harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
IS^{(\mathrm{soft})}(\kappa, \pi;u,v)= \int_{\mathcal{C}_{>}}\frac{dz}{2\pi i}\int_{\mathcal{C}_{>}}\frac{dw}{2\pi i}e^{-\frac{1}{3}(w-\kappa)^3+u(w-\kappa)-\frac{1}{3}(z-\kappa)^3+v(z-\kappa)}& \times \frac{1}{\sqrt{4zw}}\frac{w-z}{z+w}\prod_{k = 1}^{m}\frac{z-\pi_{k}}{z +\pi_{k}}\frac{w-\pi_{k}}{w + \pi_{k}},
|\phi_{(ML)}> = \prod_{(\vec{p})^{+}}\; |\phi_{M\vec{p}, L}>, \; |\phi_{M\vec{p}, L}> = ( \beta_{\vec{p}}{}^2 |1> +(-)\sqrt{2}\; \alpha_{\vec{p}}\beta_{\vec{p}}|2> + \alpha_{\vec{p}}{}^2|3> ),
\nabla^{(1, 0)}\eta(Az_\star, y_\star) = \sum_{i=1}^d ((Az_\star)^{(i)})^{p-1}y_\star^{(i)}e_d^{(i)}, \nabla^{(0, 1)}\eta(Az_\star, y_\star) = \tfrac{1}{p}\left(\sum_{i=1}^d ((Az_\star^{(i)}))^p e_d^{(i)}\right).
(x^0)^{*_\star}:=x^0,~~~~~ (x^i)^{*_\star}:=x^\mu\tau^{\mu i}(S(\beta)),~~~~~ (\xi^i)^{*_\star}:=\xi^j\tau^{ji}(S(\beta)),~~~~~ (\partial'_i)^{*_\star}:=-\tau^{ij}(\beta^{-1})\partial'_j,
\dim I_{V}\left(_{W^{1}W^{2}}^{\ \W^{3}}\right)=\dim I_{V}\left(_{W^{2}W^{1}}^{\ \W^{3}}\right),\dim I_{V}\left(_{W^{1}W^{2}}^{\ \W^{3}}\right)=\dim I_{V}\left(_{W^{1}\left(\W^{3}\right)'}^{\ \ \left(W^{2}\right)'}\right).
p_j(u'')=\begin{cases}2, \; \textnormal{if}j=1, 0, \; \textnormal{if}j=2, 1, \; \textnormal{if}3 \leq j \leq 2k-2 p_j(w^{**}-v_d)=1, 0, \; \textnormal{if}3 \leq j \leq 2k-2 p_j(w^{**}-v_d)=-10, 0, \; \textnormal{if}j=2k-1, \end{cases}
T_{13}:= \biggl|\int_{\mathcal{X}_n}\int_{B_x^c\bigl(\frac{r_{n,1}d_n}{f(x)^{1/d}}\bigr)}f(x) f(y) \log f(y) & \int_{\tilde{u}_{n,x,y}}^\infty \log \bigl(uf(x)\bigr) \, d(\tilde{F}_{n,x}- F_{n,x}^-)(u) \, dy \, dx\biggr|.
\mathcal L_g \mathcal U_{\mu,\xi}-\Delta \mathcal U_{\mu,\xi}\sim -\frac 13 \sum\limits_{a,b,i,j=1}^NR_{iab j}(\xi)x_a x_b\partial^2_{ij}U_{\mu,0}+\sum\limits_{i, l,k=1}^N \partial_l \Gamma^k_{ii}(\xi)x_l\partial_k U_{\mu,0}+{\beta_N R_g(\xi)}U_{\mu,0}.
Q_{00}(u,v) = -(\partial_t u)(\partial_t v) + \nabla_x u \cdot \nabla_x v Q_{ij}(u,v) = (\partial_{x_i}u) (\partial_{x_j}v) - (\partial_{x_j}u) (\partial_{x_i}v), i,j \in \{1,\dots,n\}Q_{0j}(u,v) = (\partial_t u)(\partial_{x_j}v) - (\partial_{x_j}u) (\partial_t v), i \in \{1,\dots,n\}.
f_1 = -(2\zeta^3+2\zeta^2+1)g_1 - g_2 + g_3, f_4 = -(2\zeta^3+2\zeta^2+1)g_4+g_5 - g_6, f_2 = (1/5)(g_2-\zeta^3g_3), f_5 = (1/5)(g_5-\zeta g_6), f_3 =(1/15)(g_2-\zeta^2g_3), f_6 = (1/10)(g_5-\zeta^4g_6).
R^\mu = \left(r^2v_\theta^2+v_r^2\right)+m_3\left(\rho^2v_\theta^2+v_{\rho}^2\right)+r^2v_\theta+\rho^2v_\phi-r^2-\rho^2-\left((x^1)^2+(y^1)^2\right)-\mu_\theta v_\theta-\mu_\phi v_\phi.
T_{v}p=[\begin{matrix}1&0&0&v_{x}\\ 0&1&0&v_{y}\\ 0&0&1&v_{z}\\ 0&0&0&1\end{matrix}][\begin{matrix}p_{x}\\p_{y}\\p_{z}\\ 1\end{matrix}]=[\begin{matrix}p_{x}+v_{x}\\p_{y}+v_{y}\\p_{z}+v_{z}\\ 1\end{matrix}]=p+v
{\cal L}_{{\rm rel}}^{(2)}= \frac{1}{2}\biggl( \mu + \frac{g^2\lambda}{8\pi r}\biggr)\,\dot{{\bf r}}^2+ \frac{1}{2}\left(\frac{g^2\lambda}{8\pi}\right)^2 \biggl( \mu+ \frac{g^2\lambda}{8\pi r}\biggr)^{-1}(\dot{\psi}+{\bf w}({\bf r})\cdot \dot{{\bf r}})^2 .
