harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\dfrac{\mu([x|n_j])}{N_{k}(x|n_j)}=\dfrac{\phi^{S_{n_j}(\mu,x)}(T_{x|n_j})}{\phi^k(T_{x|n_j})\alpha_{k+1}^{-k}(T_{x|n_j})}\geq \dfrac{\phi^{S_{n_j}(\mu,x)+\gamma}(T_{x|n_j})}{\phi^k(T_{x|n_j})\alpha_{k+1}^{-k}(T_{x|n_j})}=\alpha_{k+1}^{S_{n_j}(\mu,x)+\gamma}(T_{x|n_j}),
\chi_{0,0}^{(q)}= 0, 0\leq \chi_{n,0}^{(q)}= \int_{0}^{k}\beta_q(t_n-\zeta) h(\zeta/k) d\zeta, n\geq 1, 0\leq w_{n-l}^{(q)}\equiv \chi_{n,l}^{(q)}= \int_{-k}^{\min(k,t_{n-l})}\beta_q(t_{n-l}-\zeta) h(\zeta/k) d\zeta, n\geq 1, l\geq 1,
|F_S(x)-F_S(y)|^2\le \sum_{j\in S}(|x_{(j)}|-|y_{(j)}|)^2\le \sum_{j=1}^m (|x_{(j)}|-|y_{(j)}|)^2=\|x\|^2+\|y\|^2-2\sum_{j=1}^m |x_{(j)}y_{(j)}|\le \|x-y\|^2,
\sum_{x_i=1}^\ell \sin(x_i k_i)^2 \sin(x_i k'_i) \cos(x_i k'_i) = \sum_{x_i=1}^\ell \left[\frac{\sin(2x_i(k_i-k'_i))-\sin(2 x_i(k_i+k'_i))}{4}+ \frac{\sin(2 x_i k'_i)}{2}\right].
Volume=\frac{1}{n!}det[(\begin{matrix}v_{1}^{T}-v_{0}^{T}\\v_{2}^{T}-v_{0}^{T}\\ \vdots\\v_{n}^{T}-v_{0}^{T}\end{matrix})(\begin{matrix}v_{1}-v_{0}&v_{2}-v_{0}&\cdot\cdot\cdot&v_{n}-v_{0}\end{matrix})]^{1/2}
\Delta \rho ( k ) = \frac{1}{\pi}\frac{1}{k^{2}+ 1}\int_{1}^{\infty}d x \left[ \Theta ( \Lambda - \frac{1}{2 x^{2}}\delta - | k | ) - \Theta ( \Lambda - | k | ) \right] \left( \frac{\sqrt{x^{2}- 1}}{x^{2}}+ \frac{k^{2}}{\sqrt{x^{2}- 1}}\right)
\Delta u_1=a_{1,1}(x) -\sigma\frac{\langle x,\nabla u_1\rangle}{|x|^2}-\left(p-2\right) \frac{\langle (D^2 u_1)(\nabla u_1),\nabla u_1\rangle}{|\nabla u_1|^2}\ \mathcal G\cap B_{\rho_0/2}(x_0).
x_i = \left(a \prod_{j=1}^n u_j^{\alpha_j}+ b \prod_{j=1}^n u_j^{\beta_j}\right)^{z_i}\left(c \prod_{j=1}^n u_j^{\gamma_j}\right)^{t_i}w^{-z_i-t_i}\cdot u_i, i=1,\dots, n,
\frac{\partial^2}{\partial \eta^2}\left[ \log\left(\tfrac{1}{n}A(\tfrac{\eta}{n},\tfrac{1}{\eta})\right) \right] = \frac{\tanh''(\tfrac{\eta}{2}) \tanh(\tfrac{\eta}{2}) - \tanh'(\tfrac{\eta}{2})^2}{4 \tanh(\tfrac{\eta}{2})^2}+ \frac{\psi''(\eta) \psi(\eta) - \psi'(\eta)^2}{\psi(\eta)^2}.
\mathcal{N}_{\rm cyl}(v_{\varepsilon}+t\psi)-\mathcal{N}_{\rm cyl}(v_{\varepsilon}) =-P_{\rm cyl}v_{\varepsilon}+tP_{\rm cyl}\psi-(v_{\varepsilon}+t\psi)^{\frac{n+6}{n-6}}+P_{\rm cyl}v_{\varepsilon}+v_{\varepsilon}^{\frac{n+6}{n-6}}&=tP_{\rm cyl}\psi-(v_{\varepsilon}+t\psi)^{\frac{n+6}{n-6}}+v_{\varepsilon}^{\frac{n+6}{n-6}},
\begin{array}{lll}\alpha_0=H_{q_1}+H_{p_2}-E_{3,4,9,10}, \alpha_1=H_{q_2}-E_{15,16}, \alpha_2=H_{p_2}-E_{5,6}, \alpha_3=H_{p_1}+H_{q_2}-E_{1,2,11,12}, \alpha_4=H_{q_1}-E_{7,8}, \alpha_5=H_{p_1}-E_{13,14}\end{array}
\sum_{i:t_{i,n}\leq T}|\mathcal{I}_{i,n}|f(|\mathcal{I}_{i,n}|^{-1/2}\Delta_{i,n}X^{toy}) = \sum_{i:t_{i,n}\leq T}|\mathcal{I}_{i,n}|f(\sigma Z_i^n) \overset{\mathbb{P}}{\longrightarrow}T \mathbb{E}[f(\sigma Z)]
P_N^p(\eta_t(x)=1,\eta_t(y)=0) \geq P_N^p\Big(\eta_t(x)=1,\eta_t(y)=0,A_{x,y}(T)\Big) &=P_N^p\Big(\eta_t(x)=1,\eta_t(y)=0\Big|A_{x,y}(T)\Big)P_N^p\Big(A_{x,y}(T)\Big).
