harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e^{- 2 \phi^{\prime}}F^{2}= \frac{Q^{2}}{\rho^{4}}\Bigl ( 1 - \frac{2 G M}{\rho}\Bigr )^{- 1}\rightarrow \frac{2 G^{2}M^{2}}{\rho^{3}( 2 G M - \rho )}\propto \frac{1}{x}
{\mathcal{B}}_{i}= ( \cos \phi - 1 ) \frac{\hat{r}_{i}}{r^{2}}- \beta^{\prime}\sin \phi \sin \theta \frac{\hat{\theta}_{i}}{r}- 2 \beta^{\prime}\sin \theta \cos \theta ( \cos \phi - 1 ) \frac{\hat{\phi}_{i}}{r}
f(a_1,\dotsc,a_{i-1},a_i,a_{i+1},\dotsc,a_{j-1},a_j,a_{j+1},\dotsc,a_n) = a f(a_1,\dotsc,a_{i-1},a',a_{i+1},\dotsc,a_{j-1},a_j,a_{j+1},\dotsc,a_n) = b f(b_1,\dotsc,b_{i-1},b_i,b_{i+1},\dotsc,b_{j-1},b_j,b_{j+1},\dotsc,b_n) = c f(b_1,\dotsc,b_{i-1},b_i,b_{i+1},\dotsc,b_{j-1},b',b_{j+1},\dotsc,b_n) = d
\int_{\Omega}\partial_tu(-\Delta_N)^{-1}u\:x = \frac{}{t}\int_{\Omega}\frac{1}{2}\big|(-\Delta_N)^{-1/2}u\big|^2\:x = \frac{}{t}\frac{1}{2}\|u\|^2_{H^{-1}(\Omega)}.
b(f_{1},f_{2})=\frac{|\sum_{n}F_{n}(f_{1})F_{n}(f_{2})F_{n}^{*}(f_{1}+f_{2})|}{\sqrt{\sum_{n}|F_{n}(f_{1})|^{2}|F_{n}(f_{2})|^{2}|F_{n}^{*}(f_{1}+f_{2})|^{2}}}
n(R_2-\epsilon_n) \leq I(X_1^n;Y_2^n|W_1) = h(Y_2^n|W_1) - h(\sqrt{b}X_2^n+Z_2^n) \leq \frac{n}{2}\log(1+a\alpha P_1+bP_2 )-n\xi(b)+0.5n.
g^{ij}\, \Pi \left( F_{sij}\right) = g^{ij}\, \Pi \left( \nabla_{F_i}\nabla_{F_j}V \right) = g^{ij}\, \nabla_{F_i}^{\perp}\nabla_{F_j}^{\perp}V + g^{ij}\, \nabla_{F_i}^{\perp}\nabla_{F_j}^T V \, .
\left\langle T_k, (\tau ')^l, \frac{\alpha}{z+\psi}\right\rangle_{0,l+2,d}= \sum_{u \geq 0}\frac{(-1)^u}{z^{u+1}}\left\langle \psi^u \alpha, T_k, (\tau ')^l \right\rangle_{0,l+2,d}
\lim_{x\to\infty}\frac{h(x)}{1/x}= \lim_{x\to\infty}\frac{2\theta}{\operatorname{B}(a,b)}\frac{1}{x^3}\frac{\left[ 1-\exp\left(-\frac{\theta}{x^2}\right) \right]^{b-1}\exp\left(-\frac{a\theta}{x^2}\right)}{\operatorname{I}_{1-\exp(-\theta/x^2)}(b,a)}x.
F_{-1}=u, F_{0}=\displaystyle -u\sum_{j=1}^{n+1}\mu_{j}, F_l=\displaystyle (-1)^{l+1}u\sum_{\stackrel{j_1<\cdots<j_{l+1}}{j_i\geq 1}}\mu_{j_{1}}\ldots\mu_{j_{l+1}},0\leq l\leq n.
\mathbb{P}\left\{\mathcal{M}_i^{j, j^{\prime}}\right\}\leq c_1 \sum_{t=0}^{T-1}\exp\left(-c_2 N_in_x \left(\frac{\alpha \rho^{(i)}\|\Sigma^{(i)}_t\|}{\rho^{(i)}\|\Sigma^{(i)}_t\| + \sqrt{n_x}}\right)^2\right),
-\int\limits_X \xi \wedge \partial\overline{\rho_\omega^{2,\,0}}\wedge \omega_{n-2}= \int\limits_X \xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \partial\omega_{n-2}- \Vert \rho_{\omega}^{2,0}\Vert^2_{\omega}.
