harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left\|\nabla^\star{\boldsymbol u}\right\|_{2,B}^2=\left\|\nabla{\boldsymbol u}\right\|_{2,B}^2+\left\|\tfrac{\delta}{B}u_2\right\|_{2,B}^2+\left\|\tfrac{\delta}{B}u_3\right\|_{2,B}^2.
\mathit{CWE}(\mathcal{C})=\sum\limits_{\mathbf{t}\in \mathbb{F}_{p}^{p}\setminus \lbrace \mathbf{0}\rbrace}\mathfrak{F}(t_{0},t_{1},\cdots,t_{p-1})w_{0}^{t_{0}}w_{1}^{t_{1}}\cdots w_{p-1}^{t_{p-1}}
\langle \hat \theta_r-\theta_r,u\rangle= \frac{\langle (\hat P_r-P_r)\theta_r,u\rangle-\langle \hat \theta_r-\theta_r, \theta_r \rangle \langle \theta_r, u\rangle}{1+\langle \hat \theta_r-\theta_r, \theta_r\rangle}.
u_{\tau}^D(t):=u_{\tau,m}^D(t):=\Phi_{j,t-t_m}^D\prod_{j=1}^{m-1}\big(\Phi_{j,\tau}^S\Phi_{j,\tau}^D\big)u_\tau(0),t\in \{t_m \cup T_m\}/t_{m+1}.
{\cal V}^{H}( \mathrm{\bf r}_{1}, \mathrm{\bf r}_{2}) ={\frac{1}{2}}\left\{{\cal V}( \mathrm{\bf r}_{1}, \mathrm{\bf r}_{2}) +{\cal V}^{\dag}( \mathrm{\bf r}_{1}, \mathrm{\bf r}_{2}) \right\}. \nonumber
\psi(E_{ij})= z_{ji}E_{ji}+z_{\sigma^{-1}(j),i}E_{\sigma^{-1}(j),i}+z_{j,\sigma^{-1}(i)}E_{j,\sigma^{-1}(i)}&+z_{\sigma^{-1}(j),\sigma^{-1}(i)}E_{\sigma^{-1}(j),\sigma^{-1}(i)},
\langle 0 | \varphi^{2}( x ) | 0 \rangle = \langle 0_{{\mathrm{m}}}| \varphi^{2}( x ) | 0_{{\mathrm{m}}}\rangle + \langle \varphi^{2}( x ) \rangle_{b}^{( a )}+ \langle \varphi^{2}( x ) \rangle_{b}^{( b )}+ \langle \varphi^{2}( x ) \rangle^{( a b )},
\log(\kappa(\tfrac{\eta}{n}, \tfrac{1}{\eta})) = -\log(A(\tfrac{\eta}{n})) + \log(B(\tfrac{\eta}{n})) = - \log( A(\tfrac{\eta}{n}, \tfrac{1}{\eta})) - 2\log(\tanh(\tfrac{\eta}{2})) + \log(h(\eta)),
\begin{array}{r c l}{{{\hat{c}}^{( 6 )}{}_{i j k l m 5}}}&{{=}}&{{- 5 ( 2 \pi \alpha^{\prime}) \partial_{[ i}v^{( 0 )}c^{( 4 )}{}_{j k l m ]}- 5{\textstyle \frac{m}{2}}( 2 \pi \alpha^{\prime})^{3}v^{( 0 )}b_{[ i}\partial_{j}b_{k}\partial_{l}b_{m ]}}}\\{{}}&{{}}&{{}}\\{{}}&{{}}&{{+ v^{( 5 )}{}_{i j k l m}\, ,}}\\ \end{array}
\mathbb{E}_{f_0,T}\|\hat{f}-f_0\|_2^2 \geq \sum_{j\geq j_n}\sum_{k=0}^{2^j-1}f_{0,jk}^2\geq \sum_{k=0}^{2^j-1}2^{-j_n (2s+1)}& \gtrsim \Big(\frac{B}{\log_2 n}\Big)^{-2s}=\tilde{M}_n \Big(\frac{n^{1/(1+2s)}}{B\log_2 n}\Big)^{\frac{2s}{2+2s}}n^{-\frac{2s}{1+2s}}
\varepsilon^{2}\partial_{s}U=\partial_{\xi\xi}U+\left(\frac{d-1}{\xi}+(2\dot\varepsilon\varepsilon-\varepsilon^{2})\frac{\xi}{2}\right)\partial_{\xi}U-\frac{(d+1)}{2\xi^{2}}\sin(2U),U(0,s)=0.