D_2^2+E_2D_2=F_4, &E_1D_2+E_2D_1=F_2, &E_0D_2+D_1^2+E_1D_1+E_2D_0=F_2, &E_0D_1+E_1D_0=F_1, &D_0^2+E_0D_0=F_0.
\{\theta_1,\theta_2\}_{\mbox{c.b.}}=\beta_1\left(P-\frac{1}{N}I\right)+\left\{\theta_1[\theta_2, r]\right\}+\beta_2\left(\theta_1P\theta_2+\theta_2P\theta_1-\frac{2}{N}\theta_1\theta_1-\frac{2}{N}\theta_2\theta_2\right).
t(x) =2\frac{1+x}{1+2x}(p+x) \alpha(x)\prod_{k=1}^m\frac{(x+x_k)(x-x_k-1)}{(x-x_k)(x+x_k+1)}+(p-x-1){\tilde \delta}(x)\prod_{k=1}^m\frac{(x-x_k+1)(x+x_k+2)}{(x-x_k)(x+x_k+1)}\,.
\widehat{\Sigma}(\delta_1, \delta_2) = \left\{\gamma \in \mathcal{P}(\mathcal{S}): \boldsymbol{K}_1( \gamma_{13}, \mu_{13}) \leq \delta_1, \widehat{\boldsymbol{K}}_2( \gamma_{23}, \mu_{23}) \leq \delta_2 \right\}.
P_1^{[3]}\stackrel{(a)}\leq\sum_{\substack{(\tilde{m}_1,\tilde{w})\neq (1,1), \tilde{i}\in\mathcal{I}}}2^{-n\big(I(U_1;Y_1|U_0)-\tau^{[3]}_1(\delta)\big)}\leq2^{n(R_1+\tilde{R}+R')}2^{-n\big(I(U_1;Y_1|U_0)-\tau^{[3]}_1(\delta)\big)}=2^{n\big(R_1+\tilde{R}+R'-I(U_1;Y_1|U_0)+\tau^{[3]}_1(\delta)\big)}
|\Delta_{l,2}^{P_1,Q_i}\cap \Delta_{c}| = |\Delta_{l,2}^{P_1,R_i}\cap \Delta_{c}| = |\Delta_{l,2}^{Q_1,R_i}\cap \Delta_{c}| = = (q^5-q^2)(q^2-q)(q^2-q-2)(q^2-q-4).
{\cal L}_{1 / m^{2}}^{( 3 + 1 ) s p i n}= \frac{\alpha}{7 2 0 \pi m^{2}}\left[ 1 8 F^{\nu \lambda}F_{\nu \lambda , \mu}^{~ \mu}+ 7 F^{\nu \lambda , \mu}F_{\nu \lambda , \mu}- 2 F_{\lambda}^{\nu \lambda ,}F_{\nu \mu ,}^{\mu}\right] ,
\beta_q(\alpha) = \frac{[N^3 + s^3 p^3 - sp^2 N\, (N + s)](p - 1)(p - q) - Np\, (N - sp)[N\, (p - q - 1) + sq](\alpha_0 - \alpha)}{(N - sp)[(N - sp)\, q + sp^2](p - 1)},
c^{S_2}_0 = \frac{\lambda_0}{\alpha + \lambda_0 + \mu_1}\, \Delta h_1 c^{S_{k+2}}_k = \frac{\lambda_k}{a_{k+1}}\, \frac{\displaystyle \Delta h_{k+1}+ \frac{c^{S_{k+1}}_{k-1}}{\rho_{k-1}}}{\alpha + \lambda_k + \mu_{k+1}}, 1 \leq k \leq n-1.
\mathcal{H}_{-\tilde{A}_1,q_1}\tilde{u}_1 =e^{-i\varphi/2}\mathcal{H}_{-A_1,q_1}u_1=k^2\tilde{u}_1,\cr \mathcal{H}_{\tilde{A}_2,q_2}\tilde{u}_2 =e^{-i\varphi/2}\mathcal{H}_{A_2,q_2}u_1=k^2\tilde{u}_2,\textrm{in}\,B.
R_{s,z,f}^n \varphi(v) = \mathbb{E}_{\mathbb{Q}_{s,f}^v}\left[ e^{z \left( \log \frac{\langle f, G_n v \rangle}{\langle f, v \rangle}- nq \right)}\varphi(G_n \cdot v) \right], v \in \mathcal{S}_\epsilon.
\sigma(1,v,u,C) = \min \{|z|^p + \sum_{i=1}^s \left| a_{i\,1}z + v_i \right|^p + \sum_{i=1}^m \left| \bar a_{i\,1}z + u_i \right|^p : z \in \mathbb{Z},\, |z| \leq C \}.
\begin{array}{ll}(*_k) : \mbox{the implication}u\in Im\, \partial \Rightarrow u\in Im\, \partial\bar{\partial}\mbox{holds for all}d-closed \mbox{forms of types}& (p,q), (q,p), (p+1,q), \mbox{and}(q+1,p) &\mbox{for all}p,q \mbox{such that}p+q=k \mbox{or}p+q=2n-k\end{array}
\int_{S^2}\frac{-a^i S_i}{2}\tilde \Delta X_0^{(3)}dS^2= \int_{S^2}\frac{1}{3}(-4a^iD_{in}\tilde X^n + 4 a^iW_0 \tilde X^i + 4 a^i W_i) (-W_0 + 6 \frac{a^j P_j}{a^0}) dS^2 = 8 \frac{a^ia^j}{a^0}\int_{S^2}W_0 \tilde X^i P_j dS^2 -\frac{4}{3}a^i \int_{S^2}W_iW_0 dS^2,
\forall \tau_p\in\left[-\eta,\varepsilon\right], \forall t\in[-\ell_{k,p},\ell_{k,p}],\Im(\gamma_{k,p}(t))=t+\theta_k, \Re(\gamma_{k,p}(t))=h_{k,p}(t):=\Psi_k^{-1}\left(\Psi_k(\tau_p)-A_It^{2\mu_k}\right).
