harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}-C\leq(1-x)^{-s}I_3'-(1-x)^{-s}\int_{0}^{1}\frac{\left[1-\left(1-k\right)^s\right]^{p-1}-\left[\left(1+k\right)^s-1\right]^{p-1}}{k^{1+sp}}\mathrm{d}k\leq C, \end{aligned}
B'\begin{pmatrix}a 0 0 b \end{pmatrix}= X_{4m+4n}B \begin{pmatrix}a 0 0 b \end{pmatrix}X_{4m+4n}^* = \begin{pmatrix}X_{4m}B(a) X_{4m}^* 0 0 X_{4n}B(b) X_{4n}^* \end{pmatrix}= \begin{pmatrix}B'(a) 0 0 B'(b) \end{pmatrix}\, .
\begin{alignedat}{2}f(x_1) = a_1x_1 f(x_2) = a_2x_2 f(y_i) = b_iy_i f(v_j) = A (v_j) + a_1 (j) x_1^5 + a_2 (j) x_2^4, \, \, \, A \in{\rm{GL}}(M^\ast) f(z) = cz + y_1A_1 + y_2A_2 + y_3A_3 + y_1y_2y_3D, \,\, \, A_1, A_2, D \in \mathbb Q[x_1, x_2, v_1 \ldots, v_n] \end{alignedat}
L^x_{s,t}=\int_0^s\int_{1-t}^1\mathbf{1}_{\{W^{(0)}_{\xi,u}\leq x\}}\frac{\mathrm{d}_uW^{(0)}_{\xi,u}}{\xi}\mathrm{d}\xi+\int_0^s\int_{0}^t\mathbf{1}_{\{\widehat{W}^{(0)}_{\xi,u}\leq x\}}\frac{\mathrm{d}_u\widehat{W}^{(0)}_{s,u}}{\xi}\mathrm{d}\xi,\quad\forall\,(s,t)\in[0,1]^2.
\sum_{m}\frac{u_m(z)u_m(z')}{m^2+p^2}\longrightarrow \int_{0}^{\infty}dm~\lim_{z_c \to \infty}\frac{1}{\pi k e^{-kz_c}}\left(\frac{u_m(z)u_m(z')}{m^2+p^2}\right)
G_2^{\mathfrak H/\overline{\varGamma}_0(4)}(z)={}-\frac{1}{3}\log\frac{2^4|1-\alpha_{4}(z)|^{2}}{|\alpha_{4}(z)|}=-8\log\left\vert \frac{\eta(z)}{\eta(2z)}\right\vert,\forall z\in\mathfrak H.
\epsilon_{i,j}(v) = v_{i,j}, \alpha_k(v) = 2\epsilon_{1,k}(v) - \epsilon_{2, k+1}(v), \gamma_k(v) = \sum_{j \leq k}(3\epsilon_{0,j}(v) - 3\epsilon_{1, j+1}(v) + \epsilon_{2, j+2}(v)).
\bar{Y}^N_t = e^{t A}\, \xi + \int_0^t e^{( t - s ) A}\, F\big( \tilde{Y}^N_{\lfloor s \rfloor_{T / N}}\big) \, ds + \int_0^t e^{( t - s ) A}\, B\big( \tilde{Y}^N_{\lfloor s \rfloor_{T / N}}\big) \, dW_s ,
\mathcal{U}^\top\mathcal{U}= \begin{bmatrix}\hat{A}- A 0 0 \hat{S}\end{bmatrix}\mathcal{V}^\top \mathcal{V}= \begin{bmatrix}\hat{A}- A + B^\top \hat{S}^{-1}B B^\top \hat{S}^{-1}(\hat{S}- C) (\hat{S}- C)\hat{S}^{-1}B (\hat{S}- C)\hat{S}^{-1}(\hat{S}- C) \end{bmatrix},
\sigma_{ab}=\tilde{\sigma}_{ab}r^2+O(r^3), \,\, l_{ab}=-\tilde{\sigma}_{ab}r+O(r^2), \,\, n_{ab}=\frac{1}{2}\tilde{\sigma}_{ab}r+O(r^2), \,\, \eta_{a}=\frac{1}{3}{\beta}_a r^2+O(r^3)
d_i \leq \sup_{1\leq k\leq i_0-1}d_k+ c_{28}\sum_{k=1}^i\frac{k^{4+6(1-\gamma )}}{2^{(1-\gamma )k}}\leq \sup_{1\leq k\leq i_0-1\atop r \le \kappa^{-1}R/2}d_k(r)+ c_{28}\sum_{k=1}^{\infty}\frac{k^{4+6(1-\gamma )}}{2^{(1-\gamma )k}}=:c_{29}<\infty.
1 \geq\int_X\Phi_2(\frac{u(x). f_n(x)}{\parallel M_uf_n\parallel_{\Phi_2}})d\mu &=\int_{A_n}\Phi_2(\frac{u(x).\Phi_1^{-1}(\frac{1}{\mu(A_n)})}{\parallel M_uf_n\parallel_{\Phi_2}})d\mu &=\Phi_2(\frac{u(A_n).\Phi_1^{-1}(\frac{1}{\mu(A_n)})}{\parallel M_uf_n\parallel_{\Phi_2}})\mu(A_n).
\mu(e_1,e_1)= \mu(e_1,e_2)=e_1,\mu(e_2,e_1)=\mu(e_2,e_2)=e_2, \tilde{\alpha}(e_1 )=e_1, \tilde{\alpha}(e_2)= e_2, \tilde{\beta}(e_1 )=e_1, \tilde{\beta}(e_2)= e_1.
