harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V(\tilde{x}',\tilde{x}_m,\tilde{y})=\begin{cases}v(\tilde{x}',\tilde{x}_m,\tilde{y})\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \tilde x_m \ge 0, \sum^{k_0+1}_{i=1}c_iv(\tilde{x}',-\frac{\tilde{x}_m}{i},\tilde{y})\ \tilde{x}_m<0,\end{cases}
L_{\mathrm{s o u r c e}}^{\phi}= \sum_{i = 1}^{2}( - 1 )^{i}\rho_{\phi}^{i}( t ) \left[ r_{i}( t ) \sin ( \omega t + \delta_{i}( t ) ) + \bar{R}_{i}\phi ( ( - 1 )^{i}\bar{R}_{i}, t ) \cos ( \omega t + \delta_{i}( t ) ) \right] \, .
{\cal Z}^{(1)}=\frac{1}{N!}\sum_{\{n_i\}=-\infty}^{+\infty}\sum_{P}(-1)^{P}\int_{0}^{2\pi}\prod_{i=1}^{N}\frac{d\theta_i}{2\pi}\exp\left[-\sum_{j=1}^{N}\theta_{j}(n_j-n_{P(j)})\right]\exp\left[-{{g^2 A}\over 2}C_2(n_1,..,n_N)\right],
z(t)= C(t,0)(x_0-g(x))+S(t,0)(y_0-h(x))+\int_0^t S(t,s)f(s)ds+\int_0^t S(t,s)Bu(s,x)ds &+\sum_{0<t_i<t}C(t,t_i)I_i(u(t_i))+\sum_{0<t_i<t}S(t,t_i)J_i(u(t_i)), v(t)= B^*S^*(b,t)R(a,\Upsilon_0^b)p(x(\cdot)), p(x(\cdot))= x_b-C(t,0)(x_0-g(x))-S(t,0)(y_0-g(x))-\int_0^t S(t,s)f(s)ds &+\sum_{i=1}^n C(t,t_i)I_i(u(t_i))+\sum_{i=1}^n S(t,t_i)J_i(u(t_i)),
\hat{b}_{t_{n}. h}(x)=\frac{\sum_{k=0}^{n-1}\left(t_{n}-t_{k}\right)^{1-2 H}K\left(\left(X_{t_{k}}-x\right) / h\right)\left(X_{t_{k+1}}-X_{t_{k}}\right)}{\sum_{k=0}^{n-1}\left(t_{n}-t_{k}\right)^{1-2 H}K\left(\left(X_{t_{k}}-x\right) / h\right)\left(t_{k+1}-t_{k}\right)}.
J^{a}t_{a}=\left( \begin{array}{cccc}\frac{J^{7}}{2}+J^{8}+J^{9}J^{5}+J^{6}J^{2}+J^{3}J^{1}J^{10}+J^{11}\frac{J^{7}}{2}-J^{8}-J^{9}J^{4}J^{2}-J^{3}J^{12}+J^{13}J^{14}-\frac{J^{7}}{2}+J^{8}-J^{9}J^{5}-J^{6}J^{15}J^{12}-J^{13}J^{10}-J^{11}-\frac{J^{7}}{2}-J^{8}+J^{9}\end{array}\right)
\lfloor\frac{q}{p}\rfloor+\lfloor\frac{2q}{p}\rfloor+...+\lfloor\frac{mq}{p}\rfloor+\lfloor\frac{p}{q}\rfloor+\lfloor\frac{2p}{q}\rfloor+...+\lfloor\frac{np}{q}\rfloor=mn.
P^{\prime}_n = \overline{P}_0 e^{-\frac{\tau_n}{\Gamma}}10^{\frac{Z_n}{10}}P_n = \frac{P^{\prime}_n}{\sum^{k=N}_{k=1}P^{\prime}_k}\times P_r [mW] Z_n \sim N(0,3 ),n = 1, 2, ... N
(x\vdash y)\vdash z = x\vdash(y\vdash z) (x\dashv y)\dashv z = x\dashv(y\dashv z) (x\dashv y)\vdash z = x\vdash (y\vdash z) (x\dashv y)\dashv z = x\dashv (y\vdash z) (x\vdash y)\dashv z = x\vdash (y\dashv z)
Q =\frac{\phi'}{\phi- s\phi}, \Theta =\frac{\phi \phi' -s(\phi \phi'' +\phi'^2)}{2\phi\big[(\phi -s\phi')+(b^2-s^2)\phi''\big]}\Psi =\frac{\phi''}{2\big[(\phi -s\phi')+(b^2-s^2)\phi''\big]}.
\bar{g}( s ) = \frac{g s^{1 / 4}}{1 - \frac{\pi A ( y ) b ( y ) g}{y}( s^{1 / 4}- 1 )}= \frac{g_{I R}^{*}| \tilde{g}| s^{1 / 4}}{1 + | \tilde{g}| s^{1 / 4}}, \ \ \tilde{g}= \frac{g}{g - g_{I R}^{*}}\ \in \ \ ( - \infty , 0 ) .
\frac{p_{r}^{2}}{g_{r r}}+ \frac{p_{\theta}^{2}}{g_{\theta \theta}}- \frac{{g^{\prime}}_{t t}}{- \cal D}\left( p_{\phi}+ \frac{g_{t \phi}+ \Omega_{0}g_{\phi \phi}}{{g^{\prime}}_{t t}}E \right)^{2}= - \left( \frac{E^{2}}{{g^{\prime}}_{t t}}+ \mu^{2}\right) ,
L(b_{d_c}) =\log\left(\frac{\frac{1}{2}+\frac{1}{2}\prod_{i=1}^{d_c-1}\psi_i}{\frac{1}{2}-\frac{1}{2}\prod_{i=1}^{d_c-1}\psi_i}\right) =2\tanh^{-1}\prod_{i=1}^{d_c-1}\psi_i,
L_{\mathrm{sym}}^{2}(\Omega)^{3\times3}\coloneqq\left\{(f_{ij})_{i,j\in\{1,2,3\}}\in L^{2}(\Omega)^{3\times3}\,|\, f_{ij}=f_{ji}\quad(i,j\in\{1,2,3\})\right\}.