\left( D^{l}p\right) \left( z\right) -\left( D^{l}q\right) \left( z\right)=\sum_{k=0}^{n-1-l}\left( \left( D^{l+k}p\right) \left( 0\right) -\left(D^{l+k}q\right) \left( 0\right) \right) \frac{z^{\otimes k}}{k!}\forall z\in \mathcal{V}
\left.\begin{array}{lc}\alpha_n^{\mu}|\Omega\rangle=0 n>0 b_n|\Omega\rangle=0 n\ge 0 c_n|\Omega\rangle=0 n>0 p^{\mu}|\Omega\rangle\propto\alpha_0^{\mu}|\Omega\rangle=0. \end{array}\right.
\int_{E^{c}}S_1^{i,j}(b)(x)^{2}dx \leq 2^{j}\int_{E^{c}}\sum_{J}\sum_{R \in \mathcal{D}}\left(\sum_{P \in (R)_i}|\widehat{b_J}(P)| \right)^{2}\frac{1_R (x)}{|R|}dx = 2^{j}\sum_{J}\int_{E^{c}}\sum_{\substack{|R| > |J| R \supset J}}\left(\sum_{P \in (R)_i}|\widehat{b_J}(P)| \right)^{2}\frac{1_R (x)}{|R|}dx.
\sum_{i=1}^{n-1}\frac{i}{(v+i)^2}= \int_1^n \frac{x}{(v+x)^2}\dx+ O\left(\frac{1}{(v+1)}\right)+ O\left(\frac{1}{n}\right) &= \frac{v(1+n)}{(v+1)(v+n)}+\log\left(\frac{v+n}{v+1}\right) +O\left(\frac{1}{v+1}+ \frac{1}{n}\right)\geq\log2,
\mathcal{T}_{11}= \{T_{111}, T_{211}, T_{311}, \dots \}=(T_{n11})_{n=1}^{\infty}\mathcal{T}_{12}= \{T_{112}, T_{212}, T_{312}, \dots \}=(T_{n12})_{n=1}^{\infty}\mathcal{T}_{21}= \{T_{121}, T_{221}, T_{321}, \dots \}=(T_{n21})_{n=1}^{\infty}\mathcal{T}_{22}= \{T_{122}, T_{222}, T_{322}, \dots \}=(T_{n22})_{n=1}^{\infty}
\nu \left( A\right) =\int_{\mathcal{S}\left( \mathcal{I}\right)}P\left(A\right) d\rho \left( P\right) =\min_{\pi \in \mathrm{core}\left( \rho\right)}\int_{\mathcal{S}\left( \mathcal{I}\right)}P\left( A\right) d\pi\left( P\right) \forall A\in \mathcal{F}.
\nu\int_{\Omega}\left[a_1\vert y_1 \vert^{1 - \nu}+ a_2\vert y_2 \vert^{1 - \nu}\right]dz = \nu\| (y_1, y_2)\|_{1,p}+ \nu\| (\nabla y_1, \nabla y_2)\|_{q,\eta}- \lambda\nu(\kappa_1 + \kappa_2 + 2)\int_{\Omega}\vert y_1\vert^{\kappa_1 + 1}\vert y_2\vert^{\kappa_2 + 1}dz.
P^{\prime}( w^{\prime}) = L^{\beta k}\; \sum_{\left[ n_{i^{\prime}}\right]_{i^{\prime}= 0}^{k}}\; \sum_{\left[ X_{i}\right]_{i = 0}^{n}}\int \prod_{i = 0}^{n}d t_{i}\; \prod_{j^{\prime}= 0}^{k}\; \delta ( \sum_{{i}= m_{j^{\prime}- 1}+ 1}^{m_{j^{\prime}}}\; \; t_{i}- L^{\beta}t_{j^{\prime}}^{\prime}) \times
\iint_{\Sigma_-}\bigl(v_{kt}\varphi_i(x)\psi_m(y) -v_k(b\varphi_i'\psi_m+\varphi'''_i\psi_m +\varphi'_i\psi''_m)\bigr)\,dxdy - \iint_{\Sigma_-}f\varphi_i\psi_m\,dxdy=0,
\sum_{k=0}^{\infty}\left( \frac{1}{32}\right)^{k}\binom{2k}{k}^2 \frac{H_{k}}{k+1}= 8 - \frac{2\Gamma^{2}\left( \frac{1}{4}\right)}{\pi^{3/2}}- \frac{4 \pi^{3/2}+ 16 \sqrt{\pi}\ln(2)}{\Gamma^{2}\left( \frac{1}{4}\right)}
\frac{G_{7}^{'}\left(r\right)}{G_{8}^{'}\left(r\right)}=\frac{\pi^{2}}{4}\frac{\left[\kappa\left(r\right)+\kappa_{a}\left(r\right)\right]\left[\kappa\left(r\right)-\kappa_{a}\left(r\right)\right]}{\left[r^{'2}\kappa^{2}\left(r\right)\kappa_{a}^{2}\left(r\right)\right]\left[A\left(r\right)-1\right]}
\mathcal{Z}_{i}(t)=\sum_{j=1}^{\infty}\varepsilon^{-1}\int_{0}^{t}e^{-\varepsilon^{-2}d_{i}(t-s)\lambda_{j}}d\tilde{W}_{i,j}(s)g_{j}=\varepsilon^{-1}\int_{0}^{t}e^{\varepsilon^{-2}(t-s)\mathcal{A}_{i}}d\tilde{W}_{i}(s),
(\delta_{t}*_p \delta_{t^\prime})(f)= &=\frac{1}{\kappa_p}\int_{B_q}\int_{SU(q)}f\bigl(\arccos(\sigma_{sing}(-\sin\underline t \,w\sin\underline t^\prime \,+\,\cos \underline t\, v \cos \underline t^\prime))\bigr) \cdot\Delta(I_q-w^*w)^{p-2q}dv dw
\mathcal{I}_{\mathrm{D}}(\delta_{A}) = \inf_{\lambda \in \mathbb{R}_{+}^L}\left[ \langle \lambda, \delta_{A}\rangle + \sup_{\pi \in \Pi(\mu_1, \dotsc, \mu_L)}\int_{\mathcal{V}}g_{\lambda, A}\, d \pi \right].
