mathcal{EC}^{-1}(\{(0,M^2 /2)\})\subset C^{-1}(\{M^2 /2\})\cap U_{\beta},~\forall \beta\in (0,\beta_0),
\frac{(-1)^NN!}{(-\lambda-\frac{n-1}{2})_N}C_{2N}^{-\lambda-\frac{n-1}{2}}(x) = \sum_{j=0}^N (-1)^ja_j^{(N)}(\lambda)x^{2N-2j}, \frac{(-1)^NN!}{2(-\lambda-\frac{n-1}{2})_{N+1}}C_{2N+1}^{-\lambda-\frac{n-1}{2}}(x) = \sum_{j=0}^N (-1)^j b_j^{(N)}(\lambda)x^{2N+1-2j}.
\left. \begin{array}{l}{{\gamma = x ( \partial \Sigma )}}\\{{\hspace{3 m m}={\gamma_{\mu}}^{\prime}\circ{\gamma_{\mu - 1}}^{\prime}\circ \cdots \circ{\gamma_{3}}^{\prime}\circ{\gamma_{2}}^{\prime}\circ \gamma_{g + 1}\circ \gamma_{g}\circ \cdots \circ \gamma_{2}\circ \gamma_{1}}}\\ \end{array}\right.
\begin{array}{ccc}\left(\begin{array}{c}a b \varepsilon^{i}\rho^{i}\lambda \omega_{AB}T_{i}^{~j}\end{array}\right)~~~~ \longrightarrow ~~~~\left(\begin{array}{c}b a \rho^{i}\varepsilon^{i}-\lambda \omega_{AB}T_{i}^{~j}\end{array}\right)\end{array}
\frac{\partial}{\partial x^{\mu}}\psi^{q}= - i D_{x}( \hat{A}) \left( D_{x}^{\dagger}( \hat{A}) D_{x}( \hat{A}) \right)^{- 1}\sigma_{\mu}^{\dagger}\psi^{q}+ \psi^{r}R_{\mu}^{r q}( x ) ,
R_{a}^{+}u_{a}^{+}= e^{+ i \theta / 2}u_{a}^{+}\hspace{0 . 5 c m}R_{b}^{-}u_{b}^{-}= e^{+ i \theta / 2}u_{b}^{-}\hspace{0 . 5 c m}R_{a}^{+}v_{a}^{+}= e^{+ i \theta / 2}v_{a}^{+}\hspace{0 . 5 c m}R_{b}^{-}v_{b}^{-}= e^{+ i \theta / 2}v_{b}^{-},
G^{\epsilon}(z,\bar{z})=\lim_{|z_3 \bar{z_3}| \rightarrow \infty}|z_3\bar{z}_3|^{2\Delta_{\alpha_3}}\langle \phi_{\alpha_3}\left(z_3,\bar{z}_3\right)\phi_{\alpha_2}\left(1\right)R^{\epsilon}_\alpha \left(z,\bar{z}\right)R^{\epsilon}_{\alpha_1}\left(0\right) \rangle
1\, =\,\frac{1}{\lambda}A(k,k)+\sum_{i\neq k}\sum_{i'=1}^N\frac{x_{i'}}{\lambda^2 x_k}A(i',i)A(i,k)\cr =\,\frac{1}{\lambda}A(k,k)+\sum_{i\neq k}\frac{1}{\lambda^2}A(k,i)A(i,k)\,+\,\sum_{i,i'\neq k}\frac{x_{i'}}{\lambda^2 x_k}A(i',i)A(i,k)\,.
\widehat F(s,u,t)\cdot\widehat F(t,v,r) =(\widehat F(s),F_m(e_s,u,f_t),\widehat F(t))\cdot (\widehat F(t),F_m(e_t,v,f_r),\widehat F(r)). &=(\widehat F(s),F_m(e_s,x,e_t)F_m(e_t,v,f_r),\widehat F(r)).