f(y) =\sum_{\alpha\in\mathbb{Z}_+^n,\,|\alpha|<N}\frac{(y-z)^\alpha}{\alpha!}\partial^\alpha f(z) &\quad+\sum_{\alpha\in\mathbb{Z}_+^n,\,|\alpha|=N}\frac{(y-z)^\alpha}{\alpha!}\int_0^1\partial^\alpha f(z+t(y-z))N(1-t)^{N-1}\,dt
\begin{array}{rcl}H(t,\rho) = \displaystyle \int_{-1}^t[f(s)x(s,\rho)+g(s)]ds &= \displaystyle \left(\int_{-1}^tf(s)ds\right)\rho + \sum_{k=2}^{\infty}\left(\int\limits_{-1}^tf(s)r_k(s)ds\right)\rho^k +\int_{-1}^tg(s)ds &= \displaystyle G(t)+ F(t)\rho + \sum_{k=2}^{\infty}\left(\int\limits_{-1}^tf(s)r_k(s)ds\right)\rho^k.\end{array}
\begin{array}{c c c}{\partial_{\sigma}\varrho = 0 \hfill}{\Longrightarrow}{\varrho = \rho ( x , \tau ) , \hfill}{\partial_{\sigma}\theta = 1 \hfill}{\Longrightarrow \hfill}{\theta = \Theta ( x , \tau ) + \sigma , \hfill}\end{array}
\left| S_6 \right|^2 = \left| \sum_{M < m \leq M_1}b(m) \sum_{\substack{\frac{X_3}{m}< n \leq \frac{X_4}{m}n > v}}\Lambda(n) e(q f(mn)) \right|^2 &\leq \left(\sum_{M < m \leq M_1}|b(m)|^2 \right) \left( \sum_{M < m \leq M_1}\left| \sum_{\substack{\frac{X_3}{m}< n \leq \frac{X_4}{m}n >v}}\Lambda(n) e(q f(mn)) \right|^2 \right)
B=\left(\begin{array}{cc}0 1 -1 0\end{array}\right),\ \ \Omega(x)=\left(\begin{array}{cc}p(x) q(x) q(x) -p(x)\end{array}\right),\ \y(x)=\left(\begin{array}{c}y_1(x) y_2(x)\end{array}\right),
\begin{matrix}G(n-1)&=&p^{-1}p^{n}+&q^{-1}q^{n}+&r^{-1}r^{n}\\G(n)&=&p^{n}+&q^{n}+&r^{n}\\G(n+1)&=&pp^{n}+&qq^{n}+&rr^{n}\end{matrix}
V_\Lambda\otimes\frak{p}_{\mathbb{C}}=V_\Lambda\otimes\frak{p}_+\oplus V_\Lambda\otimes\frak{p}_-, & V_\Lambda\otimes \frak{p}_+\cong V_{r+1, s+1}\oplus V_{r-1,s+2}, & V_\Lambda\otimes\frak{p}_-\cong V_{r+1,s-2}\oplus V_{r-1,s-1}.
{\mathfrak G}_{u_{n}^{(k)}}^{(\beta-1)}(x_i=1,\,\forall\, i \ge 1) =(1+\beta)^{{n+1 \choose 2}}{\mathfrak G}_{1^{k}\times w_{0}^{(n+1)}}^{((\beta)^2-1)}(x_i=1,\,\forall\, i \ge 1),
p_\tau^k=\mathfrak{t}(e(v_\tau^k)) +\frac{2\xi_2{+}\xi_1\tau}{\xi_2{+}\xi_1\tau{+}\xi_0\tau^2}p_\tau^{k-1}-\frac{\xi_2}{\xi_2{+}\xi_1\tau{+}\xi_0\tau^2}p_\tau^{k-2}.
\Theta = \left\{1,\frac{\theta}{\theta+1},\frac{\theta}{\theta+2},\ldots\right\}; q^{\deg \mathfrak{I}}= \left\{q^{\deg P},\,\,P \in \mathfrak{I}\right\}; (q^{\deg \mathfrak{I}}-1) = \left\{q^{\deg P}-1,\,\,P \in \mathfrak{I}\right\}
f(\gamma^m)=\sum_{i=0}^{n-1}\sum_{a\in q^iK}c_i\gamma^{q^ja_0a}=\Bigl(\sum_{i=0}^{n-1}\sum_{a\in q^iK}c_i\gamma^a\Bigr)^{q^j}=f(\gamma)^{q^j}=0.
\forall\;0\le k<n,y_{k}=\sqrt[m]{\mbox{Prod}_{\mathbf{P}_{k}}\left(\mathbf{x}^{\top^{\left(m-1\right)}},\mathbf{x}^{\top^{\left(m-2\right)}},\cdots,\mathbf{x}^{\top^{j}},\cdots,\mathbf{x}^{\top^{1}},\mathbf{x}^{\top^{0}}\right)},
\begin{pmatrix}\lambda_1' + I x_1' \cr\lambda_2' + I x_2' \cr\lambda_2' + I x_2'\end{pmatrix}= e^{\left(K_1\Lambda_3 + K_2 \Lambda_8\right) I}\begin{pmatrix}\lambda_1 + I x_1 \cr\lambda_2 + I x_2 \cr\lambda_2 + I x_2\end{pmatrix}~,
J_3 =\left(\begin{array}{cccccc}b_0 a_0 0 0 0 \cdots a_0 b_1 a_1 0 0 \cdots 0 a_1 b_2 a_2 0 \cdots \vdots \vdots \vdots \ddots \end{array}\right),a_k>0,\b_k\in\mathbb{R},\k\in\mathbb{Z}_+;
{\varepsilon^*_{n-k+1,n-1}}= t_{1,n}\cdot{\varepsilon^*_{n-k,n}}\cdot t_{n,1}, id - X_{\{k,\,k+1,\,\dots ,\,n\}}= \Delta_{n-k+1,n}\cdot t_{1,n}\cdot \left(\Delta_{n-k,n}\right)^{-1}\cdot t_{n,1}.
a^{2}\nabla_{\nu}b^{2}\leqslant a^{2}\sharp_{\nu}b^{2}+(a - b)^2-\sum_{k=0}^{\infty}r_{k}\big[a^{\frac{m_k}{2^k}}b^{1-\frac{m_k}{2^k}}- a^{\frac{m_k+1}{2^k}}b^{1-\frac{m_k+1}{2^k}}\big]^{2}.