{}\begin{array}{l}\tilde{g}(u) = 2\frac{v}{u}- u \ln (e^{v(p)}-1) + \frac{u}{p}\left ( v(p)-v \right ) \tilde{f}(u) = \frac{v}{2^{3/2}\pi u}\left ( (e^{v(p)}-1) - \frac{1}{2}u^2 e^{v(p)}\right)^{-1/2}\end{array}
\Theta(a_1,\dotsc,a_m) \models\Theta^{-}(a_1,\dotsc,a_{m-1})\intertext{and}b_{h_m,c_m}= \begin{cases}\tau_{c_m}^{\gamma_q(h_m)}(1-\tau_{c_m})^{\gamma_q(h_m)}\int_{[0,1]}x^{\gamma_q(h_m)}(1-x)^{q-\gamma_q(h_m)}\,d\tau_0(x) \end{cases}
+ Q(\psi, \psi^{\alpha})]+\sum_{i=1,2}G_i^\alpha Q_{1,2}(G_i^{\alpha}, \psi)+ \psi^{\alpha}\big[-Q(G_i^\alpha, G_i) - Q(G_i,G_i^{\alpha})\big] + G_i^{\alpha}Q_{1,2}(G_i, \psi^{\alpha}),
\psi_{1}^{' \pm}= \gamma^{\mp}\left[ \begin{array}{c}{{\mp \sin \theta ( 1 + \xi^{2})}}\\{{0}}\\{{i ( 1 - \xi^{2}+ 2 i \xi_{3}) ( 1 \mp \cos \theta )}}\\{{- 2 ( 1 \mp \cos \theta ) ( \xi_{1}- i \xi_{2})}}\\ \end{array}\right]
DA_{(p)}^{\mu_1\ldots \mu_{p-2k}}=\partial^{\left[ \mu_1\right.}\eta_{(p)}^{\left. \mu_2\ldots \mu_{p-2k}\right]}+\left(p-2k+1\right) \partial_\nu \eta_{(p)}^{\nu \mu_1\ldots \mu_{p-2k}},\;k=1,\ldots ,b_p,
\begin{array}{l}{R ( \Delta ) = I ( \Delta ) = \Delta^{N - 1}( \Delta^{- 1}\Omega \Delta^{- 1})^{N - 1},}\\{R ( \Omega ) = \Delta^{N - 2}\Omega ( \Delta^{- 1}\Omega \Delta^{- 1})^{N - 1},}\\{I ( \Omega ) = \Delta^{N - 1}( \Delta^{- 1}\Omega \Delta^{- 1})^{N - 1}\Omega \Delta^{- 1}.}\\ \end{array}
I \le \frac{\Lambda_\gamma(\mathbb S^n,[g_c])}{d_{\gamma}^{*}}\left(\int_{\Gamma_{\epsilon}^{0}}w_{\mu}^{2^{*}}dx\right)^{\frac{n-2\gamma}{n}}&+\left\{\frac{1}{12n}(\overline{Ric}_{yy,yy}+8(\bar{R}_{yiyj})^2)I_{4}+\frac{1}{2}(\overline{Ric}_{yy,kk}+2\sum\bar{R}_{yiyj,ij})I_3\right\}\mu^4+\mu^4o(1).
-\int_1^\infty \pi_2(x) \mathrm{d}\left( \frac{1}{\lfloor x \rfloor}\right) = -\sum_{n \geq 1}\pi_2(n) \left( \frac{1}{n+1}- \frac{1}{n}\right) = \sum_{n \geq 1}\frac{\pi_2}{n^2},
\int_{\mathfrak{b}_{a}}\left( \left( y-y_{0}\right) \times b_{a}^{\ddagger}+\nu\times \beta_{a}^{\ddagger}\right) d\mu(y)+\int_{\partial \mathfrak{b}_{a}}\left( \left( y-y_{0}\right) \times \mathfrak{t}+\nu \times \tau \right) d\mathcal{H}^{2}=0.
\sum_{n = \frac{1}{2}}^{R / 2}\frac{\triangle k_{n}^{+}}{k_{n}^{+}}f ( k_{n}^{+}) = \sum_{n = \frac{1}{2}}^{R / 2}\frac{\triangle k_{n}^{+}}{k_{n}^{+}}e^{- \sigma^{2}/ ( \Lambda^{\prime}k_{n}^{+})}
\begin{aligned}\mathcal{M}^{\oplus_k}_{|\cdot|}= \sqrt{2}\Big(C_F \, \big(\|{\mathcal{R}_1}_k \|_{\Omega}+ \|{\mathcal{R}_3}_k \|_{\Omega}\big) +\, \|{\mathcal{R}_2}_k\|_{\Omega}+ \|{\mathcal{R}_4}_k \|_{\Omega}\Big) \end{aligned}
|\Delta_{l,1}^i| = (q^2+q+1)(q^2+q)q^2(q^4-3q^3+3q^2-q)(q+1)(q^2-q), |\Delta_{l,2}| = (q^2+q+1)(q^2+q)q^2(q^4-3q^3+3q^2-q)(q^2-q-2),
\begin{aligned}\mu_c^-(Y_1,Y_2,\dots,Y_m):= \mu(Y_1,Y_2)+\mu(Y_2,Y_3)+\dots+\mu(Y_{m-1},Y_m)+\mu(Y_m,Y_1), \mu_c^+(Y_1,Y_2,\dots,Y_m):= \mu^*(Y_1,Y_2)+\mu^*(Y_2,Y_3)+\dots+\mu^*(Y_{m-1},Y_m)+\mu^*(Y_m,Y_1).\end{aligned}
(\nabla \pi_{*})(X,Z) =\nabla^{\pi}_{X}\pi_{*}Z+\pi_{*}(\varphi\nabla^{^{M_1}}_{X}\varphi Z-\varphi(\nabla^{^{M_1}}_{X}\varphi)Z-\eta(\nabla^{^{M_1}}_{X}Z)\xi) &=\nabla^{\pi}_{X}\pi_{*}Z+\pi_{*}(\varphi(\nabla^{^{M_1}}_{X}\varphi Z_1+\nabla^{^{M_1}}_{X}\varphi Z_2)-\eta(Z)\varphi X-\eta(\nabla^{^{M_1}}_{X}Z)\xi).