\mathfrak{M}= [\mathfrak{B}_{1}, \mathfrak{C}_{1}];\ \mathfrak{C}_{1}= [\mathfrak{B}_{2}, \mathfrak{C}_{2}];\ \dots ;\ \mathfrak{C}_{\nu - 1}= [\mathfrak{B}_{\nu}, \mathfrak{C}_{\nu}];\ \dots
E[X_{L}^n] \approx \hat{E}^{N_{0},...,N_{L}}[X_{L}^{n}], \hat{E}^{N_{0},...,N_{L}}[X_{L}^{n}]:\hat{E}^{N_{0}}[X_{0}^n] + \sum_{l=1}^L \hat{E}^{N_{l}}[X_{l}^n-X_{l-1}^n],
E[(S_n)^2] \sim \begin{cases}\dfrac{n}{1-2\alpha}(-1<\alpha<1/2), begin{align*}3mm]n\log n (\alpha=1/2), begin{align*}1mm]\dfrac{n^{2\alpha}}{(2\alpha-1)\Gamma(2\alpha)}(1/2<\alpha<1).\end{cases}
\|f-\frac{p_{n-1}}{q_n}\|_2^2=\|f\frac{q_n}{\widetilde{q}_n}-\frac{p_{n-1}}{\widetilde{q}_n}\|_2^2=\|{\bf P}_-(f\frac{q_n}{\widetilde{q}_n})\|_2^2+\|{\bf P}_+(f\frac{q_n}{\widetilde{q}_n})-\frac{p_{n-1}}{\widetilde{q}_n}\|_2^2.
Q(z)=(-1)^k\sum_{1\leq s_1<s_2<\cdots<s_k\leq \mu}T_{s_1,\ldots,s_k}V(z,z_{s_1},z_{s_2},\ldots,z_{s_k})\bigg[\prod^k_{i=1}\Psi_p(z_{s_i})\bigg]^{-1}.
[\ell_{\mu}^+,\ell_{\nu}^+]=[\ell_{\mu}^-,\ell_{\nu}^-]=0,\quad [\ell^-_{\mu},\ell^+_{\nu}]=2\omega\left(\mu\cdot\nu+ \sum_{\rho\in\Delta_{+}}g_{|\rho|}\,(\rho\cdot\mu)\,(\rho^\vee\!\cdot\nu)\check{s}_{\rho}\right).
\partial_t(\frac{|\hat{U}|}{|\xi|})+\nu |\xi||\hat{U}|=Re(\frac{\hat{A}}{|\xi|}\cdot\frac{\bar{\hat{U}}}{|\hat{U}|}+\frac{\hat{\mathbb{P}F}}{|\xi|}\cdot\frac{\bar{\hat{U}}}{|\hat{U}|})
^0 \langle \, m \neq p \mid p \, \rangle = \{\sum_{n}\!^{\prime}\, [ 1 + Q ( H_{0}- E_{p})^{- 1}Q V Q ]_{m n}^{- 1}\frac{V_{n p}}{( E_{p}- E_{n}^{0})}\}^{0}\langle \, p \mid p \, \rangle
R^{i}_{k}:=\frac{1}{F^2}\big(2\frac{\partial G^i}{\partial x^k}-\frac{\partial^2 G^i}{\partial x^j \partial y^k}y^j +2G^j\frac{\partial^2 G^i}{\partial y^j \partial y^k}- \frac{\partial G^i}{\partial y^j}\frac{\partial G^j}{\partial y^k}\big).
a_1a_2a_3~ e_1e_2e_3~i_1i_2i_3 = (a_1e_1i_1)(a_2e_2i_2)(a_3e_3i_3) a_1a_2a_3 ~e_1e_2f_3~i_1i_2h_3 = (a_1e_1i_1)(a_2e_2i_2)(a_3f_3h_3) a_1a_2a_3 ~e_1f_2f_3~i_1h_2h_3 = (a_1e_1i_1)(a_2f_2h_2)(a_3f_3h_3).
\sum_{\mu: \ \mu_1=a+d+1}m_{\mathcal{M}(\alpha)}(\mu) =\sum_{\substack{(i,j)\in S_\alpha: i\geq 2j\geq 2}}2^{\max\{i-1,0\}+\max\{j-1,0\}}&+\chi((0,0)\in S_\alpha)+\chi((1,0)\in S_\alpha)+\chi((0,1)\in S_\alpha)+\chi((1,1)\in S_\alpha)
(a) \dfrac{d^2f}{dX^2_u}\leq 0,~~~~~\forall i\in I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (b) \dfrac{d^2f}{dY^2_u}\leq 0,~~~~~\forall i\in I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ (c) \dfrac{d^2f}{dX^2_u}\dfrac{d^2f}{dY^2_u}-(\dfrac{d^2f}{dXdY_u})^2\geq 0,~~~~~\forall i\in I~~~~~~~~~~~~~
[z^2-(\kappa^{(t)})^2]\tilde{\cal H}_{2k-2}^{(s,t+1)}(z)=\tilde{\cal H}_{2k}^{(s,t)}(z)+Q_{k}^{(s,t)}\tilde{\cal H}_{2k-2}^{(s,t)}(z), &[z^2-(\kappa^{(t)})^2]\tilde{\cal H}_{2k-1}^{(s,t+1)}(z)=\tilde{\cal H}_{2k+1}^{(s,t)}(z)+Q_{k}^{(s+1,t)}\tilde{\cal H}_{2k-1}^{(s,t)}(z),
\dim (\partial_A S_i \varphi^{-1}(S_i )) = \dim (S_i \partial_A \varphi^{-1}(S_i )) &= \dim (\partial_A \varphi^{-1}(S_i )) \leq \dim ([\partial_A W_1 ]_{R_{\overline{W},E}}) < \dim (A)
Z_{N}= -{\frac{\beta^{N + 1}}{\pi \omega x}}\int_{0}^{\infty}{\frac{d s}{s}}\, \Im \, \widetilde Z \oint_{C_{0}}{\frac{d w}{2 \pi i}}{\frac{1}{w^{N + 1}( w - \beta )}}F_{0}( w , s ) ,