\big|\langle\omega_0, \psi(\phi(a)\pi_s)\rangle\big| =\big|\tau(\phi(a)\pi_s)\big|=\big|\tau( \pi_s^*\pi_s\phi(a)\pi_s)\big| &=\big|\tau(\pi_s^*\phi(a)\pi_s\pi_s^*)\big|\leq \big|\tau(\pi_s^*\phi(a)^2\pi_s)\big| =0,
<\psi_f|e^{-i/\hbar(t_2-t_1)\hat{H}_{\rm eff}}|\psi_i>=i\delta_{n,m}(t_2-t_1)\int_{-\infty}^\infty d\lambda\,\tilde{\psi}^*_f(\lambda)\,e^{-i/\hbar\lambda(t_2-t_1)(\hbar\omega(n+1/2)-\Lambda)}\,\tilde{\psi}_i(\lambda)\ ,
\left| \frac{1}{(\varphi(4))^k}\mathrm{Pf}\Big[ K_{N}\Big(4N+\frac{u_i}{\varphi(4)}, 4N+\frac{u_j}{\varphi(4)}\Big) \Big]_{i,j = 1}^{k}\right| \leq (2k)^{\frac{k}{2}}C^{k}\prod_{j=1}^{k}e^{-(a-v)u_{j}}.
d(\mathcal{C}_1(p), \mathcal{C}_2(p))= \max\left\{\sup\limits_{h\in \mathcal{C}_2(p), h \neq 0}\inf\limits_{k \in \mathcal{C}_1(p)}\frac{\|h-k\|_{H}}{\|h\|_{H}}, \sup\limits_{h\in \mathcal{C}_1(p), h \neq 0}\inf\limits_{k \in \mathcal{C}_2(p)}\frac{\|h-k\|_{H}}{\|h\|_{H}}\right\}.
\dot{x}\left(t\right) =\left[\begin{array}{cc}0_{2}I_{2}0_{2}0_{2}\end{array}\right]x\left(t\right)+\left[\begin{array}{c}0 0 0 \pm1\end{array}\right]a_{p}+\left[\begin{array}{c}0 0 0 -1\end{array}\right]a_{e}, =Ax\left(t\right)+Ba_{p}+Da_{e},
\frac{d(\mu\circ\sigma_j^{-1})}{d\mu}(x_1x_2\dots)=\delta_{j,x_1}\frac{\lambda_{x_2}}{\lambda_{x_1}T_{x_1,x_2}^{(1)}}F(x_2x_3\dots)=:f_j^2(x_1x_2\dots),\, f_j\geq 0,
\lambda_{1}=\frac{\gamma+\sqrt{\gamma^2-4\omega^2}}{2},\lambda_{2}=\frac{\gamma-\sqrt{\gamma^2-4\omega^2}}{2},\quad\lambda_1+\lambda_2=\gamma,\lambda_1\lambda_2=\omega^2,\lambda_1-\lambda_2=\sqrt{\gamma^2-4\omega^2}.
T_\pi s_0T_\pi^{-1}= (s_1\cdots s_{n-1}y_n)(y_1y^{-1}_n s_{n-1}\cdots s_2s_1s_2 \cdots s_{n-1})(s_{n-1}\cdots s_1y^{-1}_1)=s_1y_2 y_1 y_2^{-1}y_1^{-1}= s_1.
\gamma>r^*(p-1)=\frac{Nr(p-1)}{N-sr}\Leftrightarrow a_0=\frac{1}{\gamma}-\frac{1}{p}+\frac{1}{p}\left(\frac{1}{r'}+\frac{1}{p-1}(\frac{ps}{N}-\frac{1}{r})\right)<0
A_N(\gamma_1,\gamma_2)^{-2}= \frac{1}{2}\int_0^1\int_0^1 ds_1ds_2 \int_{\mathbb{R}^2}dx_1dx_2 [(s_1,x_1)_N^{\gamma_1}(s_1,x_2)_N^{\gamma_2}+(s_1,x_1)_N^{\gamma_2}(s_1,x_2)_N^{\gamma_1}] \qquad\qquad\qquad~~\times[(s_2,x_1)_N^{\gamma_1}(s_2,x_2)_N^{\gamma_2}+(s_2,x_1)_N^{\gamma_2}(s_2,x_2)_N^{\gamma_1}].
\begin{aligned}{I\!I}_{13}=-4\int \nabla (u \cdot \nabla d):|\nabla d|^2 \nabla d \dx \le \int (|\nabla u| |\nabla d|^4+ |u| |\nabla d|^3 |\nabla^2 d|)dx \le C\int (|\nabla d|^2 |\nabla u|^2 +|\nabla d|^6 +|u|^2 |\nabla^2 d|^2) dx.\end{aligned}
\hat{V}^{C}_{\sigma}(C_{1},C_{2}) = \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\frac{1}{N}\sum\limits_{n=1}^N G_{\sigma}(u_{1}-x_{n}) \, G_{\sigma}(u_{1}-y_{n}) \,G_{\sigma}(u_{2}-z_{n}) \, G_{\sigma}(u_{2}-s_{n}) \mathrm{d}u_{1}\mathrm{d}u_{2}
\psi_{k+1}(\xi) \approx \frac{2\xi\exp(\eta_k \xi^2)-\frac{\partial}{\partial \xi}\exp(\eta_k \xi^2)}{\sqrt{2(k+1)}}, & = \frac{\exp(\eta_k \xi^2)}{\sqrt{2(k+1)}}\left(\xi\left(1+\frac{\xi}{2\sigma_k}\right)+\frac{\xi^2-\sigma_k^2}{2\sigma_k^2(\sigma_k+\xi)}\right).
\P\left(\sum_{i=1}^m Y_i^2 \geq \sum_{k = \log \log n}^{L}\delta 2^{k/2}\frac{\delta^2 2^k}{n^2}\right) \geq \P\left(\sum_{i=1}^m Y_i^2 \geq \frac{C \delta^3 2^{3L/2}}{n^2}\right) &\geq \P\left(\sum_{i=1}^m Y_i^2 \geq \frac{C m^{3}}{n^2 \log^c n}\right) &= 1 - O(\delta).