V_{\beta \alpha}\equiv V^\mu \left( \widetilde{\sigma}_\mu \right)_{\beta\alpha}\mbox{};\mbox{}f_{\alpha \beta}\equiv \left( \tilde{\sigma}_\mu\right)_{\alpha \beta}f^\mu \mbox{};\mbox{}f^\mu =-\frac i2\varepsilon^{\mu \nu \rho}F_{\nu \rho}\mbox{},
S ={\frac{1}{4 \pi \alpha^{\prime}}}\int d^{2}\xi \left[ \sqrt{- h}h^{\alpha \beta}g_{\mu \nu}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}- \epsilon^{\alpha \beta}{\cal F}_{i j}\partial_{\alpha}X^{i}\partial_{\beta}X^{j}\right] ,
x^{k+1}= x^{k}- \lambda \left(\nabla f_{i_k}(x^{k-d_k}) - y_{i_k}^{k-e_{k}^i}+ \frac{1}{N}\sum_{i=1}^N y_i^{k-e_k^i}\right); y_{i}^{k+1}= \begin{cases}\nabla f_{i}(x^{k-d_k}) y_i^k \end{cases}
lim_{N\rightarrow\infty}\frac{\#\{p\le N:\alpha\le\theta_{p}\le\beta\}}{\#\{p\le N\}}=\frac{2}{\pi}\int_{\alpha}^{\beta}sin^{2}\theta d\theta=\frac{1}{\pi}(\beta-\alpha+sin(\alpha)cos(\alpha)-sin(\beta)cos(\beta))
\langle A^{( 1 ) a}( k ){A^{( 1 ) b}}^{*}( k^{\prime}) \rangle = ( 2 \pi )^{4}\delta^{( 4 )}( k - k^{\prime}) \delta_{a}^{b}\langle{A^{( 1 )}}^{2}\rangle_{{\bf k}\omega},
s_{\Gamma}( \xi_{A}\partial_{\chi_{A}}\Gamma ) = ( s_{\Gamma}\xi_{A}) \partial_{\chi_{A}}\Gamma + \xi_{A}s_{\Gamma}( \partial_{\chi_{A}}\Gamma ) = \chi_{A}\partial_{\chi_{A}}\Gamma + \xi_{A}\partial_{\xi_{A}}\Gamma
\sup_{|s-t|\leq\delta}|Y_\lambda(t) - Y_\lambda(s)| =\sup_{|s-t|\leq\delta}|Z_\lambda(\Gamma_\lambda(t)) -Z_\lambda(\Gamma_\lambda(s))| \leq \sup_{|s-t|\leq\delta}|Z_\lambda(t) - Z_\lambda(s)|
W = n \sum_{\bf k}\left[ \sum_{i}\ln \left( 1 - \frac{1 6 \omega_{i}^{2}}{E_{\bf k}^{2}}\right) - \frac{1}{2}\sum_{\begin{array}{c}{\scriptstyle s_{1}, \ldots , s_{5}= \pm 1}\\{\scriptstyle s_{1}\ldots s_{5}= 1}\\ \end{array}}\ln \left( 1 - \frac{2 \sum_{i}\omega_{i}s_{i}}{E_{\bf k}}\right) \right] .
\langle \overset{1}{\mathbb{P}}\left( \tilde{W}\right), e_{n}^{(1)}\rangle = \frac{1}{\left( 1 - k^{2}\, \eta \, a^{2}\, \lambda^{(1)}_{n}\right)}\left[ a \, \langle \mathcal{P}(\cdot , 0) , e_{n}^{(1)}\rangle - \langle T, e_{n}^{(1)}\rangle \right].
N_r = x_r z \sum_{j \geq 0}\Big( \frac{N_1}{x_1 z}(N_1 + N_2 + \cdots + N_{k+1-r}) \Big)^j = \frac{x_r z}{1-\frac{N_1}{x_1 z}(N_1 + N_2 + \cdots + N_{k+1-r})}
\pi^{-\frac{rs}{2}}\Gamma\left(\frac{rs}{2}\right)\zeta(rs) E^P_{s,\varphi}(g)=\pi^{-\frac{r}{2}(1-s)}\Gamma\left(\frac{r(1-s)}{2}\right)\zeta\left(r-rs\right){E^P_{1-s,\varphi}\left((g^\top)^{-1}\right)}.
\frac{1}{\|P-Q\|}\approx \sum_{k=1}^{18}\frac{w_k}{M(k)}\sum_{j=1}^{M(k)}e^{\sigma_k [(x_3+\xi_3) - i (x_1-\xi_1) \cos \alpha_j - i(x_2-\xi_2) \sin \alpha_j ]},
\sum_{r=1}^{k-1}\left(\prod_{a=r+1}^{k}u_{i_{1},i_{a}}\right) \left(\prod_{a=2}^{r}u_{i_{1},i_{a}}\right) u_{i_{r+1},i_{1}}= - \left( \sum_{a=2}^{k}q_{i_{1},i_{a}}\left(\prod_{b =a+1}^{k}u_{i_{a},i_{b}}\right) \left( \prod_{b=2}^{a-1}u_{i_{a},i_{b}}\right) \right).
D_{\mu \nu}^{(0)}= \frac{1}{p^2 + \kappa^2}\left[(\delta_{\mu \nu}-\frac{p_{\mu}\,p_{\nu}}{p^2})- \kappa \epsilon_{\mu \nu\lambda}\,{p_{\lambda}\over p^{2}}\right] + \xi\,\frac{p_{\mu}p_{\nu}}{(p^2)^2},
e^{u\cdot Z_t}=\prod_{j}e^{u_j\cdot Z^j_t}=\prod_j e^{u_j \cdot Y^j_t}\prod_i e^{u_j \cdot X^{i,j}\circ C^i_t}=e^{u\cdot Y_t}\prod_i e^{u\cdot X^i\circ C^i_t}.
\int \delta w^1_h(x,y)\frac{(2-\sigma)\langle A^h(x)y,y\rangle\rho(x+he,y)}{|y|^{n+\sigma+2}}dy =\frac{f(x+he)-f(x)}{|h|^\beta}+k_h(x)|h|^{1-\beta}-I^1_h(x) &-\int \delta w^2_h(x,y)\frac{(2-\sigma)\langle A^h(x)y,y\rangle\rho(x+he,y)}{|y|^{n+\sigma+2}}dy &=\frac{f(x+he)-f(x)}{|h|^\beta}+k_h(x)|h|^{1-\beta}-I^1_h(x)-I^2_h(x).
\begin{bmatrix}I_\ell B_{12}B_{21}B_{22}\end{bmatrix}:= \begin{bmatrix}Q^{-1}0 0 I_{n-\ell}\end{bmatrix}\begin{bmatrix}Q_1 E 0 Q_2\end{bmatrix}\begin{bmatrix}A_{11}A_{12}A_{21}A_{22}\end{bmatrix}\begin{bmatrix}P_1 0 0 P_2\end{bmatrix}.