\ell_{R}\left(R/{I_{q^{'}}^{[p^{e}]}}\right)=\sum_{h=1}^{[\mathbb{K}_{\vartheta}: \mathbb{K}]}\# (\vartheta^{h}(R) \cap p^{e}q^{'}H) - \sum_{h=1}^{[\mathbb{K}_{\vartheta}: \mathbb{K}]}\# (\vartheta^{h}(I_{q^{'}}^{[p^{e}]}) \cap p^{e}q^{'}H)
X_i^{+}s_{\lambda}= \sum_{s \in R_{\lambda,i}}s_{\lambda \setminus s}, X_i^{-}s_{\lambda}= \sum_{s \in A_{\lambda,i}}s_{\lambda \cup s}, H_i s_{\lambda}= (\# A_{\lambda,i}- \# R_{\lambda,i}) s_{\lambda},
\phi_{1\mu}\equiv p_\mu -{{m\dot n_\mu}\over{\sqrt{\dot n^2}}}\approx 0; \phi_{2\mu}\equiv p^{(n)}_\mu -{m\over{\sqrt{\dot n^2}}}({\dot x}_\mu -{{(\dot x\dot n){\dot n}_\mu}\over{{\dot n}^2}})\approx 0;\phi_3 \equiv p^{(\lambda)}\approx 0.
\hat{g}= C_\Sigma \tilde{g}\left(\begin{matrix}\left(\dfrac{\phi_-}{\phi_+}\right)^n 1 0 \left(\dfrac{\phi_+}{\phi_-}\right)^n\end{matrix}\right) + g\left(\begin{matrix}1 0 \phi_+^{-2n}1\end{matrix}\right) - C_\Sigma\tilde{g}.
V_{\mu_p}(A_S[p^\infty])=\bigsqcup_{(\mu_1,\mu_2)\in (\mu_p+ L_p)/(L_{\mathfrak{p}}\oplus L^{\mathfrak{p}_p})}V_{\mu_1}(A[p^\infty]_{\mathfrak{p},S}) \times\bigl({\mu_2}+ L^{\mathfrak{p}}_p\bigr),
\phi_{\chi}^e(s) = \frac{\sqrt{\pi}}{4}\left[P^{-\frac{s}{2}}\frac{z(s)}{s}+ \frac{P^{-\frac{1-s}{2}}}{G\left(\overline{\chi},P\right)}\frac{z(1-s)}{1-s}\right]; \mathfrak{Re}(s) \in (0,1)
0 =2g^{ij}\partial^2_{x_dy_j}\theta_0\partial_{y_i}\theta_0+2Q(y',\partial_{y'}\theta_0)-(\partial_{x_d}\rho_0)^3-2\xi_1 \partial^2_{x_d}\rho_0\partial_{x_d}\rho_0 0 =2(\partial^2_{y'x_d}\theta_0\partial_{x_d}\rho_0)
\frac{d^{2k+1}H}{du^{2k+1}}(0^+)= \frac{d^{2k+1}H}{du^{2k+1}}(0^-) = \alpha \Big(\frac{d^{2k}H}{du^{2k}}(0^+)-\frac{d^{2k}H}{du^{2k}}(0^-)\Big)
\left|\sum_{l=0}^\infty e^{-\frac{Rl(2\lambda+l)}{2\lambda}}\mathcal K_l(t)\right| \leq\left|\sum_{l=0}^\infty e^{-\frac{Rl(2\lambda+l)}{2\lambda}}\mathcal K_l(1)\right| &\leq\left|\sum_{l=0}^\infty Rl(l(2\lambda+l))\,e^{-\frac{Rl(2\lambda+l)}{2\lambda}}\mathcal K_l(1)\right|\leq c.
\mathbb{P}(\hat{T}^{(k)}(\pi) \leq \beta m) = \mathbb{P}\left(\sum_{i=1}^{m}t^{(k)}(e_i) \leq \beta m\right) \leq e^{s\beta m}\prod_{i=1}^{m}\mathbb{E}(e^{-st^{(k)}(e_i)}).
(\frac{q}{p})^{n_{2}^{\prime}-n_{1}^{\prime}}\frac{a_{n_{2}}}{N_{n_{2}}}-\frac{a_{n_{1}}}{N_{n_{1}}}=(\frac{q}{p})^{n_{s}^{\prime}-n_{1}^{\prime}}\frac{a_{n_{s}}}{N_{n_{s}}}.
\liminf\limits_{n\rightarrow+\infty}f\left(\phi,\psi\right)(\xi_n) \geq f\left(\gamma_{2n_0-1},\gamma_{2n_0-2}\right) begin{align*}0.2cm] >(1-\gamma_{2n_0-1})\left(\frac{\gamma_{2n_0-1}}{b}-1\right)-\displaystyle\frac{m\gamma_{2n_0-2}}{\gamma_{2n_0-3}+a\gamma_{2n_0-2}}=0,
ds^2 = (\rho^2-\rho^2_+)\left( -(-\frac{\rho^2_+}{l^2}R^2-k) dT^2 + \frac{dR^2}{-\frac{\rho^2_+}{l^2}R^2-k}\right)+ \frac{d\rho^2}{\frac{\rho^2-\rho^2_+}{l^2}}+ \rho^2 d\phi^2
\tilde{\mathbf{P}}_{S_k^c S_{k}^c}^{(k)}= \mathbf{I}- (1-\beta) \mathbf{A}_{S_k^c S_{k}^c}^{(k)}\tilde{r}_{S_k^c}^{(k)}= (1-\beta) \mathbf{r}_{S_k^c}^{(k)}\boldsymbol{\beta}_{S_k^c}^{(k)}= \mathbf{1}_{S_k^c}- (1-\beta) \mathbf{w}_{S_k^c}^{(k)},
|A_d|+|B_0| \geq \left ((1-\alpha)^2 -2\alpha(1-\alpha)\lambda + \alpha^2 \right )n - \delta n = ((1-\alpha) - \alpha)^2 n + 2\alpha(1-\alpha)(1-\lambda)n -\delta n &\geq 2\alpha(1-\alpha)(1-\lambda)n - \delta n \geq \frac{1}{2}\alpha(1-\alpha)(1-\lambda)n.