\begin{array}{l l}{d s_{6}^{2}=}&{e^{2 \lambda}\left( d y^{2}+ d z^{2}\right) + e^{2 \varphi}\left( - d \tau^{2}+ \tau^{2}d \Omega_{3 , - 1}^{2}\right) ,}\\{H_{e}=}&{i d \left( e^{2 \lambda}d y \wedge d z \right) .}\\ \end{array}
\lim\limits_{x\to\infty}\frac{(\ell^{\leftarrow}_{\beta}(x))^{\alpha}}{h(\ell^{\leftarrow}_{\beta}(x))}\P\big(\eta_K(\varepsilon_1)>x\big)=\gamma_2\quad\lim\limits_{x\to\infty}\frac{(\ell^{\leftarrow}_{\beta}(x))^{\alpha}}{h(\ell^{\leftarrow}_{\beta}(x))}\P\big(\eta_K(\varepsilon_1)<-x\big)=\gamma_1,
\Gamma_b = \{(c_j; b+(j-w)e) \mid j \in [1,\, r] \}\cup \{(b-1;b-1-ie) \mid i \in [0,\, \tilde{w}) \}; \Delta_b = \{(b;b-ie) \mid i \in [0,\, w) \}\cup \{(c'_j; b-1 + (j-\tilde{w})e) \mid j \in [1,\, r-l] \}
C \geq \displaystyle\sum\limits_{j = 0}^{\infty}\; p (1-p)^j \max_{p(x): |X|^2 \le \mathcal{E}_j}I(X; Y)- H(p) & \;\;\; \displaystyle\sum\limits_{j = 0}^{\infty}\mathcal{E}_j \le B_{max}
f_{i,j+k}(L,L+M+N) + f_{j,i+k}(M,L+M+N) + f_{k,i+j}(N,L+M+N) - 2f_n(L+M+N) \leq \dim (U_1+U_2+U_3) + \dim (U_1+U_2+U_3) + \dim (U_1+U_2+U_3) - 2(\dim (U_1+U_2+U_3)) = \dim (U_1+U_2+U_3) = f_{i,j,k}(L,M,N).
\partial\cdot\partial=\partial^{\mu}\cdot\partial^{\nu}=\partial^{\mu}\eta_{\mu\nu}\partial^{\nu}=\partial_{\nu}\partial^{\nu}=\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\vec{\nabla}^{2}=(\frac{\partial_{t}}{c})^{2}-\vec{\nabla}^{2}.
f_{\mathcal{G}}(x \mid m, s)= \frac{1}{s}\, e^{(x-m)/s}\, e^{- e^{(x-m)/s}}= \frac{1}{s}\, e^{\frac{x}{s}}\, e^{-\frac{m}{s}}\, e^{- e^{\frac{x}{s}}\, e^{-\frac{m}{s}}}
\sum_{k=0}^{\infty}\frac{\binom{2k}{k}^2}{32^k}\bigg\{H_{2k}^{(2)}-\frac{1}{4}H_{k}^{(2)}\bigg\}=\Gamma\bigg(\frac{1}{4}\bigg)^2\frac{\pi^2-8G}{32\pi\sqrt{\pi}}.
\epsilon^{( 1 )}= e^{{\frac{g \eta t}{\sqrt{2}}}\epsilon_{i j}}{\cal H}^{- 1 / 8}\left[ \sqrt{f^{1 / 2}+ 1}- \sqrt{f^{1 / 2}- 1}\gamma_{\overline{{{r}}}}\right] e^{{\frac{\eta}{2}}\gamma_{\overline{{{0 \theta r}}}}\epsilon^{i j}}e^{-{\frac{1}{2}}\gamma_{\overline{{{\phi \theta}}}}}( 1 - \eta \gamma_{\overline{{{0}}}}\epsilon^{i j}) \epsilon_{0}^{( 1 )}.
u(A_r(Q)) \leq \frac{a_2\gamma_2}{\tilde c_2(1+\tilde c_2)^{k-1}}\left(\frac{2}{\kappa}\right)^{2k\delta_2}u(A_{k})=\frac{a_2(1+\tilde c_2)}{\tilde c_2}\left(\frac{2}{\kappa}\right)^{k\beta}u(A_{k}),
H = - \frac{e}{2}\pi^{2}+ \imath \pi_{\mu}\psi^{\mu}\chi - \left( \frac{1}{8 !}\epsilon_{\nu \mu \rho_{2}\ldots \rho_{9}}\pi^{\mu}\psi^{\rho_{2}}\psi^{\rho_{9}}+ \frac{\alpha}{2^{8}}\pi_{\nu}\right) b^{\nu}\; .
- \det ( \eta_{a b}I + F_{a b}) = I + \frac{1}{2}F^{2}+ \frac{1}{3}F^{3}- \frac{1}{4}\left[ F^{4}- \frac{1}{2}( F^{2})^{2}\right] + \frac{1}{5}F^{5}+ \frac{1}{1 2}( F^{2}F^{3}+ F^{3}F^{2}) ,
\texttt{C}_{(1^b2)}(\Delta)=-H_{\mathbb{RP}^2}(d; \underbrace{\Gamma,\dots,\Gamma}_{b+2},\Delta)+2H_{\mathbb{RP}^2}(d;\underbrace{\Gamma,\dots,\Gamma}_{b},(1^{d-4}2^2),\Delta)+ 2H_{\mathbb{RP}^2}(d;\underbrace{\Gamma,\dots,\Gamma}_{b},(1^{d-3}3^1),\Delta)
\left\langle \vec{w},\Upsilon \left( t\right) \vec{w}\right\rangle_{\mathbb{R}^{d}}=\int\nolimits_{\mathbb{R}}\cos \left( t\nu \right) \left\langle \vec{w},\mu_{\Upsilon}\left( \mathrm{d}\nu \right) \vec{w}\right\rangle_{\mathbb{R}^{d}}\ ,t\in \mathbb{R}\ ,\ \vec{w}\in \mathbb{R}^{d}\ .