a_{n}( \mu^{2}/ Q^{2}) = - \frac{e_{f}^{2}\mu^{4}}{1 6 \pi^{3}Q^{4}}\left( - \frac{\beta_{0}}{2}\right)^{n}n ! \left[ \frac{- 2 \beta_{1}}{\beta_{0}^{2}}( \ln n + \gamma_{E}- 1 ) - 2 \frac{p_{2}}{\beta_{0}^{2}}\right] \left( 1 + O \left( 1 / n \right) \right)
K_n=[x-\frac{1}{2n+1},x+\frac{1}{2n+1}] \ \textrm{for}\n \neq m^2 \ \textrm{where}\m \in \mathbb{N}K_n=[x-n,x+n] \ \ \ \textrm{for}\n=m^2 \ \textrm{where}\m \in \mathbb{N}
\sum_{k=0}^\infty \frac{\Gamma\left(\frac{\gamma+1+2k}{2}\right)}{k! \Gamma\left(\zeta+\frac12+k\right)}\left(\frac{1}{4t}\right)^k = \Gamma \left(\frac{1}{2}(\gamma+1)\right) \cdot \,_1{\tilde F}_1\left(\frac{1}{2}(\gamma+1);\zeta+\frac12;\frac{1}{4t}\right).
S_{\mu \nu}: = \left( \begin{array}{c}{G_{\mu \nu}}{E_{\mu \nu}}\end{array}\right) = \left( \begin{array}{c}{g_{\mu \nu}}{H_{\mu \nu}}\end{array}\right) + i \left( \begin{array}{c}{B_{\mu \nu}}{M_{\mu \nu}}\end{array}\right) \, ,
\epsilon_{i}^{a}(\theta )\approx r\,i\,m_{a}\cosh \theta-\sum_{j=1}^{k-1}\left( \delta_{ij}-2\left( A_{ij}^{su(k)}\right)^{-1}\right) L_{j}^{a}(\theta )-\sum_{j=1}^{k-1}\left( A_{ij}^{su(k)}\right)^{-1}L_{j}^{b}(\theta -\sigma )\;.
n^{-1/2}(\hat{\theta}_n -\theta_0)=\int\tilde{I}_0^{-1}\tilde{\ell}_0(x) \,\mathrm{d}\bigl\{n^{-1/2}(F_n-F_0)\bigr\}+ \mathrm{o}_P(1)\stackrel{d}{\longrightarrow}N\bigl(0,\tilde{I}_0^{-1}\bigr),
w^k(x_k + r_k x) \rightarrow w_{3/2}(x_0+ r_0 x) = r_0^{3/2}w_{3/2}(x), r^{-\frac{n+1}{2}}_k \left\| w^k\right\|_{L^2(B_{r_k}^+(x_k))}\rightarrow r^{-\frac{n+1}{2}}_0 \left\| w_{3/2}\right\|_{L^2(B_{r_0}^+(x_0))}= r^{3/2}_0,
f_{X}(x) = \sum^K_{k=1}C^k_K(-1)^{k+1}&\frac{(P_U+kx\lambda_{UU}d^\alpha_{BU}+P_U\lambda_{UU})ke^{-\frac{kxd^\alpha_{BU}}{P_U}}}{d^\alpha_{BU}(\frac{P_U}{d^\alpha_{BU}}+kx\lambda_{UU})^2}.
\alpha_{n,q}(l)= \frac{\binom{l+n}{l}\binom{l-q+n}{l-q}}{n!(q-1)!(n-q)!l(l+n+1-q)}= \frac{l^{2n-2}}{n!(n-1)!(n-q)!}\sum_{i=0}^{2n}Q_{2n-i}(q) \frac{1}{l^i}\sum_{i=0}^\infty \frac{(-1)^i(n+1-q)^i}{l^i}&=\frac{l^{2n-2}}{n!(n-1)!(n-q)!}\sum_{i=0}^\infty \sum_{j=0}^i Q_{2n-i}(q)(-1)^i(n+1-q)^{i-j}\frac{1}{l^i},
\sum_{i=0}^\infty{(3^{2n- \alpha(3)})}^i \cdot |R(t/3^i)| \leq \sum_{i=0}^\infty{(3^{2n- \alpha(3)})}^i \cdot C_J \cdot D(\Omega) \cdot t^{2n}\cdot{(3^{2n})}^{-i}< \infty.
\sum_{n\geq 0}\vert B_n \vert^{\varepsilon}\mu(B_n)=\sum_{k\geq 0}\sum_{B\in\mathcal{T}_k (\mathcal{B})}\vert B \vert^{\varepsilon}\mu(B)\leq \sum_{k\geq 0}\sum_{1\leq j\leq J_k}2^{-k\varepsilon}\sum_{B\in T_{k,j}(\mathcal{B})}\mu(B).
a_{1}\, \gamma \, - \, a_{2}\alpha \, h^{- 1}\, \left( h^{- 1}+ \mu \mathcal{F}\right)_{S}^{- 1}\, h^{- 1}\, \left[ \mathrm{d e t}\, \left( h^{- 1}+ \mu \mathcal{F}\right) \right]^{\alpha}= 0
\int\frac{x^{2}}{\sqrt{-x^{2}+3x-2}}dx=\int\frac{(\frac{-2t^{2}-1}{-t^{2}-1})^{2}\frac{2t}{(-t^{2}-1)^{2}}}{\frac{t}{-t^{2}-1}}dt=\int\frac{2(-2t^{2}-1)^{2}}{(-t^{2}-1)^{3}}dt.
\left. \begin{array}{l l l l r}{a = \sqrt{2 1 5 3 3}+ 1 1 9}&{}&{b}&{=}&{- 1 9 4}\\{c = \sqrt{2 1 5 3 3}- 1 1 9}&{}&{d}&{=}&{1 9 4}\\ \end{array}\right.