\vert F(z+h)-F(z) -L(h)\vert =\vert e \vert \leq \frac 6 5 w_x(\vert F(z+h)-F(z) \vert ) \cdot \vert F(z+h)-f(z) \vert \leq 2 w_x(\vert F(z+h)-F(z) \vert ) \cdot \vert h \vert = o(\vert h \vert).
\frac{2G_2^2\alpha_{t}\|q_{i,t+1}\|^2}{\sigma}-q_{i,t+1}^\top g_{i,t}(x_{i,t})=\Big(\frac{2G_2^2\gamma_0}{\sigma}-1\Big)\gamma_{t}\|[g_{i,t}(x_{i,t})]_+\|^2\le0,
= q^{\frac{(n-1)(n+1)}{12}}\frac{(q;q^2)_{\infty}^2}{(q^n;q^{2n})_{\infty}^{2n}}\frac{\theta_1^n \big( \frac{\pi}{2}\mid \frac{\tau'}{n}\big)}{\theta_1 \big(\frac{\pi}{2}\mid \tau')}\theta_1(nz \mid \tau').
\frac{t_{n+1}(\mp 1)}{t_n(\mp 1)}=\mp\frac{1}{2}, t_{n+1}(x)-\frac{t_{n+1}(\mp 1)}{t_n(\mp 1)}t_n(x) =\frac{1}{2^n}(T_{n+1}(x)\pm T_n(x)).
\left\lfloor\frac{2n}{d}\right\rfloor-\left\lfloor\frac{m}{d}\right\rfloor-\left\lfloor\frac{n}{d}\right\rfloor-\left\lfloor\frac{n-m}{d}\right\rfloor=\frac{m+n}{d}-\left\lfloor\frac{m}{d}\right\rfloor-\left\lfloor\frac{n}{d}\right\rfloor=1.
S_{B H}= 2 \pi \left( \sqrt{N_{2}^{( 1 )}}+ \sqrt{\bar{N}_{2}^{( 1 )}}\right) \left( \sqrt{N_{2}^{( 2 )}}+ \sqrt{\bar{N}_{2}^{( 2 )}}\right) \left( \sqrt{N_{5}^{( 1 )}}+ \sqrt{\bar{N}_{5}^{( 1 )}}\right) \left( \sqrt{N_{5}^{( 2 )}}+ \sqrt{\bar{N}_{5}^{( 2 )}}\right)
\varphi_\beta'(r)=\frac{1}{r}\frac{1}{2\beta-1}\,\varphi_\beta^2(\beta r)-\frac{1}{r}\frac{1}{2\beta-1}\,\varphi_\beta(r),\beta\in[0,1]\setminus\Big\{\frac{1}{2}\Big\},
\Re G^{R}( k ) ={\frac{N^{2}\epsilon^{2 ( \Delta - 4 )}}{8 \pi^{2}( \Delta - 3 ) !^{2}2^{2 \Delta - 5}}}| k^{2}|^{\Delta - 2}\ln{| k^{2}|}\, , \; \Delta = 2 , 3 , 4 , \ldots
E\left[S^{part,m}_{s,t}\right]=E\left[S^{part,0}_{s,t}\right] &-\frac{n}{2}E\left[ \ln(\beta(t,q_t))\right]+\frac{n}{2}E\left[ \ln(\beta(s,q_s))\right] = E\left[S^{part,0}_{s,t}\right]-\frac{n}{2}E\left[ \ln(\beta(t,q_t)/\beta(s,q_s))\right]
R ( u , \lambda ) = \sum_{i = 1}^{n}E_{i , i}\otimes E_{i , i}+ \sum_{i \neq j}\alpha ( u , \lambda_{i j}) E_{i , i}\otimes E_{j , j}+ \sum_{i \neq j}\beta ( u , \lambda_{i j}) E_{i , j}\otimes E_{j , i}
\begin{array}{lll}[Y_0,Y_1]= 2 Y_1 , [Y_0,Y_2]= -2 Y_2 , [Y_1,Y_2]= Y_0, begin{align*}2pt] [y^0,y^1]= 0 , [y^0,y^2]=0, [y^1,y^2]=0 , begin{align*}2pt][y^0,Y_0]=0, [y^0,Y_1]=y^2 , [y^0,Y_2]=-y^1, begin{align*}2pt][y^1,Y_0]=2 y^1, [y^1,Y_1]=-2 y^0 , [y^1,Y_2]=0, begin{align*}2pt][y^2,Y_0]=- 2 y^2 , [y^2,Y_1]= 0, [y^2,Y_2]=2 y^0. \end{array}
\widetilde{S}= \sum_{{\bf m}_k}p_{m_1, t'_1}\prod_{j=2}^{s_k}p_{m_{j-1}, m_j, t'_j}\cdot \sum_{\alpha_0 = m_{s_k}< \alpha_1 < \cdots< \alpha_{n_k}\leqslant n}\prod_{j=1}^{n_k}p_{\alpha_{j-1},\alpha_j,t_k},
m_{\Sigma}\left( t\right) = \frac{1}{\left( 2\pi\right)^{3/2}}{\displaystyle\int\limits_{\kappa t}^{+\infty}}{\displaystyle\iint\limits_{\;\;\left( u_{1}-\kappa t\right) T_{N}}}e^{-\frac{1}{2}u_1^2}e^{-\frac{1}{2}(u_2^2+u_3^2)}\, du_{2}\, du_{3}\, du_{1},