\int \frac{d^{4}q}{( 2 \pi )^{4}}\, \Bigg ( \frac{( k + q )^{2}+ q^{2}- k^{2}}{\Big ( ( k + q )^{2}+ M^{2}\Big )^{2}\Big ( q^{2}+ M^{2}\Big )^{2}}- \frac{4 M^{2}}{\Big ( ( k + q )^{2}+ M^{2}\Big ) \Big ( q^{2}+ M^{2}\Big )^{3}}\Bigg ) = 0 .
{\cal M}_\pi^2 (X) = M_\pi^2 (X) + \frac{\lambda}{6}\, \vec{\varphi}^{\, 2}(X) , \;\;\;\;\;\;{\cal M}_\sigma^2 (X) = M_\sigma^2 (X) + \frac{\lambda}{2}\, \vec{\varphi}^{\, 2}(X) .
G^{-1}\mathfrak{p}_{+}GA_{\boldsymbol{W}_{m}}\boldsymbol{u}& =G^{-1}\mathfrak{p}_{+}G\mathfrak{p}_{-}m_{e}T\left( m_{o}^{-1}\right)u_{2}\boldsymbol{e}_{1}+G^{-1}\mathfrak{p}_{+}G\mathfrak{p}_{-}m_{o}T\left(m_{o}^{-1}\right) u_{2}\boldsymbol{e}_{2}
\rho_{n-j}(\omega_1,\ldots,\omega_{2n+1})=\sum_{s=j}^n\rho_{s-j}(\omega_1,\ldots,\omega_{2s})\omega_{2s+1}= \sum_{1\leq s_0\leq t_1< s_1\leq t_2<s_2\leq \cdots \leq t_j< s_j\leq n+1}\omega_{2s_0-1}\prod_{k=1}^j\omega_{2t_k}\omega_{2s_k-1}.
g_{22}^{(l)}={\displaystyle \frac{1}{{\cal E}_l}}\left[{\displaystyle{-\Gamma_{\rm b}\left(-A_l\right)\frac{y^2z^2-x^2\left(A_l\rho^2+z^2\right)}{\rho^4}+\Lambda_{{\rm b}\perp}\left(-A_l\right)}}\right]
\Im G_{aa}(t,z)=\sum_{j=1}^N \frac{\eta |\langle \mathbf{e}_a, \mathbf{u}_j(t) \rangle|^2}{|\lambda_j(t)-z|^2}\prec \frac{\eta}{N}\sum_{j=1}^N \frac{1}{|\lambda_j(t)-z|^2}=\Im m_N(t,z),
\tilde{Y}_{1}\hat{R}d K_{1}\hat{R}= \hat{R}d K_{1}\hat{R}\tilde{Y}_{1}\quad \Longleftrightarrow \quad \hat{R}d K_{1}^{\dagger}\hat{R}Y_{1}= Y_{1}\hat{R}d K_{1}^{\dagger}\hat{R}\; ;
\mathcal{D}_{\nu}(x)\geq \frac{\alpha^2_{\nu,1}-x^2}{\alpha^2_{\nu,1}+x^2}~~\mbox{for all}~~|x|\leq \delta_{\nu}=\min_{n\geq 1,\nu>-1}\left\{\alpha_{\nu,1},\sqrt{\Psi_{\nu}(n)}\right\}.
G_{\mu_{1}}\left(F_{\rho_{1}}(x)\right)=a_{0}+a_{1}\alpha+\frac{a_{1}}{\sqrt[3]{\left|c_{3}\right|}}\exp\left(\frac{2\pi i-\theta i}{3}\right)(x-c_{0})^{1/3}+O\left(\left|x-c_{0}\right|^{2/3}\right)
= \int_{0}^{T}dt ~\int_{\mathbb{R}}\Bigg ( \left[\frac{1}{2}\partial_{t}A_{0}(\textbf{w}+\bar{\textbf{u}}) + \frac{1}{2}\partial_{x}A_{1}(\textbf{w}+\bar{\textbf{u}}) + D_{1}(\textbf{u}, \bar{\textbf{u}}, \textbf{w})\right]{\textbf{w}},{\textbf{w}}\Bigg ) ~ dx =: \mathrm{I}.
\frac{\lambda}{1-\xi_q^k}= \frac{1-\xi_q}{1-\xi_q^k}= \frac{1-\left(\xi_q^k\right)^{k^*}}{1-\xi_q^k}&=1+\xi_q^k+\xi_q^{2k}+\ldots +\xi_q^{k(k^*-1)}\,.
\int_{\partial S_{\varepsilon}^+}e^{zw}(u_{\varepsilon'}(z)-u_{\varepsilon_1'}(z))dz =\!\! \lim_{R\to + \infty}\int_{\partial(S_{\varepsilon}\cap \overline{D}(0,R))^+}e^{zw}(u_{\varepsilon'}(z)-u_{\varepsilon_1'}(z))dz = 0.