I = \frac{s}{4}\int_0^{\infty}\lambda^{s/4-1}\mu(\{w \in R\colon\, |T_{\mu, \delta}(1_{R}b_1, 1_Rb_2)(w)| > \lambda\})\,d\lambda &\le \frac{s}{4}\bigg[ \int_0^1 \lambda^{s/4-1}\,d\lambda + C_0 \int_1^{\infty}\lambda^{-3s/4-1}\,d\lambda\bigg]\mu(R) \lesssim \mu(R).
\left\langle U_{\lambda}\left( w \right) \right\rangle_{w_{n_{i^{\prime}}^{*}= 2}^{\prime}}^{\left({\it G}_{1}\right)_{X_{i^{\prime}}^{\prime}}l c}= 1 - \left( L^{( 2 \beta - d )}\lambda - \beta_{1}\lambda^{2}+ O ( \lambda^{3}) \right) \times
\frac{\left\||g_0|^{r-1}\right\|_{L^{s'}(\Omega)}}{\|g_0\|^{r-1}_{L^r(\Omega)}}=\left\||g_0|^\frac{p-1}{p+1}\right\|_{L^{(2^*)'}(\Omega)}=c_4\, c_2^\frac{1-p}{p+1},
W_r(R_t(t_0)) =W_r(t_1) =\Big(\frac{r}{1-e^{-r}}\Big)e^{-rt_1}= \Big(\frac{r}{1-e^{-r}}\Big)e^{rk}e^{-r(t_1+k)}&= \Big(\frac{r}{1-e^{-r}}\Big)e^{rk}e^{-r(t_0-t)}=e^{rk}e^{rt}W_r(t_0)\le e^{rt}W_r(t_0),
\left( \begin{array}{c}{{X^{0}}}\\{{X^{1}}}\\{{X^{2}}}\\{{X^{3}}}\\ \end{array}\right) ( \tau , \sigma , \zeta ) = \left( \begin{array}{c}{{\tau}}\\{{r ( \tau ) \; \sin \sigma \; \cos \zeta}}\\{{r ( \tau ) \; \sin \sigma \; \sin \zeta}}\\{{r ( \tau ) \; \cos \sigma}}\\ \end{array}\right) ,
\mathbb E\left|Y_{k+1,i}^{(t)}-Y_{k,i}^{(t)}\right|^\alpha \leq Ck^{1+\frac{2^m-1}{2^{m+1}}\alpha}\left[\max\{p^{\alpha-1}\frac{\rho(\alpha t)}{\rho(t)^\alpha},\;p^{\frac{2^l-1}{2^l}\alpha}\frac{\rho(2^lt)^{\alpha/2^l}}{\rho(t)^\alpha},l=1,2,\cdots,m+1\}\right]^k.
\delta_{\alpha,\beta}g = \mathfrak{D}_{{\rm left}, \frac{(\alpha+\beta+2)x-(\beta-\alpha)}{2(1-x^2)}}((1-x^2)^{1/2}g) -\delta^*_{\alpha,\beta}g = \mathfrak{D}_{{\rm right}, -\frac{(\alpha+\beta)x-(\beta-\alpha)}{2(1-x^2)}}((1-x^2)^{1/2}g).
\frac{dz}{dx}=\frac{dz}{dy}\frac{dy}{dx}=\frac{d(y^{3})}{dy}\frac{d(x^{2})}{dx}=\frac{3y^{2}dy}{dy}\frac{2xdx}{dx}=3y^{2}*2x=3(x^{2})^{2}*2x=6x^{5}
\kappa_{I}\epsilon^{i j}\nabla_{j}a_{0}^{I}({\bf r}, t ) + i \hbar \sum_{p}\frac{q_{p}^{I}}{2 m_{p}c}\left[ ( D_{i}\Psi_{p})^{\dagger}({\bf r}, t ) \Psi_{p}({\bf r}, t ) - \Psi_{p}^{\dagger}({\bf r}, t ) ( D_{i}\Psi_{p}) ({\bf r}, t ) \right] = 0 \, ,
\delta_{\epsilon}b = - \bar{\epsilon}\Gamma_{1 1}\Gamma_{m}d \theta \left( d X^{m}+{\frac{1}{2}}\bar{\theta}\Gamma^{m}d \theta \right) +{\frac{1}{2}}\bar{\theta}\Gamma_{1 1}\Gamma_{m}d \theta \bar{\epsilon}\Gamma^{m}d \theta .
\sigma_{i,j}\sigma_{k,l}=\sigma_{k,l}\sigma_{i,j}{\rm if}(j-l)(j-k)(i-l)(i-k) > 0, \sigma_{i,j}\sigma_{i,k}=\sigma_{j,k}\sigma_{i,j}=\sigma_{i,k}\sigma_{j,k}{\rm if}1 \le i < j < k \le n.
{\cal H}_{eff}= \frac 1 2 (\partial_i C_j)^2 - \Pi[\Delta_\perp^d]^{-1}* \left[\partial_i j^{i}+ \partial_{-}j^{-}\right] - C_i \left[j^i - 2\partial_i[\Delta_\perp^d]^{-1}*(\partial_{-}j^{-}+ \partial_j j^j) \right] - C_{-}j^{-},
\left(\begin{matrix}d^* -c^* c d\end{matrix}\right)=\left(\begin{matrix}1 x^*+x_0 0 1\end{matrix}\right)\left(\begin{matrix}a_2 0 0 a_2^{-1}\end{matrix}\right)\left(\begin{matrix}\alpha_2 \beta_2 \gamma_2 \delta_2\end{matrix}\right).