\pi_{m+n\sigma}= [\mathbf{CP}(m+n\sigma)] \in \Omega^{C_2}_* \Pi_{m+n\sigma}= [\mathbf{CP}(m+n\sigma) \to \mathbf{CP}^\infty_{C_2}] \in \Omega^{C_2}_*(\mathbf{CP}^\infty_{C_2}).
\omega_z(u_{ij}^{k}) = \varphi(\pi_z(u_{ij}^{k})1) = \varphi((u_{ij}^{k})_{(1)}(f_{1-z}\star S((u_{ij}^{k})_{(2)})) ) &= \frac{\delta_{ij}\mu_k(|q|^{\bar z}+|q|^{-\bar z})}{\mu_{k}(|q| +|q|^{-1})}.
g_{i,j}(x_2/x_1)^{\pm 1}=g_{i,j}(x)^{\pm 1}|_{x=x_2/x_1}=\iota_{x_1,x_2}\left(\frac{G^+_{i,j}(x_1,x_2)}{F^+_{i,j}(x_1,x_2)}\right)^{\pm 1}.
\partial_{t}^{k}\theta \in C^{0}\big([0, T], H^{s + 1 - k}(\Omega) \cap H^{1}_{0}(\Omega)\big) k = 0, \dots, s - 2, \partial_{t}^{s - 1}\theta \in C^{0}\big([0, T], H^{1}_{0}(\Omega)\big) \cap L^{2}\big(0, T; H^{2}(\Omega) \cap H^{1}_{0}(\Omega)\big) \partial_{t}^{s}\theta \in L^{2}\big(0, T; L^{2}(\Omega)\big).
m_{\pi}=\frac{1}{2}(\dim\pi-\chi_{\pi}(h_1)) &=\frac{1}{2}\left([n]_q!-[n-1]_q!\left(\sum_{i=1}^n\chi_i(-1)\right) \right) &=\frac{1}{2}[n-1]_q!\left(T_{n-1}- \sum_{i=1}^n\chi_i(-1)\right).
S = \sum_{\substack{r\in [5,y] b\geq 1}}f(r^b) = \sum_{\substack{b\geq 1 5\leq r\leq y}}\left(\frac{2}{r^b}+\frac{2}{r^b}\left(\frac{r-1}{r-3}-1\right)\right) & = \sum_{\substack{r\in [2,y] b\geq 1}}\frac{2}{r^b}+O\bigg(\sum_{\substack{r\geq 5 b\geq 1}}\frac{1}{r^{b+1}}\bigg) & = 2\log\log y+O(1),
(f_2\psi_M)^1(\gamma)(m) - (\psi_Nf_1)^1(\gamma)(m) = f^0(m w_{1,2}\gamma) - f^0(m) w_{1,2}\gamma & + \sum_{i,j=1}^{n}(f^1(w_2 \gamma_j)(m w_{1,2}\gamma_i)-f^1(w_1\gamma_i)(m) w_{1,2}\gamma_j)\gamma(p_jp_i).
\frac{1}{k}>\|t_k\|_{L^\infty(\Omega_k)}, &|w_k(0)|+|\nabla w_k(0)|+|\nabla w_{ik}(0)|+|\nabla w_{ek}(0)|>\frac{1}{2}, &\displaystyle\sup_{x\in\Omega_k}\left(|w_k(x)|+|\nabla w_k(x)|+|\nabla w_{ik}(x)|+|\nabla w_{ek}(x)|\right)=1,
\frac{\partial}{\partial t}|\nabla^2 w|^2= (1+\frac{\alpha'}{2}R)\Delta|\nabla^2 w|^2-2(1+\frac{\alpha'}{2}R)|\nabla^3w|^2-2(R+\frac{3}{4}\alpha'R^2+r)|\nabla^2w|^2 +2R(1+\frac{\alpha'}{2}R)(R-a)^2+\alpha'\nabla R\nabla R\nabla^2w-2a\nabla^2f\nabla^2w.
\omega^{ij}_{g_ih_i}(g_j) =\tau_j(g_j)^{-1}\tau_i(g_ih_i)^{-1}\tau_j(g_j)\tau_i(g_ih_i) & = \tau_j(g_j)^{-1}\tau_i(h_i)^{-1}\tau_j(g_j)(\tau_j(g_j)^{-1}\tau_i(g_i)^{-1}\tau_j(g_j)\tau_i(g_i))\tau_i(h_i) & = \omega^{ij}_{g_i}(g_j)\omega^{ij}_{h_i}(g_j).
\int_{Q_R}f^l \, d\mu\leq (c_0 \lambda_0 M)^l \mu (Q_R) & + \left[ (M^l -1) (c_0 \lambda_0)^l \mu\big(E_f(Q_{2 R}, c_0 \lambda_0 )\big) + M^l \int_{Q_{2R}}g^l \, d\nu + M^l\frac{M^l-1}{(M^{\frac{l}{\hat p}}-1)^{\hat p}}\Big(\int_{Q_{2R}}\hat g^{\frac{l}{\hat p}}\, d\hat \nu \Big)^{\hat p}\right]\sum_{j=1}^\infty (\alpha M^l )^j.