\begin{array}{c}\widetilde{\tau}=\displaystyle\frac{\tau}{\cosh\omega}+\Theta(\widetilde{\tau},x)\tanh\omega,\hfill begin{align*}3mm]\widetilde{\Theta}=\displaystyle\frac{\Theta(\widetilde{\tau}, x)}{\cosh\omega}-\tau\tanh\omega,\hfill\end{array}
\begin{aligned}S(B_1(k),B_2(k)) =-y^kB_1(k)+x^{k-1}B_2(k)+\left (x^{k-1}\cdot \frac{y^k-1}{y-1}-y^k\cdot\frac{x^{k-1}-1}{x-1}\right )\cdot B_3(k) S(B_1(k),B_3(k)) =B_2(k)+\frac{y^k-1}{y-1}\cdot B_3(k) S(B_2(k),B_3(k)) =B_1(k)+\frac{x^{k-1}-1}{x-1}\cdot B_3(k).\end{aligned}
\mathit{deg}(\sigma_{(p,q,n)}(t))=\begin{cases}\frac{(N-1)p(q-1)}{8}\ \ (p,q), &\frac{(N-1)(p-1)q}{8}\ \ (p,q), &\frac{(N-1)(p-1)(q-1)}{8}\ \ (p,q,n), &\frac{N(p-1)(q-1)}{8}\ \ (p,q,n). \end{cases}
O_{B}^{H}(M) =\min\left\{\sum_{i=1}^kh^ix_i \mid \sum_{i=1}^k (h^i-h+1)x_i=M, \x_i\in\mathbb{N}, 1\leq i\leq k\right\}&=\min\left\{M+(h-1)\cdot \sum_{i=1}^kx_i \mid \sum_{i=1}^k (h^i-h+1)x_i=M, \x_i\in\mathbb{N}, 1\leq i\leq k\right\}.
\omega v_n-\Delta v_n =\frac{\lambda}{1+\beta\lambda}\left(c_r^2 h_n+\frac{1}{\lambda}\Delta g_n-c_r^2\lambda v_n\right)+\omega v_n,\x\in\Omega, \partial_\nu v_n =-\frac{\lambda}{1+\beta\lambda}\left(c_r v_n+\frac{1}{\lambda}\partial_\nu g_n\right),\x\in\partial\Omega.
-\left( \mathbf{curl}\,\sigma(\mathbf y)\times \mathbf p_m, e_j\right)+\left(\mathbf{curl}\left(\sigma\left(\mathbf y\times \mathbf p_m\right)\right), e_j\right) =b\left( p_m, e_j, \sigma(y) \right)-b\left( e_j,p_m, \sigma(y) \right) &+b\left(p_m, y, \sigma( e_j) \right)-b\left( y,p_m, \sigma( e_j) \right).
\begin{cases}n\equiv 1 \mod 3: \tau^\prime_n \mid \tau^\prime_m \Leftrightarrow d \mid (m-n), n\not\equiv 1 \mod 3: \tau^\prime_n \mid \tau^\prime_m \Leftrightarrow d \mid (m-n) \frac{m-n}{d}\not\equiv 1 \mod 3.\end{cases}
\limsup_{n\to\infty}\frac{1}{a_c^{(n)}}\log P(S_n(\kappa_n(x))\leq\lfloor\kappa_n(x)\rfloor-a_n) \leq\limsup_{n\to\infty}\frac{1}{a_c^{(n)}}\log P\left(\frac{S_n(\kappa_n(x))}{(1-r^{-1})^{-1}h(x)a_c^{(n)}}\leq\frac{\varepsilon}{h(x)}\right) &\leq-(1-r^{-1})^{-1}h(x)\inf_{y\in \left(-\infty,\frac{\varepsilon}{h(x)}\right]}H(y).
\P_x(\rho > n) = \mathfrak{X}\Delta(x) n^{-d(d-1)/4}\bigg( 1 + \sum_{r=1}^{\infty}U_r(x) n^{-r}\bigg)+ O\left(\frac{1+(x_d - x_1)^{2N + 1}}{n^{N+1/2}}\right)
\big(1-e^{\xi(y,t)-\xi(x,t)}\big)^2 \leq |\xi(y,t)-\xi(x,t)|^2 e^{2|\xi(y,t)-\xi(x,t)|}= \frac{M^2}{(\lambda T-t)^2}|\rho(y)-\rho(x)|^2 e^{\frac{2M}{\lambda T-t}|\rho(y)-\rho(x)|}\leq \frac{\beta^2R_0^{2\beta-2}M^2}{(\lambda T-t)^2}e^{\frac{2\beta R_0^{\beta -1}M d(x,y)}{T(\lambda-1)}}d(x,y)^2,
\left. \left[ \prod_{k \neq i , j}\, K ( X_{k}, s D ) \right] \frac{\partial}{\partial D_{b}}K ( X_{i}, s D ) \, \frac{\partial}{\partial D_{b^{\prime}}}K ( X_{j}, s D ) \leq \exp{\left[ - M \, \sum_{k}| X_{k}| \right]}\ . \right.