\prod_{pprime}\frac{1}{1-p^{-s}}=\frac{1}{1-2^{-s}}\cdot\frac{1}{1-3^{-s}}\cdot\frac{1}{1-5^{-s}}\cdot\frac{1}{1-7^{-s}}\cdot\frac{1}{1-11^{-s}}\cdot\cdot\cdot\frac{1}{1-p^{-s}}\cdot\cdot\cdot
\int_{t-a}^{t-b}\tilde{M}(W)^m(-dW) =-\left(\int_{-\infty}^{t-b}\tilde{M}(W)^mdW - \int_{-\infty}^{t-a}\tilde{M}(W)^mdW\right) &=\int_{0}^{t-a}\tilde{M}(W)^mdW - \int_{0}^{t-b}\tilde{M}(W)^mdW &=\frac{1}{m+1}\left( \tilde{M}(t-a)^{m+1}-\tilde{M}(t-b)^{m+1}\right).
H^s(\Psi(Z(v)) = \sum_{v_i \in \Lambda^0}A_1(v, v_1) \cdots A_k (v_{k-1}, v_k) H^s(\Psi(Z(v v_1 \cdots v_k))) &= \sum_{v_i}A_1(v, v_1) \cdots A_k(v_{k-1}, v_k) N^{-ks}H^s(\Psi(Z(v_k))) &= \sum_{v_k \in \Lambda^0}A_1 A_2 \cdots A_k (v, v_k) N^{-ks}H^s(\Psi(Z(v_k))).
d s^{2}= d \chi^{2}+ \sinh^{2}\chi \, d \varphi^{2}- \cosh^{2}\chi \, d \tau^{2}= \left({\frac{2}{1 - \rho^{2}}}\right)^{2}( d \rho^{2}+ \rho^{2}d \varphi^{2}) - \left({\frac{1 + \rho^{2}}{1 - \rho^{2}}}\right)^{2}d \tau^{2}.
\lambda_0\|b\|_{X^\ast}\le\|Bc\|_1\|b\|_{X^\ast}+\frac{\kappa}{1-\kappa}\left(\lambda_0+\|Bc\|_1\right)\|b\|_{X^\ast}&=\left(\frac{\kappa}{1-\kappa}\lambda_0+\frac{1}{1-\kappa}\|Bc\|_1\right)\|b\|_{X^\ast}.
\Big\|(I\otimes M_v)\Big(D_m+\sum_{k=1}^{d}e_k\otimes M_{w_k}+I\otimes M_{w_{d+1}}-iI\Big)^{-1}\Big\|_p &\quad\le c_{d,p}\big(1+\max_{1\le k\le d+1}\|w_k\|_{L^\infty}\big)\begin{cases}\|v\|_{\ell^p(L^2)}\; 1\le p<2 \|v\|_{L^p}\; 2\le p<\infty.\end{cases}
T_{i-1}T_{i+1}(E_{i}) &=E_{i+1}E_{i-1}E_{i}-q^{-1}E_{i-1}E_{i}E_{i+1}-q^{-1}E_{i+1}E_{i}E_{i-1}+q^{-2}E_{i}E_{i-1}E_{i+1}.
\nu_+ - \nu_- = \frac{N}{24 \cdot 8 \pi^2}\int_{\cal M}\hbox{tr}(R \wedge R) - \frac{g^2 N_f}{32 \pi^2}\int_{\cal M}F^{a}_{\mu \nu}\tilde{F}^{\mu \nu a}d^4 x,
\frac{1}{f_j^k}\frac{df_{ij}}{f_{ij}^{k+1}}+ \left( \frac{1}{f_{ij}^k}- 1 \right) \frac{df_j}{f_j^{k+1}}= \frac{1}{f_{ij}^{k+1}}\underbrace{\left( \frac{df_{ij}}{f_j^k}+ \left( f_{ij}-{f_{ij}^{k+1}}\right) \frac{df_j}{f_j^{k+1}}\right)}_{\Theta}\, .
( b_{i})^{n}\psi_{p}= \sum_{{{1 \leq j_{k}\neq j_{l}\leq N \atop j_{k}, j_{l}\neq i}\atop ( 1 \leq k < l \leq n )}}\prod_{k = 1}^{n}\frac{1}{z_{i}- z_{j_{k}}}\psi_{p},
\int e^{i \vec{q}\cdot \vec{\varrho}}\frac{\vec{q}}{\vec{q}^{2}+ \mu^{2}}\frac{d^{2}\vec{q}}{( 2 \pi )^{2}}= i \frac{| \mu |}{2 \pi}K_{1}( | \mu | \varrho ) \hat{\varrho},
\frac{f_{2}^{\prime \prime}}{f_{2}^{\prime}}\left( \mu_{1}+\mu_{2}\frac{f_{2}^{\prime}}{f_{2}}\right) =\left( \mu_{3}\frac{f_{2}^{\prime}}{f_{2}}+\mu_{4}\left( \frac{f_{2}^{\prime}}{f_{2}}\right)^{2}\right) .
\sum_{n_0=1}^{+\infty}P(\vec n, \vec m) = \frac{(1-q)^{-\ell-1}}{4^{-\ell}Z_{\ell}(q)}\left(\frac{\rho_b}{1-\rho_b}\right)^{x_{\ell}}\sum_{k=0}^{\ell}\frac{(-1)^k}{1-q^k R}e_k\left( \left\lbrace q^{x_i}\right\rbrace_{0\leq i\leq \ell}\right) P^{SSRW}_{0,0}(\vec x, \vec m).
u (zq^n)D(f,g)(zq^{n}) - u(-q^n) D(f,g)(-q^n) = & \left[f(zq^n )-f(-q^n)\right] g(zq^{n+1}) - f(zq^{n+1}) \left[ g(zq^n) - g(-q^n) \right] & + f(-q^n) \left[ g(zq^{n+1}) - g(-q^{n+1})\right] - \left[ f(zq^{n+1}) - f(-q^{n+1}) \right] g(-q^n).