\sum_{k = 1}^{m}{\frac{1}{\bar{T}_{i}\bar{T}_{k}}}\left( \bar{T}_{i}\delta_{i k}+ X_{i k}^{( m )}\right) A_{k j}\ = \ \delta_{i j}+{\frac{1}{{\bar{T}}_{i}}}\left( X_{i j}^{( m )}- X_{i j}- \sum_{k = 1}^{m}{\frac{1}{{\bar{T}}_{k}}}X_{i k}^{( m )}X_{k j}\right) = \delta_{i j}\ .
w(t_1)=\frac{1}{5},w(t_2)=\frac{2}{5},w(t_3)=1, g_1 = \big({-}t_1^6 - 15t_1^4t_2 + 15t_1^2t_2^2 + 10t_2^3 + 30t_1t_3\big)/30, g_2 = \big(5t_1^7 + 3t_1^5t_2 + 15t_1^3t_2^2 - 5t_1t_2^3 + 6t_2t_3\big)/6, g_3 = \big({-}105t_1^{10}+ 200t_1^8t_2 + 350t_1^6t_2^2 + 175t_1^2t_2^4 - 14t_2^5 + 20t_3^2\big)/40.
\mathbb{K}_{2}(s):=\frac{15}{8}\left( 1-s^{2}\right)^{2}\mathbf{1}_{\left\{0\leq s<1\right\}},\mathbb{K}_{3}(s):=\frac{35}{16}\left(1-s^{2}\right)^{3}\mathbf{1}_{\left\{0\leq s<1\right\}}.
\Delta_{1 2}^{i j}= - [ ( \pi . \Lambda ) g^{i j}- \pi^{i}\Lambda^{0 j}+{\frac{e}{m}}\epsilon^{j b c}F_{b}^{i}S_{c}] = - \Delta_{2 1}^{j i}; \pi . \Lambda = \pi_{a}\Lambda^{0 a}
E_{ab}z^{k}\mapsto \colon\sum_{l\in\mathbb{Z}}\big(e_{a}z^{k+l}\wedge\big)\big(i\big(e_{b}z^{l}\big)\big)\colon=\sum_{l\in\mathbb{Z}}\colon{}_{a}\psi^{+}_{(k+l)}{}_{b}\psi^{-}_{(-l-1)}\colon.
1= \prod_{j=2}^{N}\Big[\chi_{1,j}^{0}+\chi_{2,j}^{0}\Big] =\sum_{p_0\cup q_0=\{2,\ldots,N\},p_0\cap q_0=\emptyset}\Big[\prod_{j\in p_0}\chi_{1,j}^{0}\Big] \Big[\prod_{j\in q_0}\chi_{2,j}^{0}\Big]\,.
F^{\prime}\left(g_{s,1,2}\right)= F\left(g_{e,2,1}\circ g_{s,1,2}\right)\overset{(1)}{=}F\left(g_{e,3,1}\circ g_{e,2,3}\circ g_{s,1,2}\right) \overset{(2)}{=}F\left(g_{e,1,3}\circ g_{s,2,3}\circ g_{e,1,2}\right)=F^{\prime}\left(g_{s,2,3}\right).
\left(\frac{1}{8}\right)\left((B^2-k)^{1/2}(2B^3-kB)-k^2\ln\left(B+(B^2-k)^{1/2}\right)\right) &\qquad=\frac{\lambda_1\lambda_2^2\lambda_3\omega^{1/2}}{\lambda_4}t+k_1,
\frac{1 0 6}{1 1 1}= \frac{1 0 . 6 g}{x}- \frac{1 0 6}{1 0 0}= \frac{1 0 . 6 g}{y}\frac{1 0 6}{1 1 7}= \frac{1 0 . 6 g}{z . \cdot}
\bigg|x^{s+(1/2)}\int_x^1 dt \, t^{-s - (1/2)}g(t)\bigg| \leq x^{s + (1/2)}\bigg(\int_x^1 dt \, t^{-2s -1}\bigg)^{1/2}\|g\|_{L^2((0,1);dt)}& = (2s)^{-1/2}x^{1/2}\big[1 - x^{2s}\big]^{1/2}\|g\|_{L^2((0,1);dt)}, x \in (0,1),
1 = \sum_{i=1}^n(\gamma_i-\alpha_i) z_i = \sum_{i=1}^n(\gamma_i-\alpha_i) \sum_{k=1}^{N_1}u_k e_{ki}= \sum_{k=1}^{N_1}u_k \sum_{i=1}^n (\gamma_i-\alpha_i) e_{ki}= \sum_{k=1}^{N_1}u_k e_k,
|\tau \rangle =\sum_{h_{1},...,h_{\mathsf{N}}=0}^{p-1}\prod_{a=1}^{\mathsf{N}}Q_{\tau ,a}^{(h_{a})}\prod_{1\leq b<a\leq \mathsf{N}}(X_{a}^{(h_{a})}-X_{b}^{(h_{b})})|h_{1},...,h_{\mathsf{N}}\rangle ,
\tilde{G}_{\mu_{1}\ldots \mu_{p}}^{a}\equiv p ! \frac{\partial L}{\partial \bar{F}^{a \mu_{1}\ldots \mu_{p}}}, \tilde{\bar{G}}_{\mu_{1}\ldots \mu_{p}}^{a}\equiv p ! \frac{\partial L}{\partial F^{a \mu_{1}\ldots \mu_{p}}}.