\begin{aligned}\phi_{\zeta}^+(\omega_1) = \zeta^{\frac{5}{2}}\omega_1, \phi_{\zeta}^+(\omega_2) = \zeta^{\frac{3}{2}}\omega_2, \phi_{\zeta}^+(\omega_3) = \zeta^{\frac{1}{2}}\omega_3, \phi_{\zeta}^+(\omega_4) = \zeta^{-\frac{1}{2}}\omega_4. \end{aligned}
\begin{aligned}\lambda_{i,k}' = -\lambda_{i,k}+ \sum_{r\neq k}\lambda_{i,r}\max(0,-b_{r,k}) i \in [m]\backslash \{k\}, \lambda_{k,j}' = -\lambda_{k,j}- \sum_{r\neq k}\lambda_{j,r}\max(0,-b_{r,k}) j \in [m]\backslash \{k\}.\end{aligned}
logP=\sum_{n=0}^{N}(-\lambda+\lambda-\frac{\lambda^{2}}{2}+\cdot\cdot\cdot)=\sum_{n=0}^{N}(-\frac{\lambda^{2}}{2}+\cdot\cdot\cdot)=-50\sum_{n=0}^{N}(\begin{matrix}N\\n\end{matrix})^{2}p^{2n}(1-p)^{2(N-n)}+\cdot\cdot\cdot
BF_{kl}=\frac{m(\mathbf{y}| M_{k})}{m(\mathbf{y}| M_{l})}= \frac{\lambda_{n,k}\int \prod_{i=1}^{n}p(y_{i})\Pi_{n,k}(p)}{\lambda_{n,l}\int \prod_{i=1}^{n}p(y_{i})\Pi_{n,l}(p)}.
\Lambda_{\ \nu}^{\mu}=\left(\begin{array}[c]{cccc}\gamma -\beta\gamma 0 0 -\beta\gamma \gamma 0 0 0 0 1 0 0 0 0 1\end{array}\right) ,\qquad\quad\beta=\frac{v}{c},\quad\gamma=\frac{1}{\sqrt{1-\beta^{2}}}
{e}_{\hat{0}}^{\; \; 0}{\bar{e}}_{\alpha}^{\; \; i}\left[ \frac{1}{2}\frac{\partial \Gamma^{( 2 )}}{\partial w^{i}}\right] = -{e}_{\hat{0}}^{\; \; 0}{\bar{e}}_{\alpha}^{\; \; i}\left[ B_{i}^{w ( 2 )}\right] +{e}_{\alpha}^{\; \; \mu}{e}_{\hat{0}}^{\; \; \nu}K_{\mu \nu}^{b},
{\cal L}= -{1\over 4}({\cal F}_{\mu\nu}-iegW^*_\mu W_\nu)^2 -{1\over 2}|{\cal D}_\mu W_\nu -{\cal D}_\nu W_\mu |^2 + G^2 \phi^2 |W_\mu|^2 -{1\over 2}(\partial_\mu \phi)^2 - V(\phi)
[\begin{matrix}T_{1}&T_{2}&T_{3}\end{matrix}]=[\begin{matrix}n_{1}&n_{2}&n_{3}\end{matrix}]\cdot[\begin{matrix}\sigma_{11}&\sigma_{21}&\sigma_{31}\\ \sigma_{12}&\sigma_{22}&\sigma_{32}\\ \sigma_{13}&\sigma_{23}&\sigma_{33}\end{matrix}]
i_{1,p+q-1}=i_{p,q}U_{p,q}U_{p,q}S_{1,p+q-1}=S_{p,q}U_{p,q}U_{p,q}D_{1,p+q-1}=D_{p,q}(U_{p,q}\otimes U_{p,q}) U_{p,q}e_{1,p+q-1}=e_{p,q}
C_{0}:=\mathrm{Cone}\left\{\begin{aligned}(-1,\;\;\,0,-1,0,1,0,0,1,0), (-1,-1,\,\;\;0,1,0,0,1,0,0), (\;\;\,0,-1,-1,0,0,1,0,0,1), (-1,-1,-1,1,0,1,0,1,0), (-1,-1,-1,0,1,0,1,0,1)\,\end{aligned}\right\}.
\varphi_{2}^{\prime}( N ) = \left( \frac{j ( N - j + 1 )}{( 2 j + 1 ) ( N + 2 )}\right)^{1 / 2}\varphi_{+}^{\prime}( N ) - \left( \frac{( j + 1 ) ( N + j + 2 )}{( 2 j + 1 ) ( N + 2 )}\right)^{1 / 2}\varphi_{-}^{\prime}( N ) .
\theta_1(\rho)= \Bigl(-\int_0^\rho m'(y)(1-m'(y)\bar\sigma)dy\Bigr)m'(\rho)\int_0^\rho\frac{1}{m'(y)^2}dy &+ \Bigl(\int_0^\rho m'(y)(\int_0^y \frac{1}{m'(z)^2}dz) (1-m'(y)\bar\sigma)dy \Bigr)m'(\rho).
\left\vert \left\langle \nabla \ln S,\nabla \psi \right\rangle \right\vert =\frac{e^{\frac{f}{\sqrt{T-\sqrt{T}}}}}{\sqrt{T-\sqrt{T}}e^{\sqrt{T-\sqrt{T}}}}\left\vert \left\langle \nabla S,\nabla f\right\rangle \right\vert S^{-1}.
{\cal E}_{\nu}( \eta , \eta^{\prime}) = ( r r^{\prime})^{- \frac{d - 1}{2}- i \nu}{W}_{\nu}^{\infty}({\omega},{\omega}^{\prime}) = \frac{\Gamma \left( \frac{d - 1}{2}+ i \nu \right) \Gamma ( - i \nu )}{4 \pi^{\frac{d + 1}{2}}}\ \ \left[ - ( \eta - \eta^{\prime})^{2}\right]^{- i \nu - \frac{d - 1}{2}}
L=\mbox{Tr}\left\{-\frac{1}{4}G^{\mu \nu}G_{\mu \nu}+\frac{1}{2}D_{\mu}S^{\dagger}D^{\mu}S\right\}+\frac{1}{2}D_{\mu}\phi^{\dagger}D^{\mu}\phi -V(S,\phi )\, .