\sum_{b=0}^{N-1}\phi_{j}^{(0)}(b|u,v,w)L(a,b|u,v,w)\,{\bf p}_{j+1}=\phi_{j}^{(1)}(a|u,v,w)\,{\bf p}_{j}\quad \mbox{for}\quad a\in{\bf Z}/N{\bf Z}.
\Big|\int_{B_R}u [H_V, B] u{\rm d}x\Big|= \Big|\int_{B_R}\left(4u\Delta^3u + [V, A\Delta]u\right){\rm d}x\Big| &= \Big|\int_{B_R}\left( 4u\Delta (-Vu + \lambda u) + [V, A\Delta]u \right){\rm d}x\Big| &\leq C\lambda \|u\|^2_{H^2(B_R)}\leq C\lambda^2.
\| E_{\mathbb{P}^{d}}g \|_{L^p(B_{N^2})}\lesssim \| w_{B_{N^2}}E_{\mathbb{P}^{d}}g \|_{L^p(\mathbb{R}^{d+1})}= \| \sum_{\theta: \frac1N\times \cdots \times \frac1N}w_{Q_{N^2}}E_{\mathbb{P}^{d}}g_\theta \|_{L^p(\mathbb{R}^{d+1})}=: \| \sum_{\theta: \frac1N\times \cdots \times \frac1N}f_\theta \|_{L^p(\mathbb{R}^{d+1})},
0 \le \lim_{n\to\infty}\frac{1}{n\pi}\int_{-a_n}^{a_n}\frac{Q(s)}{\sqrt{a_n^2-s^2}}\, ds \le \frac{Q(a_n)}{n\pi}\int_{-a_n}^{a_n}\frac{ds}{\sqrt{a_n^2-s^2}}= \frac{Q(a_n)}{n}\le \frac{C}{\sqrt{T(a_n)}}\to 0
|\mathfrak{J}| \leq \frac{2}{s}\Big[\frac{D(s)- E(s)}{E(s)}\Big]+ \frac{1}{s}\cdot \frac{s^2\sup_{b\leq s}|\nabla u|^2}{E(s)}\cdot \Big[s^{-n}\int_{b\leq s}\Big|Hess(b^2)- 2g\Big|\Big] &= \mathfrak{J}_1+ \mathfrak{J}_2
J_{\mu}(z)=\sqrt{\frac{2}{\pi z}}\left [\cos\left (z-\frac{1}{2}\mu\pi-\frac{1}{4}\pi\right ) -\frac{1}{2z}\left(\mu^2-\frac{1}{4}\right)\sin\left(z-\frac{1}{2}\mu\pi-\frac{1}{4}\pi\right)+O\left (z^{-2}\right )\right ],
I(\dot w, \dot \rho) \phi = I(\dot w, \dot \rho) (t^\vee \wedge \phi_0) &= w(t^\vee) \wedge I(\dot w, \dot \rho) \phi_0 &= w(t^\vee) \wedge \left( \frac{w(\hat f)}{\hat f}\ast \phi_0 \right) &= t^\vee \frac{w(t^\vee)}{t^\vee}\wedge \left( \frac{w(\hat f)}{\hat f}\ast \phi_0 \right) &= t^\vee \wedge \phi_0 = \phi.
\langle n_1, \mu_1, \epsilon_1, n_2, \mu_2, \epsilon_2 | n_{12}, \mu_{12}, \epsilon_{12}\rangle = 2^{-n_1}\sqrt{\frac{w_j [n_2]_{\mu_2}!}{\kappa_0 [n_1]_{\mu_1}! [n_1+n_2]_{\mu_2}!}}R_{n_1}^{(-1)}(y_j ; \mu_2, \mu_1, n_1 + n_2),
F_\lambda(\zeta,\mu) :=\,k\,a_0+a_1\sum_{i=1}^k\Bigl(\mu_i\,g_{\lambda}(\zeta_i)-\sum_{j\neq i}\mu_i^{1/2}\,\mu_j^{1/2}\,G_{\lambda}(\zeta_i,\zeta_j)\Bigr)+a_2\,\lambda\,\sum_{i=1}^k \mu_i^2 &-a_3\,\sum_{i=1}^k\Bigl(\mu_i\,g_{\lambda}(\zeta_i)-\sum_{j\neq i}\mu_i^{1/2}\mu_j^{1/2}\,G_{\lambda}(\zeta_i,\zeta_j)\Bigr)^2 ,
\sigma'(c) = (f \circ \rho \circ \sigma \circ \rho^{-1}(P_1),\dots,f\circ \rho \circ \sigma \circ \rho^{-1}(P_n)) & = (h \circ \sigma (\rho^{-1}(P_1)),\dots, h \circ \sigma (\rho^{-1}(P_n)))
T_{\alpha,\beta, m}[y](x) =\frac{1}{W_{\alpha,\beta, m}(x)}\left[\left(W_{\alpha,\beta, m}(x)(1-x^2)y'(x)\right)'\right. & \qquad\qquad\left.+W_{\alpha,\beta, m}(x)\left(m(\alpha-\beta-m+1)-2\beta(1-x)\left(\log(P_m^{(-\alpha-1,\beta-1)}(x))\right)'\right)y(x)\right]
g(z,\bar{z}_1z_2 + \bar{z}_1^2) = \bar{z}_1 f(z,\bar{z}_1z_2 + \bar{z}_1^2)+ \bar{z}_1^2 + r(z_1,\bar{z}_1,f(z,\bar{z}_1z_2 +\bar{z}_1^2),z_3,\ldots,z_n) .
d_{Q}({\bf x},\Sigma) =|x_j| \ \j\in Q=\{1,\ldots,M\}\intertext{or}d_{Q}({\bf x},\Sigma) =\tfrac{1}{\sqrt{2}}|x_j-x_k| \ \ \text{for some}j\in Q,k\in\{M+1,\ldots,N\}\,.