V^{( r )}\hspace{- . 0 7 c m}( m , \Lambda ) - V^{( r )}\hspace{- . 1 2 c m}\left( m , \Lambda^{\prime}\right) ={\delta V^{( r )}\hspace{- . 0 7 c m}( m , \Lambda , \Lambda^{\prime}) - \sum_{s = 2}^{r - 1}B_{r - s , s}V^{( r - s )}\hspace{- . 0 7 c m}( m , \Lambda )},
u_t-a(t,x)u_{xx}=f(t,x), (t,x)\in (0,T)\times (-\ell,\ell), u_t\pm b(t,x)u_x =g(t,x), \qquad\phantom{bla}t\in (0,T), x=\pm \ell, u(0,x) =u^0(x), \qquad\quad\;\;\;\, x\in [-\ell,\ell]
D_{- \frac{i}{2}\Lambda - 1}( - ( 1 + i ) \tau ) , \; \; D_{- \frac{i}{2}\Lambda - 1}( ( 1 + i ) \tau ) , \; \; D_{\frac{i}{2}\Lambda}( ( 1 - i ) \tau ) , \; \; D_{\frac{i}{2}\Lambda}( - ( 1 - i ) \tau ) ,
\Gamma_{d i v}^{l g}= \frac{1}{4 \pi \epsilon}\int d^{d}x \, \sqrt{- g}\, \left\{\frac{2 4 - n}{6}R - \left[ 8 - n \Phi^{\prime}( \phi )^{2}\right] g^{\mu \nu}\partial_{\mu}\phi \partial_{\nu}\phi + 2 \Lambda e^{2 \phi}\right\},
L_{2}^{\pm}(u)^{-1}=\begin{pmatrix}* -e_{1}^{\pm}(u)k_{2}^{\pm}(u)^{-1}0 0 -k_{2}^{\pm}(u)^{-1}f_{1}^{\pm}(u) k_{2}^{\pm}(u)^{-1}0 0 0 0 * -e_{1}^{\pm}(u)k_{2}^{\pm}(u)^{-1}0 0 -k_{2}^{\pm}(u)^{-1}f_{1}^{\pm}(u) k_{2}^{\pm}(u)^{-1}\end{pmatrix}
\lim_{N\to\infty}\dfrac{1}{N^2}\sum_{i=0}^{N-1}\sum_{j=0}^{N-1}f(a^{p^iq^j}) =\lim_{N\to\infty}\dfrac{1}{N^2}\sum_{i=0}^{N-1}\sum_{j=0}^{N-1}f(c^{p^iq^j}) &=\lim_{N\to\infty}\dfrac{1}{N^2}\sum_{i=0}^{N-1}\sum_{j=0}^{N-1}f(b^{p^iq^j})
\tau_{n}^{\beta\alpha}=\phi_{n}^{\beta}\circ\left( \phi_{n}^{\alpha}\right)^{-1}:\phi_{n}^{\alpha}\left( U_{n}^{\alpha}\cap U_{n}^{\beta}\right)\longrightarrow\phi_{n}^{\beta}\left( U_{n}^{\alpha}\cap U_{n}^{\beta}\right).
{\cal L}= \sqrt{- \hat{G}}\, [{\cal R}_{\hat{G}}+ \hat{G}^{M N}\partial_{M}\Phi \partial_{N}\Phi -{\textstyle{\frac{1}{1 2}}}\hat{H}_{M N P}\hat{H}^{M N P}-{\textstyle{\frac{1}{4}}}\hat{F}_{M N}^{I}\hat{F}^{I \, M N}] ,
I_2 =-\partial_x \left(\left\langle A(v)\left(\partial_x v,\partial_x v\right),\partial_x v\right\rangle\right)+\left\langle A(v)\left(\partial_x v,\partial_x v\right),\partial_{xx}^2 v\right\rangle &=\left\langle A(v)\left(\partial_x v,\partial_x v\right),\partial_{xx}^2 v\right\rangle.
0\neq \frac{10}{7}F\left(\frac{8}{3},\frac{1}{3},\frac{10}{3};\frac{1}{\lambda}\right)(\lambda^2-\lambda+1)-\frac{1}{6}F\left(\frac{5}{3},\frac{1}{3},\frac{7}{3};\frac{1}{\lambda}\right)\left(\lambda+1 \right)(8\lambda^2-11\lambda+8)+ &F\left(\frac{2}{3},\frac{1}{3},\frac{4}{3};\frac{1}{\lambda}\right)\lambda(1-\lambda)^2,
\prod_{j=1}^{n_i}c_{i,j}^{\mu_{i,j}}=1 \ \ (i=1,2), \prod_{j=1}^{n_3}c_{3,j}^{\mu_{3,j}}=\zeta_d^{\beta\mu_3}, c_{i,j}\ne c_{i,j'}\ \mbox{if $j\ne j'$},
\int_\Omega \omega(y) (x\cdot y)\, dy = -\int_\Omega \omega(y) (x\cdot y)\, dy - \int_\Omega (\omega(y) \times y) \times x\, dy = - \frac 12 \left( \int_\Omega (\omega(y) \times y)\, dy \right) \times x.
\frac{1}{2}\frac{d}{dt}\int_{\mathbb T^2}\varphi^2(\theta)dx+\int_{\mathbb T^2}\nabla\cdot\left(\frac{1}{2}\varphi^2(\theta)u\right)\, dx +\nu\int_{\mathbb T^2}\Lambda\theta\varphi(\theta)\, dx=\int_{\mathbb T^2}f\varphi(\theta)\, dx.
\displaystyle \prod_{n = 1 \atop n \equiv \pm ( j + 1 ) ( \mathrm{m o d}5 )}^{\infty}( 1 - q^{n})^{- 1}= \displaystyle \sum_{r = 0}^{\infty}\frac{q^{r^{2}+ j r}}{( q )_{r}}, j = 0 , 1 ,
e^{-\rho x/2}\int_0^x e^{\rho\xi/2}\alpha(\xi)d\xi=\int_0^x e^{-\rho \eta/2}\alpha(x-\eta)d\eta \le \int_0^1 e^{-\rho \eta/2}\alpha(x-\eta)d\eta+\sum_{j=1}^{\infty}\int_j^{j+1}e^{-\rho \eta/2}\alpha(x-\eta)d\eta\le
\mu_* = \inf_{\overset{u \in W^{1,p(x)}_0 (\Omega)}{u\neq 0}}\frac{\displaystyle\int_{\Omega}\frac{|\nabla u|^{p(x)}}{p(x)}}{\displaystyle\int_{\Omega}\frac{|u|^{p(x)}}{p(x)}}, \overline{\mu}_* = \inf_{\overset{u \in W^{1,p(x)}_0 (\Omega)}{u\neq 0}}\frac{\displaystyle\int_{\Omega}|\nabla u|^{p(x)}}{\displaystyle\int_{\Omega}| u|^{p(x)}}.