\delta_{h}S =\int_{{\cal M}^{1+1}}\left({1 \over 2}\hat{E}^{I}\wedge \hat{E}^{--}i_\delta f^{++I}-{1 \over 2}\hat{E}^{I}\wedge \hat{E}^{--}i_\delta f^{++I}+m j_1 \wedge \hat{E}^{i}i_\delta f^{i}\right)
\mathbf{\Lambda}=\mbox{diag}(\lambda_1,...,\lambda_{2N}), \mathbf{V}=\left[\mathbf{v}_1,...,\mathbf{v}_{2N}\right],\mathbf{v}_j=\left(\begin{array}{c}\mathbf{e}_j \lambda_j \mathbf{e}_j\end{array}\right),
{u}_t-{\hat u}_t = D\Delta (u-\hat u)+H( u)-H( \hat u) x\in \Omega,\quad t>0{v}_t-{\hat v}_t =\tilde D\Delta_{M}( v-\hat v)+F( u,v)-F( \hat u, \hat v) x\in M,\quad t>0 D\frac{\partial (u-\hat u)}{\partial \eta}=G( u, v)-G( \hat u,\hat v) x\in M, t>0 u-\hat u =0 x\in\Omega , t=0 v-\hat v =0 x\in M, t=0
S_{\textrm{GA}}(R) = \sum_{i=1}^{N_{max}}\frac{R}{i\cdot\pi}\cdot &\int_{0}^{\infty}\bigg[ \mathcal{I}m \Big\{\Psi_{\textrm{GA}}(t;i, R) - \Psi_{\textrm{GA}}(t;i-1,R) \Big\}\bigg] dt,
Op_h^w\Big(\chi(h^{\beta}\xi)\gamma(\frac{x+p'(\xi)}{\sqrt{h}})\Big)v^{\Sigma}= v^{\Sigma}_{\Lambda}- Op_h^w\Big((1-\chi(h^{\beta}\xi))\gamma(\frac{x+p'(\xi)}{\sqrt{h}})\Big)v^{\Sigma}\,.
\overline{{\Gamma}}_{5}= - \varepsilon^{\widetilde{0}, 0}- \varepsilon^{0 , \widetilde{0}}- \varepsilon^{\widetilde{\mu}, \mu}- \varepsilon^{{\mu}, \widetilde{\mu}}- \frac 1 2 e_{\mu \nu \rho \omega}\varepsilon^{[ \mu \nu ] , [ \rho \omega ]}.
\langle\delta s(\mu_{\underline 0}),\delta s(\mu_{\underline 0})\rangle=4(-1)^{M}(2M+2)^2(2M+1)^2(2M+3)^{-1}\binom{2M+2}{k_{\rm id}-1}^{-1}\binom{2M}{k_{\rm id}-2}^2\binom{2M+2}{M+1}^{-1}
\mathrm{T r}\, d \Phi \, d \Phi^{\dagger}= \sum_{i}d \lambda_{i}^{2}+ \sum_{\alpha , \beta}d t_{\alpha}\, d t_{\beta}\, \left( \sum_{i}\lambda_{i}\alpha_{i}\right) \left( \sum_{j}\lambda_{j}\beta_{j}\right) \, \mathrm{T r}\, E_{\alpha}E_{\beta}.
m^2_G\;=\; - 2 e^2 \int_{{\mathcal R}}\frac{d^2p}{(2\pi)^2}\,\frac{\partial}{\partial p_\mu}(\frac{p_\mu}{p^2}) + 2 e^2 \int_{{\mathcal R}}\frac{d^2p}{(2\pi)^2}\,\frac{\partial}{\partial p_\mu}(\frac{p_\mu}{p^2+\Lambda^2})\;,
\|(w_\ell,r_\ell)\|_{2,\ell}= \sup_{0 \neq (v_\ell,q_\ell) \in V_\ell \times Q_\ell}\frac{\mathcal{B}_\ell((w_\ell,r_\ell),(v_\ell,q_\ell))}{\|(v_\ell,q_\ell)\|_{0,\ell}}\forall \, (w_\ell,r_\ell) \in V_\ell \times Q_\ell.
\int_{\mathbb{R}}e^{-i\xi x}\,dx=2\pi\delta_0(\xi),\int_{\mathbb{R}}e^{-i\xi x}\frac{1}{x}\,dx=-i\pi\,\mathrm{sgn}(\xi),\qquad\int_{\mathbb{R}}\frac{e^{-i\xi x}-1}{x^2}\,dx=-\pi\,|\xi|,
\frac{\delta \Delta_{\rm C}}{\delta B}= 0,\qquad\frac{\delta \Delta_{\rm C}}{\delta \bar{C}}+ 2 \partial_\mu \frac{\delta \Delta_{\rm C}}{\delta A_\mu^*}= 0,\qquad\frac{\delta \Delta_{\rm C}}{\delta \eta}= 0,\qquad\frac{\delta \Delta_{\rm C}}{\delta \bar{\phi}}+ 2 \partial_\mu \frac{\delta \Delta_{\rm C}}{\delta \psi_\mu^*}= 0
Q_1(M,P_i,\tau)=2^6\left\{e^{\frac{1}{24}E_2(\tau)A_3}\left(\prod_{j=1}^{6}\frac{x_j\theta'(0,\tau)\theta_1(x_j,\tau)}{\theta(x_j,\tau)\theta_1(0,\tau)}\right)\right. &\left.\frac{1}{2}\left(\prod_{l=1}^8\theta_1(y_l^i,\tau)+\prod_{l=1}^8\theta_2(y_l^i,\tau)+\prod_{l=1}^8\theta_3(y_l^i,\tau)\right)\right\}^{(12)},
\mathcal{E}:= \Big\{b\in L^{\infty}(I;L^1(\Omega)); \; b> 0 \textrm{a.e. in}Q, \; \ln b \in L^{\infty}(0,T; L^1(\Omega)), &(1+b)^\lambda \in L^2(I; W^{1,2}(\Omega)) \textrm{for all}\lambda \in (0,1), &b-b_{\Gamma}\in L^1(0,T;W^{1,1}_{\Gamma}(\Omega))\Big \}.