(N+\delta+1-n)R_{n}(x)-(N-n) R_{n+1}(x) =\frac{(x+\gamma+1)(x+\delta+1)}{(\gamma+1)}\hat R_n(x), (n+1) \hat R_n(x)- (n+\gamma+2) \hat R_{n+1}(x) = -(\gamma+1) R_{n+1}(x).
\Psi = \mathrm{T r}\int \left( \bar{c}\, \, d^{\ast}\! \! \, A + \bar{b}\wedge \, \, d^{\ast}\, \! B + \bar{a}\, \, d^{\ast}\! \, \Omega + \bar{\theta}\, \, d^{\ast}\, \! b - \, d^{\ast}\, \! \bar{b}\wedge \Xi \right)
L((A,A')),A'') = \begin{cases}(-1)^{\tau(A, A \cup A')(t+1)}f_{A,S}(A' \setminus A) \mbox{if $A'' = A$}(-1)^{\tau(A', A \cup A')(t+1)+1}f_{A',S}(A \setminus A') \mbox{if $A'' = A'$}0 \mbox{otherwise}, \end{cases}
\frac{\mathrm{d}}{\mathrm{d}t}\bar{\boldsymbol u}= \frac{1}{|\Omega|}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\boldsymbol u \mathrm{d}x = \bar{\theta}\boldsymbol e_2, \frac{\mathrm{d}}{\mathrm{d}t}\bar{\theta}= \frac{1}{|\Omega|}\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Omega}\theta \mathrm{d}x = \bar{f},
I(z) = \frac{{(n - 1)!}}{{(2\pi z)^{n}}}\sum_{m = 1}^\infty{\frac{1}{{m^{n + 1}}}}- \frac{(n - 1)!}{(2\pi z)^n}\sum_{j = 0}^{n - 1}{\frac{(2\pi z)^{j}}{j!}\sum_{m = 1}^\infty{\frac{e^{- 2\pi m z}}{m^{n - j + 1}}}}.
S_{i j}^{k \ell}\left( \begin{array}{c c}{a}&{d}\\{b}&{c}\\ \end{array}\right) :{\cal V}_{a , b}( \theta_{1}) \otimes{\cal V}_{b , c}( \theta_{2}) \rightarrow{\cal V}_{a , d}( \theta_{2}) \otimes{\cal V}_{d , c}( \theta_{1})
\Big [ ( \sum_{n = 1}^{\infty}R_{2 k - 1 , 2 n}x_{2 n}^{\mu}) , ( \sum_{m = 1}^{\infty}p_{2 m}^{\nu}T_{2 m , 2 l - 1}) \Big ]_{\star}= i \delta_{k , l}\eta^{\mu \nu}
X =H_{0}^{s}\left( \Omega\right) V=L^{2}(\Omega) \varphi^{1}(u) =\frac{\gamma}{2}\left\Vert u\right\Vert_{L^{2}(\Omega)}^{2}+\frac{1}{2}\iint_{Q}\frac{|u(x)-u(y)|^{2}}{|x-y|^{d+2s}}\mathrm{d}x\mathrm{d}y \varphi^{2}(u) = 0 f(u)=g.
\partial_{x}(A_{0}(\textbf{w}+\bar{\textbf{u}})\partial_{t}\textbf{w}+ A_{1}(\textbf{w}+\bar{\textbf{u}})\partial_{x}\textbf{w}) \cdot \partial_{x}\textbf{w}=[\partial_{x}(A_{0}D)\textbf{w}+ (A_{0}D)\partial_{x}\textbf{w}] \cdot \partial_{x}\textbf{w}.
a=\mathrm{D}^-_+\mathrm{d}t+\mathrm{D}^-_i\mathrm{dA}^i+\mathrm{D}^k_i\mathrm{dN}^i_k+\mathrm{D}^k_+\mathrm{dA}^\star_k \equiv \mathrm{D}^\kappa_\iota\mathrm{d}\Lambda^\iota_\kappa,