\gamma^0 = \left( \begin{array}{cc}0 1 1 0 \end{array}\right), \hspace*{0.5cm}\gamma^1 = \left( \begin{array}{cc}0 -1 1 0 \end{array}\right), \hspace*{0.5cm}\gamma^5 = \left( \begin{array}{cc}1 0 0 -1 \end{array}\right).
\begin{bmatrix}n_{11}n_{12}n_{13}n_{21}n_{22}n_{23}n_{31}n_{32}n_{33}\end{bmatrix}\begin{bmatrix}2 -1 1 -1 2 1 -1 -1 1\end{bmatrix}= \begin{bmatrix}-16 8 22 8 -16 22 8 8 22\end{bmatrix}
M_{d_0}(p,H)=\frac{2\pi^2\mu(d_0)\phi(d_0)}{d_0^2p}\sum_{\delta\mid d_0}\frac{\delta\mu(\delta)}{\phi(\delta)}S(H_\delta,\delta p),\hbox{where}S(H_\delta,\delta f) =\sum_{h\in H_\delta}s(h,\delta f),
x_i \cdot T_{\ell_1,\ell_2, \cdots, \ell_{i-1},{\ell_{i}-1},{\ell_{i+1}+1},\cdots,\ell_m}- x_{i+1}\cdot T_{\ell_1,\ell_2, \cdots,\ell_{i-1},\ell_{i},\ell_{i+1}, \cdots, \ell_{m}}=0
\mathcal{L}^\vee \otimes \iota^*\Big( \bigoplus_{i\in Q_0}\mathcal{U}_i^\vee\boxtimes\mathcal{U}_i \Big) = \bigoplus_{i\in Q_0}((\iota^*\circ p_1^*)(\mathcal{U}_i) \otimes \mathcal{L})^\vee \otimes(\iota^*\circ p_2^*)(\mathcal{U}_i)
\begin{aligned}- \int (v_1-v_2) g_B \partial_x \psi - \int \nabla f(v_1) \nabla \phi(x+y)h[g_B](y,v_2) \psi(x,v_1,v_2) &- \int \nabla f(v_2) \nabla \phi(-x+y)h[g_B](y,v_1) \psi(x,v_1,v_2)= \int (\nabla_{v_1}-\nabla_{v_2})[f \otimes f ]\nabla \phi(x) \psi,\end{aligned}
R_\nabla^{}(q_1,q_2)X = -\nabla_X[q_1,q_2] + [q_1,\nabla_Xq_2] +[\nabla_Xq_1,q_2] + \nabla_{\nabla_{q_2}^X}q_1 - \nabla_{\nabla_{q_1}^X}q_2.
\phi_{\mathcal{Q}}^{\mathrm{tw}}\varpi_{\jmath}-\varpi_{\jmath}=-\sum_{\jmath'\in{{}^{\jmath}B}}\alpha_{\jmath'}, (\phi_{\mathcal{Q}}^{\mathrm{tw}})^{-1}\varpi_{\jmath}-\varpi_{\jmath}=-\sum_{\jmath'\in B^{\jmath}}\alpha_{\jmath'}.
I_{a,b,d,(K_s,J'_s)}:CF^{[a,b)}(H^{(k)},J)\rightarrow CF^{[a+d,b+d)}(H^{(k)},J^{+\frac{1}{k}}) I_{a,b,d,(K_s,J'_s)}(z_-)=\sum_{z_+\in \widetilde{P}(H^{(k)})}\sharp \mathcal{N}(z_-,z_+,K_s,J'_s)\cdot z_+
\dot\mu(t,x)=C\int_{-\delta}^\delta (\mu'(t,x)-\mu'(t,\xi))\log \left(\frac{(x+\xi)^2+(\mu(t,x)+\mu(t,\xi))^2}{(x-\xi)^2+(\mu(t,x)-\mu(t,\xi))^2}\right)d\xi
f(x) = f(z = x^1 - i x^2, \bar{z}= x^1 + i x^2) \longrightarrow\hat{f}(a, a^\dagger) = \int \frac{d^2 x d^2 y}{(2 \pi)^2}f(x)e^{i[\bar{p}(\sqrt{2\theta}a - z) + p(\sqrt{2\theta}a^\dagger -\bar{z})]},
\frac{\partial^n\beta}{\partial v^n}(u,v)=Q_0+\int_v^u\left\{Q_1\frac{\partial^n\alpha}{\partial v^n}+Q_1'\frac{\partial^n\beta}{\partial v^n}+Q_1'''\frac{\partial^nt}{\partial v^n}\right\}(u',v)du'.
\,\int_t^1 u_0(1-\tau^2,\varphi) \frac{\tau^2-t^2}{2\tau}d\tau begin{align*}5pt] \quad+\frac{1}{2\pi}\int_0^{2\pi}\int_t^1 u_0\Bigl(\frac{\rho^2-t^2}{\rho}, \theta\Bigr) \sum_{j=1}^{\infty}s_j F_j(1, \varphi) \overline{Q_j(\rho,\theta)}\rho d\rho d\theta=0.
\eta_{B_R,p}^{(n)}= \sqrt{\frac{2}{\pi k_p}}^{i\frac{\pi}{4}}i \Sigma_n(1,1),\varphi_{B_R,p}^{(n)}(\hat{x})=^{in\theta_x}, \eta_{B_R,s}^{(n)}= \sqrt{\frac{2}{\pi k_s}}^{i\frac{\pi}{4}}i \Sigma_n(2,2),\varphi_{B_R,s}^{(n)}(\hat{x}) =^{in\theta_x}.
\max\biggl \{\dfrac{q}{2n-k+2},\dfrac{q-n+2k-1}{n+k+1}\biggl\}\ge \dfrac{q}{2n-k+2}\cdot t+\dfrac{q-n+2k-1}{n+k+1}\cdot\lambda &=\dfrac{2q-n+2k-1}{3n+3}\ge\dfrac{2q-n+1}{3n+3}.