\Phi_{j}:\mathbb{C}^{2}\rightarrow\mathbb{C}^{n+1}\ ,\ \mathbb{C}^{2}\supset\S^{3}(1)\rightarrow S^{2n+1}(1)\subset\mathbb{C}^{n+1}, \mathbf{z}=(z_{1},z_{2})\mapsto\Phi_{j}(\mathbf{z})=\tilde{Z}=(z_{1}^{n},\sqrt{\binom{n}{1}}z_{1}^{n-1}z_{2},..,\sqrt{\binom{n}{k}}z_{1}^{n-k}z_{2}^{k},..,z_{2}^{n})
\frac{\partial U_{\Lambda}(\varphi,T)}{\partial \Lambda}=-\frac{K_{3}}{2}2\pi T^{2}\Lambda\,g\left(\frac{\Lambda}{T}\right)\ln\det\left[\Lambda^2 +\frac{\partial^{2}U_{\Lambda}(\varphi)}{\partial\varphi_{i}\partial\varphi_{j}}\right]
V_{M}(x_{1},x_{2})=\log \left|\dfrac{\vartheta_{3}\left( \left.\frac{\pi z}{2a}\right| i\right)}{\vartheta_{1}\left( \left.\frac{\pi z}{2a}\right| i\right)}\right|^{2}+2 \log \left( \dfrac{\pi}{2a}\dfrac{\vartheta'_{1}\left( 0 \vert i\right)}{\vartheta_{3}\left( 0 \vert i\right)}\vert z\vert\right) .
\gamma_{p}:=p \eta_{1}+ (p-1)\eta_{2}+ c_{p}\big(p\hat{\eta}_{1}^{2}+ 2(p-1)\hat{\eta}_{1}\hat{\eta}_{2}+ (p-2)\hat{\eta}_{2}^{2}\big) \delta :=\eta_{2}+ 2 c_{p}\big(\hat{\eta}_{1}\hat{\eta}_{2}+ \hat{\eta}_{2}^{2}\big).
{V}_{imp}\equiv \int d^{2}z\,\partial_{\alpha}([u_{i}X^{0}]\Theta(X^{0})\partial_{\alpha}X^{i})=\int d\tau \,u_{i}\left( X^{0}\Theta(X^{0})\right) \partial_{n}X^{i})~;\qquad i=1,\ldots ,9.
\mathsf{ev}^{(1)}_{x}(e_{0})=F_{1}, \mathsf{ev}^{(1)}_{x}(e_{1})=x E_{1}, \mathsf{ev}^{(1)}_{x}(h_{0})=-H_{1}, \mathsf{ev}^{(1)}_{x}(h_{1})=H_{1}, & \mathsf{ev}^{(2)}_{x}(e_{0})=0, \mathsf{ev}^{(2)}_{x}(e_{1})=x E_{2}, \mathsf{ev}^{(2)}_{x}(h_{0})=-H_{2},\mathsf{ev}^{(2)}_{x}(h_{1})=H_{2}.
{\bf D}_{\beta}{\bf F}^{\beta \alpha}= g{\bf J}^{\alpha}\ , \mathrm{w i t h}\quad{\bf J}^{\alpha}= - i \bigl [ \Phi ,{\bf D}^{\alpha}\Phi \bigr ] \ , \quad{\mathrm{a n d}}\quad{\bf D}^{\alpha}{\bf D}_{\alpha}\Phi = 0 \ .
[\begin{matrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{6}\end{matrix}]=[\begin{matrix}\frac{E_{1}}{1-\nu_{12}\nu_{21}}&\frac{E_{2}\nu_{12}}{1-\nu_{12}\nu_{21}}&0\\ \frac{E_{2}\nu_{12}}{1-\nu_{12}\nu_{21}}&\frac{E_{2}}{1-\nu_{12}\nu_{21}}&0\\ 0&0&G_{12}\\ \end{matrix}][\begin{matrix}\epsilon_{1}\\ \epsilon_{2}\\ \epsilon_{6}\end{matrix}]
\sum_{|i-i_0| + |j-i_0| \leq N^{\nu}}\mathbb{E}[ O ( \rho_T (E) N (\mu_i (T) - E), \rho_T (E) N (\mu_j (T) - E )) | \{\lambda_k (0) \}_k] = \sum_{|i-i_0| + |j-i_0| \leq N^{\nu}}\mathbb{E}[ O ( \rho (E) N (\mu_i (T) - E), \rho(E) N (\mu_j (T) - E )) | \{\lambda_k (0) \}_k] + \O ( N^{- c_6/2})
\dot{z}_i =A_i^0z_i+f_i(x_i), \dot{\hat x}_{i}=S\hat x_i-\theta_i d d^\top P\hat x_{vi}-\theta_i d\mbox{sgn}[(d^\top P\hat x_{vi})]+\bar g_i(\bar z_i,\,z_i,\,x_i) \dot{\theta}_i =||d^\top P\hat x_{vi}||_2^2+||d^\top P\hat x_{vi}||_1 \dot{\xi}_i =A_i^0\xi_i+f_i(x_i) \dot{w}=Sw+dv
\Lambda_{m}(\mathcal{O}_{\boldsymbol{K}, q}) = \left\{H = (h_{jl}) \in M_{m}(\boldsymbol{K}\otimes_{\mathbb{Q}}\mathbb{Q}_{q}) \bigm | H^{*}= H, \h_{jj}\in \mathbb{Z}_{q}, \ \sqrt{-D_{\boldsymbol{K}}}\,h_{jl}\in \mathcal{O}_{\boldsymbol{K}, q}\right\}.