Z_j(t,\frac{1}{2})=\frac{\left(\cosh(\frac{\pi}{4a_j}t)-\sinh(\frac{\pi}{4a_j}t)\right)}{\sinh(\frac{\pi}{2a_j}t)}\stackrel{\textrm{as}\, t\rightarrow 0^+}{\sim}\frac{a_j}{\pi t}-\frac{1}{4}.
\varrho({\beta^*_{n-k+1}})\,e_{\underline{j}}= \varrho \left( \sum_{k\le m\le n}t_{m,k}^* \right) e_{\underline{j}}= \left(\sum_{k\le m\le n}\varrho(t_{m,k}^*)\right) e_{\underline{j}}&= \left(\sum_{k+1\le m\le n}\varrho(t_{m,k}^*) + \varrho(id) \right) e_{\underline{j}}
\frac{E_{0}(\alpha_{N})+N\log N}{N}=\log(2)\,\frac{\sum_{i=2}^{t}(n_{1}-n_{i}+2-2i)\,2^{n_{i}-n_{1}}}{\sum_{i=1}^{t}2^{n_{i}-n_{1}}}+\log\left(\sum_{i=1}^{t}2^{n_{i}-n_{1}}\right).
Q_{3,\tilde{q},\tilde{p}}=\underbrace{\sigma_2^{\lambda(1)}\sigma_1^{\lambda(2)}\sigma_2^{\lambda(\frac{\tilde{q}-3}{2})}\sigma_1^{\lambda(\frac{\tilde{q}-1}{2})}}_{P}\sigma_2^{-\lambda(\frac{\tilde{q}-1}{2})}\sigma_1^{-\lambda(\frac{\tilde{q}-3}{2})}...\sigma_2^{-\lambda(2)}\sigma_1^{-\lambda(1)}
2 \Re \langle \mathcal{L}_{\varepsilon}X, X \rangle = 2 \Re \langle X, \mathcal{L}_{\varepsilon}X \rangle = \langle \mathcal{L}_{\varepsilon}X, X \rangle + \langle X, \mathcal{L}_{\varepsilon}X \rangle \leq 0,
C_j^{(\nu)}(z) = \frac{(2\nu)_j}{(\nu+1/2)_j}P_j^{(\nu-1/2,\nu-1/2)}(z) = \frac{(2\nu)_j}{j!}{}_2F_1\left(-j, j+2\nu, \nu+\frac{1}{2}, \frac{1-z}{2}\right).
\frac{1+\gamma(p)}{2\gamma(p)}\left(\bigg(\frac pq\bigg)^{1/p}+\bigg(\frac qp\bigg)^{1/q}\right) =\frac{8+q(q-1)}{2}(q-1)^{1/q-1}(p-1) &\le (q+3) (q-1)^{1/q-1}(p^*-1).
\prod_{q=1}^r \iota'_i(v_q)^{l_q-1}-\prod_{q=1}^r \iota'_k(v_q)^{l_q-1}=\sum_{p l_p=0}\prod_{q \neq p}\left(\frac{\iota'_i(v_q)}{\iota'_i(v_p)}-\frac{\iota'_k(v_q)}{\iota'_k(v_p)}\right)^{l_q-1}.
T^\mu\_\nu(\eta)=\left(\begin{array}{cc}H(\eta)c^{-1}(\eta) 0 0 T^{\beta}_{\beta}-c^{-1}(\eta)H(\eta) \ \end{array}\right)+c^{-1}\left(\begin{array}{cc}\zeta \alpha' -\alpha' -\zeta \ \end{array}\right)
I_{\pm}(x) :=\lim_{\lambda \to \pm\Lambda\pm 0}\int_{-\pi}^{\pi}\frac{-i\sin k}{2\phi_{2,1}\phi_{2,2}\cos k-\lambda}e^{ikx}\frac{dk}{2\pi}&= 2\lim_{\lambda \to \pm\Lambda\pm 0}\int_0^{\pi}\frac{\sin k\sin xk}{2\phi_{2,1}\phi_{2,2}\cos k-\lambda}\frac{dk}{2\pi}.
\begin{aligned}|[\bold{S}_{j-1}^{-1}]^{(1,2)}_{m,n}| \le \max_{a,b}| [\bold{\Gamma}_{j-1}]_{a,b}| \| (\bold{I}_{j-1}-\bold{\Pi}_{j-1})^{-1}_{(n)}\|_1 & \| (\bold{I}_{j-1}-\bold{\Pi}_{j-1}-\bold{\Xi}_{j-1}(\bold{I}_{j-1}-\bold{\Pi}_{j-1}^{T})^{-1}\bold{\Gamma}_{j-1})^{-1}_{(m)}\|_{1}\le \frac{K}{M},\end{aligned}
\left|\pi[f - f_R] \right| \leq \pi\big[ |f - f_R|\big] = \pi\big[ |f|\mathbf{1}_{|f| \geq R}\big] - R\pi[\{|f| \geq R\}] \leq \sqrt{\pi[f^2]}\sqrt{\pi[\{|f| \geq R\}]}- R\pi[\{|f| \geq R\}] \leq \frac{1}{4R}.
R^{n,3}_\tau = \frac 1n \sum_{j=0}^{\lfloor n\tau \rfloor}\int_{j}^{(j+1) \wedge n\tau}\gamma_{\frac xn}(\sigma'\sigma)^2(X_{\frac xn}) \sum_{i=0}^{\lfloor x \rfloor}\int_{i}^{(i+1) \wedge x}[ (x-i)^\alpha - (x-v)^\alpha]^2 dvdx.