\sum_{j}[\frac{1}{2}m_{j}v_{j}^{\prime2}+m_{j}U\cdot v_{j}^{\prime}+\frac{1}{2}m_{j}U^{2}]=\sum_{j}[\frac{1}{2}m_{j}u_{j}^{\prime2}+m_{j}U\cdot u_{j}^{\prime}+\frac{1}{2}m_{j}U^{2}]
\widehat{u}(\xi, t) =\frac{r_0e^{r_1 t}-r_1e^{r_0 t}}{r_0-r_1}\widehat{u_0}(\xi) +\frac{e^{r_0t}-e^{r_1 t}}{r_0-r_1}\widehat{u_1}(\xi) =e^{t\varphi(\xi)}\left[\frac{\varphi(\xi)\sin(t\phi(\xi))}{\phi(\xi)}+\cos(t\phi(\xi))\right] \widehat{u_0}(\xi) +e^{t\varphi(\xi)}\frac{\sin(t\phi(\xi))}{\phi(\xi)}\widehat{\Delta u_1}(\xi).
S = \frac{1}{2 \kappa^{2}}\int d^{5}x \sqrt{- g}\left({\cal R}^{5}+ \frac{1 2}{l^{2}}\right) - \sigma \int d^{4}x \sqrt{- g_{b r a n e}}+ \int d^{4}x \sqrt{- g_{b r a n e}}{\cal L}_{m a t t e r},
\zeta(u^*)=v^*,\ \H(u,v)=v-\zeta(u)\ \{\rm for}\ \u\geq0,\v\geq0\ \{\rm or}\zeta(v^*)=u^*,\ \H(u,v)=\zeta(v)-u\ \{\rm for}\ \u\geq0,\v\geq0.
-2\int_0^t \operatorname{Im}\langle (1-\mathfrak{s}\kappa | z|^2)z, z\d W(s)\rangle_{L_x^2}-2 \mathfrak{s}\int_0^t\operatorname{Im}\langle \partial_x z,\partial_x (z\d W(s))\rangle_{L_x^2}= -2 \mathfrak{s}\int_0^t\operatorname{Im}\langle \partial_x z,z\partial_x \d W(s)\rangle_{L_x^2},
[ \hat{S}_{0}, \hat{S}_{+}] = ( \hat{C}_{2}- \hat{C}^{\prime}) \hat{J}^{+}\qquad [ \hat{S}_{0}, \hat{S}_{-}] = - ( \hat{C}_{2}- \hat{C}^{\prime}) \hat{J}^{-}\qquad [ \hat{S}_{+}, \hat{S}_{-}] = 2 ( \hat{C}_{2}- \hat{C}^{\prime}) \hat{H}
E ( v q ) - E ( v ) E ( q ) = \frac{< \{\hat{Q}, \hat{P}\}>_{\psi}}{2}\, ; \; \; \hat{Q}= \hat{q}- < \hat{q}>_{\psi}\, , \; \; \hat{P}= \frac{\hat{p}- < \hat{p}>_{\psi}}{m}\, ,
B^{(2)}= \Omega\bigg( \ln \frac{\mathsf{2\cdot Num}\cdot 2d}{\xi}\bigg)^{d}; \ \delta^{(2)}= \bigg( \frac{\xi \cdot \vartheta^{1/2d}}{2d \cdot B^{(2)}\cdot \sqrt{2 \cdot \mathsf{Num}}\cdot \mathsf{Coeff}^{2/d}}\bigg)^{2d}; \c_{(2)}= \frac{2 \cdot \mathsf{Num}}{\delta^{(2)}\cdot \sqrt{\xi}}.
\#\mathcal{Z}\leq\sum_{\lambda,\mu,\nu}\sum_{m\leq M_\lambda}\sum_{\substack{q\leq Q_\mu (q,mp_0)=1}}\:\sideset{}{^*}\sum_{\chi\pmod{qm}}N(1-\tfrac{c^*\log\log X}{\log X},T_\nu,\chi).
a_{m_*}=C\mathcal{H}^{n-1}(\partial E_m) & \approx \mathcal{H}^{n-1}(\partial \mu_m K) &= \mu_m^{n-1}\mathcal{H}^{n-1}(\partial K)= (\frac{m}{|K|})^{\frac{n-1}{n}}\mathcal{H}^{n-1}(\partial K)
P_{K}( V ;{\bf a}_{1}, \ldots ,{\bf a}_{K}, \rho_{1}, \ldots , \rho_{K}) = \frac{1}{Z ( V )}\, \frac{C^{K}}{K !}\prod_{j = 1}^{K}\rho_{j}^{\alpha}\ \Theta ( \{{\bf a}_{j}\}, \{\rho_{j}\}) \, .
\begin{array}{l l}{\varphi ( r ) = \eta ,}&{r \rightarrow \infty}\\{\varphi ( r ) = 0}&{r \rightarrow 0}\\ \end{array}\begin{array}{l l}{H ( r ) = 1}&{r \rightarrow \infty}\\{H ( r ) = 0}&{r \rightarrow 0 .}\\ \end{array}
\displaystyle{\lim_{\substack{v^{\, \prime}\to v \tau \downarrow 0}}}\, \displaystyle{\frac{\varphi(\bar{w}+ \tau \, v^{\, \prime}) - \varphi(\bar{w}) - \tau \, \varphi^{\, \prime}(\bar{w};v^{\, \prime})}{\tau^2}}\, = \, \Phi^{\, \prime}(\bar{w};v)^T F^{\, \prime}(\Phi(\bar{w});\Phi^{\, \prime}(\bar{w};v)) + F(\Phi(\bar{w}))^T \Phi^{(2)}(\bar{w};v).
M_n(x;\alpha,c) = \frac{(\alpha-\beta)_n}{(\alpha)_n}\sum_{k=0}^{n}\frac{(\beta)_k(-n)_k}{k!(\beta-\alpha-n+1)_k}F_1\!\left(\!-n+k,-x,x;\beta-\alpha-n+k+1;\frac{1}{c},\frac{1}{d}\right)M_k(x;\beta,d),
\langle T_{\mu\nu}\rangle = \lim_{x'\rightarrow x}\biggl({2\over3}D_{;\nu'\mu}-{1\over3}D_{;\nu\mu}-{1\over6}g_{\mu\nu}D^{;\sigma'}{}_\sigma+{1\over3}g_{\mu\nu}D^{;\sigma'}{}_{\sigma'}+{1\over6}(R_{\mu\nu}-{1\over2}Rg_{\mu\nu})D\biggr).