\frac{1}{R^{4}}\sum_{a = 0}^{2 l}\sum_{b = 0}^{2 l}\sum_{m_{a}= - a}^{a}\sum_{m_{b}= - b}^{b}= \frac{1}{{\pi}^{2}}\int_{\Lambda}d^{2}\vec{p}_{a}\int_{\Lambda}d^{2}\vec{p}_{b},{\Lambda}= \frac{2}{\theta},
\mathcal{E}_s=\sum_{k=0}^s \Big\{\int \frac{1}{A}|D_t\theta_k|^2+i\theta_k \overline{\partial_{\alpha}\theta_k}d\alpha+\int \frac{1}{A}|D_t\sigma_k|^2+i\sigma_k \overline{\partial_{\alpha}\sigma_k}d\alpha\Big\},
\Pi\Bigl(\sum_{i=1}^N(\kappa_if_i- \kappa_if_{0,i})^2 \leq \epsilon^2/2\Bigr) \geq \exp\Bigl(-\Bigl(1+2\alpha+2p+\frac{\log 2}{2}\Bigr)N\Bigr) \exp\Bigl(-N^{(1+2\alpha-2\beta)\vee 0}\|f_0\|_\beta^2\Bigr) \times{}\Pr\Bigl(\sum_{i=1}^N V_i^2 \leq 2\delta^2N^{1+2\alpha+2p}\Bigr).
\Vert T_{m\vert_{(K_N^{-1}\mathbb{Z})^2}}^{(N+1)}: L_{p_1}(\mathbb{T}, S_{p_1}^{N+1}) \times L_{p_2}(\mathbb{T}, S_{p_2}^{N+1}) \rightarrow L_{1}(\mathbb{T}, S_{1}^{N+1})\Vert\leq B_{p_1, p_2,N}.
\Lambda \partial_{\Lambda}{\cal{G}}_{\mathrm{i r}}^{\Lambda , \Lambda_{0}}\{\varphi_{\bf{k}}\}= - \partial_{t}{\cal{G}}_{t}\{\tilde{\varphi}_{\bf{q}}\}+ \hat{\cal{R}}_{s}{\cal{G}}_{t}\{\tilde{\varphi}_{\bf{q}}\}+ \frac{N_{t}}{2}\left[ \eta_{t}+ D \ln Z_{t}\right] ,
\iota(h) (k, g) = (1, W(1, 1)^{-1}h)(k, g) begin{align*}1mm] = (k,\, W(1, k) V(k)(W(1, 1)^{-1}h) g) begin{align*}1mm] = (k,\, W(1, k) V(k)(W(1, 1))^{-1}V(k)(h) g) begin{align*}1mm] = (k,\, V(k)(h) g),
Y=\sum_{i_{1}=0}^{p_{1}}\sum_{i_{2}=0}^{p_{2}}\cdot\cdot\cdot\sum_{i_{D}=0}^{p_{D}}(a_{i_{1}i_{2}\cdot\cdot\cdot i_{D}}\times u_{1}^{i_{1}}u_{2}^{i_{2}}\cdot\cdot\cdot u_{D}^{i_{D}})
q(\hat{A}_4=0)=\oint K^\alpha \,dS_\alpha =-\left. \int \frac{\,d^3\vec{x}}{16\pi^2}\epsilon^{ijk}\,{\rm tr\,}\left( \hat{A}_i\,\hat{F}_{jk}+ \frac{2i}3\hat{A}_i\hat{A}_j\hat{A}_k\right) \right|_{x_4=-\infty}^{x_4=+\infty}.
\mu_K(A) =\sum_{\substack{\zeta^{(k)}\in A k\in \mathcal{K}}}\mu_{f}(I_k)\Big/\sum_{k\in \mathcal{K}}\mu_{f}(I_k)= \sum_{\substack{\zeta^{(k)}\in A k\in \mathcal{K}}}\sum_{\xi \in I_k}|a_{\xi}|^2\Big/\sum_{k\in \mathcal{K}}\mu_{f}(I_k) .
- \Bigl [ \left[ Q ( \xi ) , Q ( \eta ) \right] , a_{-}( k ) \Bigr ] = \Bigl [ \left[ Q ( \eta ) , a_{-}( k ) \right] , Q ( \xi ) \Bigr ] + \Bigl [ \left[ a_{-}( k ) , Q ( \xi ) \right] , Q ( \eta ) \Bigr ] .
{\bf D}= (\delta{}_{v_1}) \otimes \cdots \otimes (\delta{}_{v_N}) , D^{\prime}_1 = (\delta{}_{o(e_1 )}) \otimes \cdots \otimes (\delta{}_{o(e_M)}) , D^{\prime}_2 = (\delta{}_{t(e_1 )}) \otimes \cdots \otimes (\delta{}_{t(e_m)}) , B^{\prime}= ( \beta{}_{e_1}) \otimes \cdots \otimes ( \beta{}_{e_m}) .
\left\lbrace\begin{array}{lll}\displaystyle \phi_r'' -\Delta \phi_r + \sum_{s=1}^p\alpha_{rs}\phi_s=0 \hbox{in}(0,+\infty) \times \Omega, \displaystyle \partial_\nu\phi_r +\sum_{s=1}^p\beta_{rs}\phi_s=0\hskip1.5cm \hbox{on}(0,+\infty) \times \Gamma, t=0:\phi_r= (E_r, \widehat{U}_0),\phi_r'=(E_r, \widehat{U}_1) \hbox{in}\Omega, \end{array}\right.