E = -{\frac{1}{4 \pi a}}\int_{- \infty}^{\infty}d y e^{i y \delta}\sum_{l = 1}^{\infty}( 2 l + 1 ) x{\frac{d}{d x}}\ln [ 1 - \xi^{2}( ( s_{l}e_{l})^{\prime})^{2}] ,
M^{-1}(M^T + N) M^{-T}= (I-G) \mathcal{A}^{-1}+ G \mathcal{A}^{-1}G^T &= \mathcal{A}^{-1}- G \mathcal{A}^{-1}G^T &= \sum_{l=0}G^l \Sigma (G^T)^l - \sum_{l=1}^\infty G^l \Sigma (G^T)^l &= \Sigma.
\Big [ \frac{t_{n}^{a}\, t_{m}^{a}}{z_{n}- z_{m}}\, , \, \frac{t_{m}^{a}\, t_{\ell}^{a}}{z_{m}- z_{\ell}}\Big ] + \mathrm{p e r m u t a t i o n s ~ c y c l i q u e s}\, = \, 0 \, .
D_{X}(t) = D_{X}(0)+\int_{0}^{t}\frac{dD_{X}(s)}{ds}ds \leq D_{X}(0)+\int_{0}^{t}D_{V}(s)ds &\leq D_{X}(0)+\int_{0}^{t}D_{V}(0)\exp\left(-\frac{\kappa_1\mathcal{A}(v)(0)\phi(D_X^\infty)}{T_M^\infty}s\right)ds &\leq D_{X}(0)+\frac{T_M^\infty}{\kappa_1\mathcal{A}(v)(0){\phi(D_X^\infty)}}\cdot D_{V}(0)< D_{X}^\infty, \forall t\in [0,t^*],
P e^{\int d \tau M ( \tau )}= \sum_{N = 0}^{\infty}\int d \tau^{1}\dots d \tau^{N}\theta ( \tau_{1 2}) \dots \theta ( \tau_{N - 1 , N}) M ( \tau_{1}) \dots M ( \tau_{N}) , \ \theta ( \tau_{1 2}) = \theta ( \tau_{1}- \tau_{2}) \ .
L^{+ 4}( x , \theta^{+}, u ) = \mathrm{T r}( A^{+ +}\partial^{+ \alpha}A_{\alpha}^{+}-{\frac{1}{2}}A^{+ \alpha}D^{+ +}A_{\alpha}^{+}+ A^{+ +}A^{+ \alpha}A_{\alpha}^{+}) \ .
d_i (\omega_0, \dots, \omega_{n-1}, \omega) = \begin{cases}(\omega_0, \dots, \omega_i \omega_{i+1}, \dots, \omega_{n-1}, \omega) i < n (\omega_0, \dots, \omega_{n-1}\omega) i = n (\omega_1, \dots, \omega_{n-1}, \omega f(\omega_0)) i = n+1 \end{cases}
[{\bf{w}}_t^{opt},{\bf{w}}_r^{opt}] = \arg \mathop{\max}_{{{\bf{w}}_{\bf{t}}}{\bf{,}}{{\bf{w}}_{\bf{r}}}}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}\sum\nolimits_{l = 1}^N |{{\bf{w}}_r^T{{\bf{g}}_l}\lambda_l}{{\bf{h}}_l^T{{\bf{w}}_{\bf{t}}}}|^2.
[\begin{matrix}-2+2i&7+3i&7+3i&-5+4i\\ 3+3i&-2+4i&6+2i&-1+4i\\ 2+2i&-8+0i&-9+1i&-7+5i\\ 8+2i&-9+0i&6+3i&-4+4i\end{matrix}],
\pi^{O(2p,2q)}_{_\ell}|_{U(p)\times U(q)}= \sum_{\substack{0\le b,c begin{align*}.2ex] b+c \ge \ell+2q}}\sum_{\substack{0\le b',c' begin{align*}.2ex] b'+c' \le b+c -\ell-2q begin{align*}.2ex] b'+c' \equiv b+c-\ell \pmod{2}}}\pi^{U(p)}_{b,c}\otimes \pi^{U(q)}_{b',c'}.
\tilde{\mathcal{Q}}\triangleq \left\{\tilde{\mathbf{q}}_{\tilde{v}}\left|\tilde{\mathbf{q}}_{\tilde{v}}=\left[\tilde{f}_{\tilde{v}}(-N_{\rm cp}),\tilde{f}_{\tilde{v}}(-N_{\rm cp}+1),\ldots, \right. \tilde{f}_{\tilde{v}}(-N_{\rm cp}+L-1)\right]^{\rm T}, \tilde{v}\in \mathcal{U}_{2V}\right\}.