h^\mu_{~\nu}= \left (\begin{array}{ccccc}h_1 h_3 0 0 0 -d^2 h_3 /f h_2 0 0 0 0 0 h^\chi_{~\chi}h^\chi_{~\theta}h^\chi_{~\phi}0 0 h^\theta_{~\chi}h^\theta_{~\theta}h^\theta_{~\phi}0 0 h^\phi_{~\chi}h^\phi_{~\theta}h^\phi_{~\phi}\end{array}\right )
\frac{(-1)^{|\pi(\tau)|}x^\tau \langle V,z_\tau\rangle}{x_{j_{s+1}}U}= -\prod_{r=1}^{s}D_{j_{r}-1}~-~\sum_{q=1}^{s}\left(\prod_{r=2}^q D_{j_{r-1}}\right)x_{j_q}\left(\prod_{r=q+1}^{s}D_{j_{r}-1}\right)~+~\prod_{r=2}^{s+1}D_{j_{r-1}}
P(X_{q}=1)=\frac{\#\{o\in O_{\mathbb{T}}:T_{\mathbb{T}}(o)=t,A\cup\{a\}\subseteq A_{\mathbb{T}}(o)\}}{\#\{o\in O_{\mathbb{T}}:T_{\mathbb{T}}(o)=t,A\subseteq A_{\mathbb{T}}(o)\}}
\begin{pmatrix}1 0 0 0 0 0 \frac{1}{2}\frac{1}{2}0 1 1 -1 \frac{1}{2}0 \frac{1}{2}1 1 -1 -\frac{3}{4}\frac{3}{8}\frac{3}{8}\frac{3}{2}-1 1 0 0 0 0 \frac{3}{2}-1 0 0 0 0 -1 \frac{3}{2}\end{pmatrix}
\delta_{\epsilon}\mathrm{{\bf ~ A}}= \left[ \begin{array}{c c}{\epsilon^{k}\bar{\psi}_{k}- \psi^{k}\bar{\epsilon}_{k}}{D \epsilon_{j}}{- D \bar{\epsilon}^{i}}{\bar{\epsilon}^{i}\psi_{j}- \bar{\psi}^{i}\epsilon_{j}}\end{array}\right] ,
\delta \Psi \bigg ( \phi^{A}, \, \phi_{A}^{*}; \, \frac{\delta_{l}}{\delta \phi^{A}}, \, \frac{\delta}{\delta \phi_{A}^{*}}\bigg ) = \delta \Psi \bigg ( \varphi^{i}, \, \psi^{\alpha}, \, \phi_{A}^{*}; \, \frac{\delta_{l}}{\delta \varphi^{i}}, \, \frac{\delta_{l}}{\delta \psi^{\alpha}}, \, \frac{\delta}{\delta \phi_{A}^{*}}\bigg )
u_{(k)}(x,t):= \int\limits_{A-1}^{0}\eta_{\frac{1}{k}}(t-\tau)\int\limits_{\mathbb{R}^{3}_{+}}\omega_{\frac{1}{k}}(x-y)\phi_{k}(y)u^{(\frac{2}{k})}_{}(y,\tau)dyd\tau
\sum_{k=1}^n a_k b_k = a_n \sum_{k=1}^n b_k - \sum_{k=1}^{n-1}\sum_{m=1}^k b_m(a_{k+1}-a_k) &= \frac{1}{n}\int_1^{n+1}f ( \{x\}, \{\nu x\}) \mathrm{d}x + \sum_{k=1}^{n-1}\frac{1}{k(k+1)}\int_1^{k+1}f ( \{x\}, \{\nu x\}) \mathrm{d}x
\textstyle\big\{\; \{4k+1,4k+2,4k+5\}, \{4k+1,4k+3,4k+6\}, \{4k+1,4k+4,4k+7\}, \{4k+2,4k+3,4k+7\}, \{4k+3,4k+4,4k+5\}, \{4k+5,4k+6,4k+7\}, \{4k+2,4k+4,4k+6\}\; \big\}, (k\geq 1).
S_{FP}=\int d^4 x\biggl[-{1\over 4}F_{\mu\nu}F^{\mu\nu}+{\bar\psi}(i\gamma^\mu D_\mu -m)\psi-{1\over 2\xi}(\partial_\mu A^\mu)^2+i{\bar C}\partial_\mu\partial^\mu C\biggr] \,,
\left|T(f,g;x)\right| \leq \left\|L_n\right\|(1+\left\|L_n\right\|)\cdot K(f,1;C,Lip_1)\cdot K(g,1;C,Lip_1) &=\left\|L_n\right\|(1+\left\|L_n\right\|)\cdot\frac{1}{2}\cdot \widetilde{\omega}(f;2)\cdot\frac{1}{2}\cdot\widetilde{\omega}(g;2) &=\frac{1}{4}\left\|L_n\right\|(1+\left\|L_n\right\|)\omega(f;2)\cdot\omega(g;2).
\begin{aligned}f_0 = \rho, f_{e_1}= f_{e_2}= f_{e_3}= 0, \sum_{d=1}^3 f_{2e_d}= 0, \sigma_{ij}= (1+\delta_{ij}) f_{e_i+e_j}, q_i = 2 f_{3e_i}+ \sum_{d=1}^3 f_{2e_d+e_i}, i,j=1,2,3,\end{aligned}
G_{\alpha}^{( 1 )}\equiv \{\tau_{\alpha}^{( 1 )}, \tilde{H}^{( 0 )}\}+ \{\tau_{\alpha}^{( 0 )}, \tilde{H}^{( 1 )}\}+ \{\tau_{\alpha}^{( 2 )}, \tilde{H}^{( 1 )}\}_{( \Phi )}\nonumber
\lim_{k\rightarrow\infty}\frac{n_k}{\sum_{j=1}^{k-1}N_jn_j+n_0'}=0,\ \lim_{k\rightarrow\infty}\frac{\tilde{n}_k}{\sum_{j=1}^{k-1}\tilde{N}_j\tilde{n}_j+\tilde{n}_0'}=0