\begin{gathered}\omega^{(g)}_{m,0}=\sum_{k_1+\dots+k_m=3g-3+m}\langle \tau_{k_1}\dots\tau_{k_m}\rangle_g\;\textstyle\prod_{i=1}^m\frac{(2k_i+1)!!}{z_i^{2k_i+2}}dz_i, \langle \tau_{k_1}\dots\tau_{k_m}\rangle_g=\int_{\overline{\mathcal M}_{g,mn}}\psi_1^{k_1}\dots\psi_n^{k_m}\;.\end{gathered}
\varphi ( s , r ) = \frac{( 2 \pi )^{( k_{1}- k_{2})^{2}}}{2^{2 k_{1}k_{2}}k_{1}! k_{2}!}\, \frac{\det \left[ J_{0}( s_{p 1}r_{p^{\prime}1}) \right]_{p , p^{\prime}= 1 , \ldots , k_{1}}\det \left[ J_{0}( s_{q 2}r_{q^{\prime}2}) \right]_{q , q^{\prime}= 1 , \ldots , k_{2}}}{B_{k_{1}k_{2}}( s^{2}) B_{k_{1}k_{2}}( r^{2})}\ ,
v_p\left(\sum\limits_{k_1=0\atop p\neq 2k_1+1}^{n-r}\frac{c_{k_1}}{(2k_1+1)^{s_1}}\right)\geq \min\limits_{0\leq k_1\leq n-r\atop p\neq 2k_1+1}v_p\left(\frac{c_{k_1}}{(2k_1+1)^{s_1}}\right)=\min\limits_{0\leq k_1\leq n-r\atop p\neq 2k_1+1}v_p(c_{k_1})>v_p\left(\frac{c_{\frac{p-1}{2}}}{p^{s_1}}\right).
0 = \nabla_{( \alpha}\overline{{\xi}}_{\overline{{\beta}}) a}= \partial_{( \alpha}\overline{{\xi}}_{\overline{{\beta}}) a}- H_{\alpha \overline{{\beta}}}^{\overline{{\gamma}}}\overline{{\xi}}_{\overline{{\gamma}}a}+ H_{\alpha \overline{{\beta}}}^{\gamma}\xi_{\gamma}.
\phi ( x^0\,,\,\vec{x}\,)\,=\,c\,\int d^n x^{\prime}{(x^0)^n \over \left( (x^0)^2 \,+\,(\vec x \,-\,{\vec x}^\prime \,)^2\,\right)^n}\phi_0 ({\vec x}^\prime )\,\,.
T_{1}( p , q ) ={\frac{C_{1}( q )}{\alpha^{2}}}\left({\frac{p}{\alpha}}\right)^{- 2}+{\frac{A_{1}( q )}{\alpha^{2}}}\left({\frac{p}{\alpha}}\right)^{- ( 3 / 2 ) + ( 1 / 2 ) \eta}+{\frac{B_{1}( q )}{\alpha^{2}}}\left({\frac{p}{\alpha}}\right)^{- ( 3 / 2 ) - ( 1 / 2 ) \eta},
\Pi ( \tau ) = \frac{2 n_{c}\rho_{c}^{2}\zeta}{m^{* 2}\tau^{4}\sqrt{\pi}}\int_{0}^{\infty}d \alpha \int_{0}^{\infty}d \beta \mathrm{e}^{- \zeta^{2}{t^{2}}}\left( \zeta^{2}t^{3}- \frac{3}{2}t \right) \cosh \alpha \cosh \beta ,
\eta(x)\equiv (\exists u,t)( x=u+t\, \wedge \,(\exists y,z,y_1,z_1)(u=y_1-z_1\, \wedge\, y_1(y^3-2)=1\, \wedge \,z_1(z^3-2)=1) &\wedge(\exists y,z,y_1,z_1)(t=0\, \vee\, (t=y_1z_1 \,\wedge\, y_1(y^3-2)=1 \,\wedge\, z_1(z^3-2)=1))
\alpha = A u_{0}{\frac{2^{\nu}\Gamma ( \nu + 1 ) \Gamma ( \nu ) \Gamma ( c ) u_{0}^{- \nu}}{\Gamma ( c - a ) \Gamma ( c - b )}}, \beta = A u_{0}{\frac{\Gamma ( - \nu + 1 ) \Gamma ( - \nu ) \Gamma ( c ) u_{0}^{\nu}}{2^{\nu}\Gamma ( a ) \Gamma ( b )}}.
\sup_{T\leq1}(T^{\frac{1}{4}})^{2 - 2\alpha}\|(a \partial_1^2 w^{\tau})_T - a \partial_1^2 w^{\tau}_T \| = \| [a, (\cdot) ] \diamond \partial_1^2 w^{\tau}\|_{2\alpha -2}\lesssim [a]_{\alpha}[w^{\tau}]_{2\alpha}.
{\ell}(C) \leq{\ell}(p_{1}s_{1}^{-1}){\ell}(s_{1}){\ell}(D_{1}){\ell}(s_{1}^{-1}){\ell}(s_{1}p_{2}s_{2}^{-1}){\ell}(s_{2}){\ell}(D_{2}){\ell}(s_{3}^{-1})\cdots &{\ell}(s_{t-1}p_{t}s_{t}^{-1}){\ell}(s_{t}){\ell}(D_{t}){\ell}(p_{t+1}) \leq{\ell}(p_{1}){\ell}(D_{1}){\ell}(p_{2}){\ell}(D_{2})\cdots{\ell}(D_{t}){\ell}(p_{t+1})={\ell}(p)
= \frac{1}{2}\partial_{x}((A_{1}\partial_{xx}\textbf{w}) \cdot \partial_{xx}\textbf{w}) + \frac{3}{2}(\partial_{x}A_{1}\partial_{xx}\textbf{w}) \cdot \partial_{xx}\textbf{w}+ (\partial_{xx}A_{1}\partial_{x}\textbf{w}) \cdot \partial_{xx}\textbf{w}.
x_{i+1}(t)-x_i(t) > \dfrac{1}{R_i}t \in \left[0,t_1\right[, &x_{i+1}(t_1)-x_i(t_1) = \dfrac{1}{R_i}, 0<x_{i+1}(t)-x_i(t) < \dfrac{1}{R_i}t \in \left]t_1, t_2\right].