\sum^{\lfloor n/e\rfloor}_{\substack{r=1 (r,n)=1}}\frac{1}{r^{2}}\equiv -J_{e}(n)n^{\phi(n)-2}\phi_{J_{e}}^{(2-\phi(n))}(n)\frac{B_{\phi(n)-1}(\frac{1}{e})}{\phi(n)-1}\pmod{n},
\phi_{i}([x_{0}:\cdot\cdot\cdot:x_{i}:\cdot\cdot\cdot:x_{n}])=(\frac{x_{0}}{x_{i}},...,\frac{x_{i-1}}{x_{i}},\frac{x_{i+1}}{x_{i}},...,\frac{x_{n}}{x_{i}})=\mathbb{C}_{i}^{n}
n \: C \alpha_{1}^{1 - p}\: \beta_{1}^{1 - q - \gamma}\left( \prod_{j = 2}^{p}\: \alpha_{1}^{a_{j}- \alpha_{j}}\: \alpha_{j}^{- a_{j}}\right) \beta_{i_{0}}^{\gamma}\left( \prod_{k = 2}^{q}\beta_{1}^{b_{k}- \beta_{k}}\: \beta_{k}^{- b_{k}}\right) ,
Q_1= \frac{1}{|Dw|^2\log|Dw|^2}\bigg[ 2(1-G_u)^2u_{11}^2+2(1-G_u)^2(v^2+1)\sum_{2\leq i\leq n}u_{1i}^2 &+4(1-G_u)\frac{u_1}{v^2}u_{11}\sum_{1\leq k\leq n}w_ku_{k1}+4(1-G_u)u_1\sum_{2\leq i\leq n}u_{1i}\sum_{1\leq k\leq n}w_ku_{ki}&+2(1-G_u)^2v^2\sum_{2\leq i\leq n}u_{ii}^2\bigg];
\frac{(\Gamma n^0 \, \Gamma u^0_3)^2}{2 (\Gamma n^0 + N^0)^2}\: + \: T^i \ln(\Gamma n^0 + N^0) \: = \: \Phi^0 \:+ \: \frac{1}{2}(\Gamma u^0_3)^2 + + T^i \ln(\Gamma n^0).
\mathcal{D}\left((s^{(1)}_{k})_{k=0}^{2m-1}, (1,0,\ldots,0)\right) \stackrel{a\to0+}{\longrightarrow}\mathcal{D}\left((s_{k+1})_{k=0}^{2m-1}, (1,0,\ldots,0)\right) = (-1)^{m}\mathcal{E}\left((s_{k+1})_{k=0}^{2m-1}, (1,0,\ldots,0)\right).
\max_{\boldsymbol{Q}}\textrm{tr}( (\boldsymbol{R}+\boldsymbol{V})\boldsymbol{Q}) & ~~\textrm{s.t.}\quad~ \boldsymbol{Q}\succcurlyeq 0; Q_{n,n}=1, \forall n = 1,\ldots, N+1.
\sum_{m=1}^{n}\int_{0}^{t}\int_{\mathbb{S}(\mathbb{C}_{n})}\int_{\mathbb{R}}\Bigg[f(\lambda_{m}^{(n)}(X_{s-}^{(n)}+rvv^{\ast}))-f(\lambda_{m}^{(n)}(X_{s-}^{(n)})) & -f^{\prime}(\lambda_{m}^{(n)}(X_{s-}^{(n)}))\mathrm{tr}(D\lambda_{m}^{(n)}(X_{s-}^{(n)})rvv^{\ast})1_{\{|r|\leq1\}}\Bigg]\rho_{n}^{\alpha}(dr)\pi(dv)ds.
|Law(X(t))- Law(X(s))| = \sqrt{\sum_{x=1}^d \left[e_x \cdot (Law(X(t))- Law(X(s)))\right]^2}&\leq \sqrt{\sum_{x=1}^d |t-s|^2 4 |e_x|_{\infty}^2 \nu(U)^2}= 2 \nu(U)\sqrt{d}|t-s|,
V_2^* V_1 = ( I_{H^2}\otimes P U + M_z^*\otimes P^\perp U )(I_{H^2}\otimes P^\perp U+ M_z\otimes PU) & = I_{H^2}\otimes P U{P^\perp}U + M_z \otimes P U P U + M_z^*\otimes{P^\perp}U{P^\perp}U + I_{H^2}\otimes{P^\perp}U P U
F^{1 / 2}= \frac{2^{1 / 2}N^{1 / 2}g_{s}( M_{s}^{2}A_{T})}{5 \pi \sqrt{3}( 2 \pi )^{3}( 1 . 9 1 \times 1 0^{- 5}) ( 2 \pi R M_{s})^{6}}.
ug^{ij}g^{kl}\nabla_jh_{il}f_k = u(\nabla^i h^j_i)f_j = u(\nabla^j h^i_i+\overline{R}^{\,ij}_{\, \,\,\,li}N^l)f_j = 2uH^jf_j - uf_j \overline{R}^{\,j}_l N^l = 2uH^jf_j - uk_0f_jg^j_l N^l= 2u\langle\nabla H,\nabla f\rangle - uk_0\langle \nabla f,\mathbf{N}\rangle = 2u \langle \nabla H,\nabla f \rangle,
\int_{\Omega}u (-\Delta \phi_1(q_0))\, dx + \int_{\Omega}q_0 u\phi_1(q_0) \,dx = \lambda_1(q_0) \int_{\Omega}u\phi_1(q_0) \,dx- \int_{\Omega}u^\frac{p+1}{p-1}\phi_1(q_0) \,dx,
X_t =e^{A(t-s)}X_s + \int^t_s e^{A(t-r)}Y_r \, dr + \int^t_s e^{A(t-r)}Z_r \, dW_r &=e^{A(t-s)}\mathcal{X}_s + \int^t_s e^{A(t-r)}Y_r \, dr + \int^t_s e^{A(t-r)}Z_r \, dW_r.
g = \Delta_{1, 2, 3, 5}\Delta_{1, 2, 3, 7}\Delta_{4, 6, 8, 9}- \Delta_{1, 2, 3, 4}\Delta_{1, 2, 3, 7}\Delta_{5, 6, 8, 9}-\Delta_{1, 2, 3, 5}\Delta_{1, 2, 3, 6}\Delta_{4, 7, 8, 9}+ \Delta_{1, 2, 3, 4}\Delta_{1, 2, 3, 6}\Delta_{5, 7, 8, 9}.