\aligned g_0(t) := - 4(e^{t/2}+e^{-t/2}-2), g_1(t) := \sum_{n \leq e^{|t|}}\frac{\Lambda(n)}{\sqrt{n}}(|t|-\log n), g_\infty(t) := - \frac{|t|}{2}\left[ \frac{\Gamma'}{\Gamma}\left(\frac{1}{4}\right) - \log \pi \right] -\frac{1}{4}\left( C - e^{-|t|/2}\Phi(e^{-2|t|},2,1/4) \right) \endaligned
\mbox{}\int_{\Bbb R^N}\left( \big| \nabla u_n \big|^{p(x)}+ \big( \lambda V(x) + Z(x) \big) | u_n |^{p(x)}\right) \eta_R = \phi_\lambda'(u_n) \left( u_n \eta_R \right) - \int_{\Bbb R^N}u_n \big| \nabla u_n \big|^{p(x)-2}\nabla u_n \cdot \nabla \eta_R + \int_{\Bbb R^N \setminus \Omega'_\Upsilon}\tilde{f}(x,u_n) u_n \eta_R.
\frac{1}{n}\sum_{i=0}^{n-1}\frac{1}{\lambda^i}\mathcal P^i\left(h(\cdot) \frac{1}{\lambda^{n-i}}\mathcal P^{n-i}(\cdot,M)\right) \xrightarrow{n\to\infty}\eta \int_M h(y) \eta(y) \mu(\d y), L^1(M,\mu).
\begin{array}{r c l}{{\displaystyle \sum_{k}K_{k}^{j}( z ) \phi^{k}( z ) | \mathrm{v a c}\rangle_{i}}}&{{=}}&{{\nu^{( i )}( z ) \phi^{j}( - z ) | \mathrm{v a c}\rangle_{i},}}\\{{{}_{i}\langle \mathrm{v a c}| \displaystyle \sum_{k}\phi_{k}( z ) K_{j}^{k}( z )}}&{{=}}&{{\nu^{( i )}( z ){}_{i}\langle \mathrm{v a c}| \phi_{j}( - z ) ,}}\\ \end{array}
\tilde A_1^\dagger \vert n_1,n_2\rangle_q = \sqrt{[n_1 +1]_q}\,\vert n_1 +1,n_2\rangle_q; \tilde A_2^\dagger \vert n_1,n_2\rangle_q = \sqrt{[n_2 +1]_q}\,\vert n_1,n_2 + 1\rangle_q,
\#\left(P_{g\oplus h}^{-1}\alpha\right) =\#\left((a\,c) \left(P_{g}^{-1}\oplus P_{h}^{-1}\right)\left(\gamma_{\, r}\oplus \beta\right) \right) =\#\left(\left(P_{g}^{-1}\oplus P_{h}^{-1}\right)\left(\gamma_{\, r}\oplus \beta\right)\right)\pm 1.
h(t)=\Phi_{105}(t) =t^{48}+t^{47}+t^{46}-t^{43}-t^{42}-2t^{41}-t^{40}-t^{39}+t^{36}+t^{35}+t^{34}+t^{33}&+t^{32}+t^{31}-t^{28}-t^{26}-t^{24}-t^{22}-t^{20}+t^{17}+t^{16}+t^{15}+t^{14}+t^{13}&+t^{12}-t^{9}-t^{8}-2t^{7}-t^{6}-t^{5}+t^{2}+t+1.
1\leq |\phi(0)|\leq \frac{|\tilde{P}_{m_n-1}(\theta_1)|}{|\tilde{P}_{q_n-1}(\theta_{m_n-q_n+1})|}\prod_{j=m_n-q_n+1}^0|\cos(\pi \theta_j)| \cdot |\phi(m_n-q_n)| &+\frac{|\tilde{P}_{-m_n+q_n-1}(\theta_{m_n-q_n+1})|}{|\tilde{P}_{q_n-1}(\theta_{m_n-q_n+1})|}\prod_{j=0}^{m_n-1}|\cos(\pi\theta_j)| \cdot |\phi(m_n)|
\left\| A \right\|_{1, 2}={\bf max}_{1 \le i \le n}\left\| A(:,i) \right\|_2, \left\| A \right\|_{1, \infty}={\bf max}_{i, j}| a_{i, j}|, \left\|AB\right\|_{1,2}\le \left\|A\right\|_2 \left\|B\right\|_{1,2}, \left\|AB\right\|_{1,2}\le \left\|A\right\|_{1,2}\left\|B\right\|_1,
\Phi_{ij}(\epsilon) =\begin{cases}{1 \over \lambda_i(\epsilon)}- \langle a_{i}^{\epsilon}| G^{(n)}_{0}| a_{i}^{\epsilon}\rangle \textrm{if $i = j$}- \; \langle a_{i}^{\epsilon}| G^{(n)}_{0}| a_{j}^{\epsilon}\rangle \textrm{if $i \neq j$}.\end{cases}\;.