y^{[\sf d]}_{ij}[t] =\sum_{k=1}^{K}\mathbf{{g}}_{ij,k}[t]{\mathbf{x}}^{[\sf bs]}_{k}[t]+\sum_{k=1}^{K}\sum_{l=1}^{N}h_{ij,kl}[t]x^{[\sf u]}_{kl}[t]+z_{ij}^{[\sf d]}[t]
\widetilde{G}_{1}\left( x,t\right) =\left( G_{1,1}-G_{1,2}\right) \left(x,t\right) ,\widetilde{G}_{2}\left( x,t\right) =\left(G_{2,1}-G_{2,2}\right) \left( x,t\right) ,\left( x,t\right) \in Q_{T}.
n^{v(F' \cup F'')}p^{e(F' \cup F'')}= n^{2v(F_1) - v(S)}p^{2e(F_1) - e(S)}= \frac{O(n^{2(k - 1/m_k(F_2))})}{n^{v(S)}p^{e(S)}}\stackrel{\eqref{eq:delta_S}}{=}O(n^{k - 1/m_k(F_2) - \delta/2}),
k_{i,j}{}{}= (x \otimes t^i) \wedge (h \otimes t^j) \otimes v_{-1}+ (h \otimes t^i) \wedge (x \otimes t^j) \otimes v_{-1}{}{}+ (x \otimes t^i) \wedge (y \otimes t^j) \otimes v_1+ (y \otimes t^i) \wedge (x \otimes t^j) \otimes v_1- (h \otimes t^i) \wedge (h \otimes t^j) \otimes v_1, \ 1 \leq i < j.
\left \|\sum\limits_{j=N}^{\infty}\lambda_{j}a_{j}\right \|_{h_{\omega}^{p}}\leq C\left \|\sum\limits_{j=N}^{\infty}\frac{\lambda_j \chi_{Q_j}}{\omega(Q_j)^{\frac{1}{p}}}\right \|_{L_{\omega}^{p}};
H_{*}(X) = H_0(X) \oplus H_2(X) \oplus H_4(X) \oplus H_6(X) &= \langle 1\rangle \oplus \langle e_1, e_2, e_{12}\rangle \oplus \langle e_1\otimes e_2, e_1 \otimes e_{12}, e_2 \otimes e_{12}\rangle \oplus \langle e_1 \otimes e_2 \otimes e_{12}\rangle H_*(C) = H_*(X) \oplus H.
\begin{array}{cclcccl}\{b_k,a_k\}_2 = -b_k a_k, \; \{a_k,b_{k+1}\}_2 = -a_k b_{k+1}, \{b_k,b_{k+1}\}_2 = -a_k, \; \{a_k,a_{k+1}\}_2 = -a_k a_{k+1}.\end{array}
\hat{v}_\alpha^q(\tilde{y}_\alpha)+\left\vert\frac{\nabla\hat{v}_\alpha(\tilde{y}_\alpha)}{\hat{v}_\alpha(\tilde{y}_\alpha)}\right\vert^n+\left\vert\frac{\nabla^2\hat{v}_\alpha(\tilde{y}_\alpha)}{\hat{v}_\alpha(\tilde{y}_\alpha)}\right\vert^\frac{n}{2}=1.
=-x^+\ln\frac{x^+}{c_{+}(i,a)\Delta t}+\tilde x^+\ln\frac{\tilde x^+}{c_{+}(i,\tilde a)\Delta t}+2(x^+-\tilde x^+)+\Delta t\left(c_{+}(i,\tilde a)-c_{+}(i,a)\right)-
G'_{\alpha}(\vec{\beta},{\bf w}) \, = \, -\alpha ( U \otimes (w - w_n) + (w - w_n) \otimes U ) - (\beta - \beta_n) ( V \otimes w + w \otimes V ) ~~~ - \beta_n ( V \otimes (w - w_n) + (w - w_n) \otimes V ) - w \otimes (w - w_n) - (w - w_n) \otimes w_n \,.
\chi^{k,n}= \sum_{(m_1,\dots, m_r)\models n -k}\sum_{(o_1,\dots,o_r)}(\chi^{(m_1)}\circ \chi^{o_1}) \cdot (\chi^{(m_2)}\circ \chi^{o_2})\cdot \cdots \cdot (\chi^{(m_r)}\circ \chi^{o_r}),
\big(\big(K_1^2 \partial_i\bar{\partial}_j \log K_1\big)(z,w) \big)_{i,j=1}^m &\quad\quad = \big(\big(K_2^2\partial_i\bar{\partial}_j\log K_2\big)(z,w) \big)_{i,j=1}^m + \big(\big(K_3^2\partial_i\bar{\partial}_j\log K_3\big)(z,w)\big)_{i,j=1}^m + \big(\left\langle \gamma_j(w), \gamma_i(z)\right\rangle\big)_{i,j=1}^m.
\mathbf{E}\left(F\left((\eta^L_t)_{t\in[0,L]}\right)\right)=\frac{\mathbf{E}\left(\nu(\eta_0)^{-1}F\left((\eta_t)_{t\in[0,L]}\right)\:\vline\:\eta_L=\eta_0,\:S_L>0\right)}{\mathbf{E}\left(\nu(\eta_0)^{-1}\:\vline\:\eta_L=\eta_0,\:S_L>0\right)},
\mathrm{G}_{d}= ( - 1 )^{n}\varepsilon_{\mu_{1}\nu_{1}\cdots \mu_{n}\nu_{n}}\varepsilon^{\alpha_{1}\beta_{1}\cdots \alpha_{n}\beta_{n}}\prod_{i = 1}^{n}R_{\alpha_{i}\beta_{i}}^{\mu_{i}\nu_{i}};
\frac{\partial^{2}}{\partial x_{1}^{2}}f(x_{1},x_{2},...,x_{n}),\frac{\partial^{2}}{\partial x_{1}x_{2}}f(x_{1},x_{2},...x_{n}),...,\frac{\partial^{2}}{\partial x_{n}^{2}}f(x_{1},x_{2},...,x_{n}).