\tfrac{1}{2}\cdot \delta^{\epsilon}|B| \leq |B_{c}\cap B_{0}| = \sum_{j = 1}^{N}|B_{c}\cap B_{j}| \leq \sum_{j \notin \mathcal{J}}\delta^{\bar{\epsilon}}|B_{j}| + \sum_{j \in \mathcal{J}}|B_{j}| \leq \tfrac{1}{4}\cdot \delta^{\epsilon}|B| + \sum_{j \in \mathcal{J}}|B_{j}|.
\zeta(s,t)=\sum_{n>m\ge1}\frac{1}{n^{s}m^{t}}=\sum_{n=2}^{\infty}\frac{1}{n^{s}}\sum_{m=1}^{n-1}\frac{1}{m^{t}}=\sum_{n=1}^{\infty}\frac{1}{(n+1)^{s}}\sum_{m=1}^{n}\frac{1}{m^{t}}
\bigg\|\partial_r \frac{f}{|x|}\bigg\|_{2}^2 = \mathbf{H}_{3}^2 \bigg\| \frac{f}{|x|^2}\bigg\|_{2}^2 + \bigg\|\partial_r \frac{f}{|x|}+ \mathbf{H}_{3}\frac{f}{|x|^2}\bigg\|_{2}^2.
\Pi_{\mu \nu}^{M}( k ) = \frac{i e^{2}}{( 2 \pi )^{3}}\sum_{i = 0}^{n_{f}}c_{i}\int d^{3}p \frac{P ( M_{i})}{({M_{i}}^{2}-{p_{1}}^{2}) ({M_{i}}^{2}-{p_{2}}^{2})}\, \, ,
C^n_\mathrm{mRBA}((A,R), (M,S)) = \begin{cases}M n =0, C^n (A, M) \oplus C^{n-1}(A_R,\widetilde{M}) = \mathrm{Hom}(A^{\otimes n}, M) \oplus \mathrm{Hom}(A^{\otimes n-1},{M}) n \geq 1.\end{cases}
\sigma_{\kappa}^{2}=\frac{1}{\beta}\frac{2+\kappa}{2-\kappa}\frac{4\kappa}{4-9\kappa^{2}}[\frac{\Gamma(\frac{1}{2\kappa}+\frac{1}{4})}{\Gamma(\frac{1}{2\kappa}-\frac{1}{4})}]^{2}
\sum_{i,j=1}^{m}b_{ij}t^{i-1}s^{j-1}=\frac{s^{m}\hat{\mathbf{n}}\left(1/s\right) \hat{\mathbf{d}}\left( t\right) -s^{m}\hat{\mathbf{d}}\left(1/s\right) \hat{\mathbf{n}}\left( t\right)}{1-st}.
\lim_{\substack{X\rightarrow P X\in \Gamma_{\gamma}(P)}}\left\langle\nabla\left(\int_{\partial D}\mathcal{K}_{1}(X, Q)f(Q)d\sigma(Q)\right),\,\, n_{P}\right\rangle =-\frac{1}{2}f(P)+T^{*}f(P),
f(x,m) = \frac{z_--m}{z_--y_+}(x-y_+)^\rho + \frac{m-y_+}{z_--y_+}(z_--x)^\rho - \frac{z_+-m}{z_+-y_-}(x-y_-)^\rho - \frac{m-y_-}{z_+-y_-}(z_+-x)^\rho .
{\l}^{A}\hat{\bar{\mu}}^{\dot{B}}P_{A \dot{B}}+ \l^{A}\hat{\mu}^{B}Z_{A B}= 0 , \qquad \hat{\mu}^{A}{\bar{\l}}^{\dot{B}}P_{A \dot{B}}+ \bar{\l}^{\dot{A}}\hat{\bar{\mu}}^{\dot{B}}\bar{Z}_{\dot{A}\dot{B}}= 0 ,
\times \exp \left( \frac{1}{2}\sum_{b=1}^N \sum_{j=1}^{m_b}\sum_{j^\prime=1}^{m_{b}}\Big(1- \delta_{j j^\prime}\Big)\Big[ \ln(y^{(b)}_j - y^{(b)}_{j^\prime})^2 + 2\gamma - \ln(4)\Big] \right) \; ,
\|S_r(E)\|_{\infty}=\|\tilde P_r-P_r-L_r(E)\|_{\infty}\leq \|\tilde P_r-P_r\|_{\infty}+ \|L_r(E)\|_{\infty}\leq 1+ \frac{\sqrt{2}}{\bar g_r}\|E\|_{\infty}.
(-h^2 \Delta + W(h^{-1}x)\mathbf{1}_{B(0,2)}(x) )\chi_h(x) \widetilde{u}(x) = \chi_h(x)(-h^2 \Delta + W(h^{-1}x))\widetilde{u}(x)+ [-h^2\Delta,\chi_h(x)]\widetilde{u}(x) =[-h^2\Delta,\chi_h(x)] \widetilde{u}(x).
R_{mn}{}^{pq}\rightarrow W_{mn}{}^{pq}- \frac{16}{d-2}\delta_{[m}{}^{[p}(\bar\psi_{|r|}\Gamma^{|r|}\psi_{n]}{}^{q]}- \bar\psi^{|r|}\Gamma^{q]}\psi_{n]r}) \, .
\begin{array}{r c l}{{\Phi}}&{{=}}&{{\displaystyle \frac{1}{N}\sum_{n = 0}^{\infty}\left\langle \mathrm{T r}\; ( u U + v V )^{n}\right\rangle}}\\{{}}&{{=}}&{{\displaystyle \frac{1}{N}\sum_{w ( u , v )}w ( u , v ) \left\langle \mathrm{T r}\; w ( U , V ) \right\rangle \ .}}\\ \end{array}