- 2\beta \sum\limits_{i \in B}{F^{ii}u_{ii}d_i^2}\leq - 2\beta \sum\limits_{i \in B, u_{ii}< 0}{F^{ii}u_{ii}d_i^2}\leq - \frac{2}{n}\sum\limits_{i \in B,u_{ii}< 0}{F^{ii}u_{ii}}\leq - \frac{2}{n}\sum\limits_{u_{ii}< 0}{F^{ii}u_{ii}}= - \frac{2}{n}[k-l - \sum\limits_{u_{ii}> 0}{F^{ii}u_{ii}}].
e^{-X}de^{X}=dX^{i}e_{i}-\frac{1}{2!}X^{i}dX^{j}{f_{ij}}^{k}e_{k}+\frac{1}{3!}X^{i}X^{j}dX^{k}{f_{jk}}^{l}{f_{il}}^{m}e_{m}-\cdot\cdot\cdot,
e^{-\frac{x^{2}}{2}}\cdot H_{n}(x)=\pi^{\frac{1}{4}}2^{\frac{n}{2}+\frac{1}{4}}\sqrt{n!}n^{-\frac{1}{12}}(Ai(2^{\frac{1}{2}}n^{\frac{1}{6}}t)+O(n^{-\frac{2}{3}})),
\bigcup_{i=1}^t I'_i \ = \I'_{T-1}\cup I'_T \ = \ [1,n] \smallsetminus [j_{T-1}+1, j_{T}]\,, I'_{T-1}\cap I'_{T}= \emptyset\,, \mbox{\quad{and}}\bigcap_{i=1}^t I_i \ = \I_{T-1}\cap I_T \ \subseteq \ [j_{T-1}+1, j_{T}]\,,
h(m) + \int F'\,d\mu \leq h(m) \leq H_m(\alpha) \leq -m(A_0)\log m(A_0) - \sum_{n=1}^{+\infty}m(A_n)\log m(A_n) \leq - (1-\epsilon)\log(1-\epsilon) - \sum_{n=1}^{+\infty}\frac{\epsilon}{2^{n-1}}\log\frac{\epsilon}{2^{n-1}}= - (1-\epsilon)\log(1-\epsilon) - 2\,\epsilon\log\epsilon +4\,\epsilon\log 2 < \eta\;.
{\displaystyle \sum_{\emptyset\neq u\subseteq\{1,\cdots,N\}}S_{u}}={\displaystyle \sum_{\emptyset\neq u\subseteq\{1,\cdots,N\}}S_{u,v}}+{\displaystyle \sum_{\emptyset\neq u\subseteq\{1,\cdots,N\}}S_{u,c}}=1.
S ={\frac{1}{2 \pi}}\int d^{2}x \sqrt{- g}\left\{e^{- 2 \Phi}\left( R + 4{\left( \nabla \Phi \right)}^{2}+ 4 \lambda^{2}\right) -{\frac{1}{2}}\sum_{i = 1}^{N}{\left( \nabla f_{i}\right)}^{2}\right\},
\partial_{r}\left(\frac{1}{r^{2}}\int_{S_{t,r}}\rho d\sigma_{S_{t,r}}\right)=\partial_{r}\left(\int_{\mathbb{S}^{2}}\rho d\sigma_{\mathbb{S}^{2}}\right)=\int_{\mathbb{S}^{2}}\partial_{r}\rho d\sigma_{\mathbb{S}^{2}}=\frac{1}{r^{2}}\int_{S_{t,r}}\partial_{r}\rho d\sigma_{S_{t,r}}.
0\leq \int_{\Omega}u(t)+\int_{\Omega}v(t) \leq \max \left\{\int_{\Omega}u_0+\int_{\Omega}v_0, \,\,2|\Omega|\frac{\max\{a_0,b_0\}}{\min\{\alpha,\beta\}}\right\},\forall t \in (0, T_{max}),
\widetilde{S}_{s}=\left(\begin{array}{cccccc}\beta_{st}I_q K^1 J^{1}H^1 &\ddots \vdots \vdots \vdots & \beta_{st}I_q K^p J^{p}H^p * \cdots * \beta_{st}I_p L^{st}M^{ts}* \cdots * * \beta_{st}I_q M^{st}* \cdots * * * \beta_{st}\end{array}\right)\succeq0,
\Delta^{\mp , a , p}( x , x^{\prime}) = \int_{- \infty}^{+ \infty}\, d z \, \int_{- \infty}^{+ \infty}\, \frac{d p_{D}}{2 \pi}\, e^{- i p_{D}y^{D}}\Delta_{Q}^{\mp , a , p}\; , \quad y_{\mu}= x_{\mu}- x_{\mu}^{\prime}\; ,
\tau_{j}^{(t+1)}=\frac{\sum_{i=1}^{n}T_{j,i}^{(t)}}{\sum_{i=1}^{n}(T_{1,i}^{(t)}+T_{2,i}^{(t)})}=\frac{1}{n}\sum_{i=1}^{n}T_{j,i}^{(t)}.
0=[[\beta_{-1}, \beta_{-m+1}],\beta_n] +(m+n)(m-2) \beta_{n-m}+ \sum_{j=1}^{m-2}(n+1)j \alpha_{n-1-j}\alpha_{-m+1+j}&+ \sum_{j=1}^{m-2}j \alpha_{n-j}\alpha_{-m+1+j}\beta_{-1}-\sum_{j=1}^{m-2}j \alpha_{n-j}\alpha_{-m+1+j}\beta_{-1}&-\sum_{j=1}^{m-2}j(-m+1+j) \alpha_{n-j}\alpha_{-m+j}-\sum_{j=1}^{m-2}j(n-j) \alpha_{n-j-1}\alpha_{-m+1+j}.
\int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}F ( \tau ) = \int_{\bigcup \gamma_{c d}{\cal F}}\frac{d^{2}\tau}{\tau_{2}^{2}}F ( \tau ) = \sum_{( c , d ) = 1}\int_{\gamma_{a b}{\cal F}}\frac{d^{2}\tau}{\tau_{2}^{2}}F ( \tau ) \; .
\mathcal{P}u_s (z) + \lambda u_s (z) = D_{v_i}[ f^i (z) \zeta (v_d - s) - a^{i d}(z) \zeta' (v_d - s) u (z)] &- f_d (z) \zeta' (v_d - s)+ g \zeta (v_d - s)- a^{d j}(z) \zeta' (v_d - s) D_{v_{j}}u (z).