F^{-1}= \left( \begin{array}{c|c|c}f^{-1}-\sum_{i=1}^{2}f^{-1}A_i C^{-1}_i \tilde{A}_i f^{-1}-f^{-1}A_1 C^{-1}_1 -f^{-1}A_2C^{-1}_2 \hline C^{-1}_1 \tilde{A}_1 f^{-1}C^{-1}_1 0 \hline C^{-1}_2 \tilde{A}_2 f^{-1}0 C^{-1}_2\end{array}\right).
\frak{d}(\mathcal{R}_1)= \mathfrak{d}(\mathfrak{S}_{1,\delta=0}) = \frac{p-3}{(p-1)^2}\frak{d}(\mathcal{R}_2) = \mathfrak{d}(\mathfrak{S}_{3,\delta=0}) = \frac{p^2-p-1}{(p-1)(p^2-1)}.
\boldsymbol{d}_{\mathcal{V}}((s_1,s_2), (s_1^\prime, s_2^\prime)) = \left[ \boldsymbol{d}_{\mathcal{S}_1}(s_1,s_1^\prime)^p + \boldsymbol{d}_{\mathcal{S}_2}(s_2, s_2^\prime)^p \right]^{1/p}.
\begin{bmatrix}\begin{array}{c|c}Q_0 row(Q_{-j})_{j\geq 1}\hline col(Q_j)_{j\geq 1}T_Q \: \otimes \: I_d \end{array}\end{bmatrix}- \begin{bmatrix}\begin{array}{c|cc}A B^* 0 \hline B C 0 0 0 0 \end{array}\end{bmatrix}\geq 0 .
\lim_{M\rightarrow\infty}\left\Vert \left( \boldsymbol{E}_{M}\oplus\boldsymbol{E}_{M}\right) \left( \boldsymbol{P}_{M}\oplus\boldsymbol{P}_{M}\right) \left( u\oplus v\right) -\left( u\oplus v\right) \right\Vert=0,
A_{|\alpha\beta\gamma|\cdot\cdot\cdot}B^{\alpha\beta\gamma\cdot\cdot\cdot}=A_{\alpha\beta\gamma\cdot\cdot\cdot}B^{|\alpha\beta\gamma|\cdot\cdot\cdot}=\sum_{\alpha<\beta<\gamma}A_{\alpha\beta\gamma\cdot\cdot\cdot}B^{\alpha\beta\gamma\cdot\cdot\cdot}
\int d^{6}\xi \, \left( \sum_{a = 0}^{5}\left| ( \partial_{a}+ i A_{a}- i A_{a}^{\prime}) \chi \right|^{2}+ \biggr ( \frac{1}{4 g_{\mathrm{Y M}, p}^{2}}+ \frac{1}{4 g_{\mathrm{Y M}, p^{\prime}}^{2}}\biggl ) \sum_{I = 1}^{3}( \chi^{\dagger}\tau^{I}\chi )^{2}\right) \ ,
P_{t-1}= P_{n-1}\prod_{i=n}^{t-1}a_i = P_{n-1}\prod_{i=n}^{t-1}\left(p_{\pi(i)+1}-i \right) & = \prod_{k=1}^{\pi(t)-1}d_k! \prod_{j=1}^{t-p_m}\left(p_{m+1}-p_m-j+1 \right) & = \prod_{k=1}^{\pi(t)-1}d_k! \prod_{k=1}^{t-p_{\pi(t)}}\left(p_{\pi(t)+1}-p_{\pi(t)}-k+1 \right)
R \,=\, \begin{cases}\|\nabla f\|_{L^2(\Omega)}^{2/3}\|f\|_{L^1(\Omega)}^{-2/3}\hbox{if}~\, \|\nabla f\|_{L^2(\Omega)}\le \|f\|_{L^1(\Omega)}~, begin{align*}1mm] \|\nabla f\|_{L^2(\Omega)}^{1/2}\|f\|_{L^1(\Omega)}^{-1/2}\hbox{if}~\, \|\nabla f\|_{L^2(\Omega)}\ge \|f\|_{L^1(\Omega)}~, \end{cases}
[ a_{i}, a_{j}^{\ast}] = \delta_{i j}, \; [ c_{i}, a_{j}] = [ c_{i}, a_{j}^{\ast}] = [ a_{i}, a_{j}] = [ a_{i}^{\ast}, a_{j}^{\ast}] = 0 , \; \mathrm{f o r a l l ~}i , j .
\frac{d}{d\alpha}\int_{a(\alpha)}^{b(\alpha)}f(x,\alpha)dx=\frac{db(\alpha)}{d\alpha}f(b(\alpha),\alpha)-\frac{da(\alpha)}{d\alpha}f(a(\alpha),\alpha)+\int_{a(\alpha)}^{b(\alpha)}\frac{\partial}{\partial\alpha}f(x,\alpha)dx
\frac{1}{2}\frac{d}{dt}\|\sqrt{\rho}\nabla\times\psi\|^2+\nu\|\Delta\psi\|^2 = \int_{\Omega}-[-\Delta,\rho]u^{(m)}_t\Delta\psi dx+\int_{\Omega}-[-\Delta,\rho u\cdot\nabla]u\Delta\psi dx = R_1+R_2.
d s^{2}= - \frac{t^{2}}{( t^{2}+ r_{-}^{2}) ( t^{2}- r_{+}^{2})}d t^{2}+ t^{2}( d \phi + \frac{r_{+}r_{-}}{t^{2}}d r )^{2}+ \frac{( t^{2}+ r_{-}^{2}) ( t^{2}- r_{+}^{2})}{t^{2}}d r^{2}.