\partial_tR^{\pm}+\frac{1}{\partial_1\hat{\Phi}^{\pm}}\Big\{(\hat{\mathbf{w}}^{\pm}\cdot\nabla R^{\pm})+R^{\pm}\mathrm{div}\hat{\mathbf{v}}^{\pm}\Big\}=\mathcal{F}^{\pm}\Omega_T,
\partial_x^m\,\mathbb{E}f\bigl(Y^n_t(x)\bigr) =\partial_x^{m-1}\mathbb{E}f'\bigl(Y^{n}_{t}(x) \bigr) +\frac{1}{2}\sqrt{t}\,\partial_x^{m-1}\mathbb{E}\bigl(\xi g_1\bigl(x,\xi \sqrt t, Y^{n-1}_{t}\bigr) \bigr) &=:P_m(t,x)+R_m(t,x).
|d(\psi(y_1), \psi(y_2)) - d(y_1, y_2)| \leq |d(\phi^{-1}(\pi(y_1)), \phi^{-1}(\pi(y_2))) - d(\pi(y_1), \pi(y_2))| + |d(\pi(y_1), \pi(y_2)) - d(y_1, y_2)| &\leq \epsilon + 6\epsilon &= 7\epsilon.
b(s)=-\frac{e^{si\pi/2}}{2^{s}-1}\int_{0}^{\infty}\frac{st^{s}}{sinh\pi t}\frac{dt}{t}=\frac{2e^{si\pi/2}}{2^{s}-1}\int_{0}^{\infty}\frac{e^{\pi t}st^{s}}{1-e^{2\pi t}}\frac{dt}{t}.
ds^{2}=-\frac{1}{(1-\frac{2m}{r}-\frac{\epsilon^{2}}{2r^{2}})}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta d\varphi^{2})+(1-\frac{2m}{r}-\frac{\epsilon^{2}}{2r^{2}})dt^{2},
\sqrt{\beta_K}\tilde{\mathbf{G}}= \mathbf{Z}_{p}\boldsymbol{\Psi}_p^{\dagger}(\boldsymbol{\Psi}_{p}\boldsymbol{\Psi}_p^{\dagger})^{-1}= \mathbf{Z}_{p}\boldsymbol{\Psi}_p^{\dagger}/\left(m_0m_1\rho_p\right).
E^\gamma E^{e_I}=\sum_{v_2=i_2}^n\cdots \sum_{v_t=i_s}^n\prod_{u=2}^s\gamma_{v_ui_u}^{(1-\delta_{v_ui_u})}E^{\gamma+\sum_{u=2}^s e_{v_ui_{u-1}}-e_{v_ui_u}}.
(x,t)(A)=(\frac{(-\frac{L^{\prime}}{2})+u(A-\frac{L^{\prime}}{2c})}{\sqrt{1-u^{2}/c^{2}}},\frac{(A-\frac{L^{\prime}}{2c})+(-\frac{L^{\prime}}{2})u/c^{2}}{\sqrt{1-u^{2}/c^{2}}})
d s^{2}= g_{t t}d t^{2}+ g_{r r}d r^{2}+ 2 g_{t \phi_{i}}d t d \phi_{i}+ g_{\phi_{i}\phi_{j}}d \phi_{i}d \phi_{j}+ g_{\theta \theta}d \theta^{2}+ 2 g_{\theta \psi_{i}}d \theta d \psi_{i}+ g_{\psi_{i}\psi_{j}}d \psi_{i}d \psi_{j},
\begin{array}{l}{{\Phi_{j}\bar{\Phi}_{i}= \delta_{i j}+ q R_{k i j l}\bar{\Phi}_{l}\Phi_{k},}}\\{{\Phi_{l}\Phi_{k}= q^{- 1}R_{j i k l}\Phi_{j}\Phi_{i}, \qquad i , j = 1 , 2 , . . . , N ,}}\\ \end{array}
\sum_{j=1}^n \bigg\|\frac{L_j f}{|x|}\bigg\|_{2}^2 = - \bigg\langle f , \frac{1}{|x|^4}\Delta_{\mathbb{S}^{n-1}}f \bigg\rangle = -\int_0^\infty \langle f(r(\cdot)) , \Delta_{\mathbb{S}^{n-1}}f(r(\cdot)) \rangle_{L^2(\mathbb{S}^{n-1})}r^{n-5}\, \mathrm{d}r
\ddot{\bf r}= \omega^{2}{\bf r}+ 2 \omega J \dot{\bf r}-\frac{C_{13}m_1 ({\bf r}-{\bf s}_1)}{\|{\bf r}-{\bf s}_1\|^{3}}- \frac{C_{23}m_2 ({\bf r}-{\bf s}_2)}{\|{\bf r}-{\bf s}_2\|^{3}},
f_i \pi := ( x_1,\,\ldots,\,x_{p},\,r_ix_{p+1},\,\dots,\, r_ix_{q},\,r_ix_{q+1},\,x_{q+1},\,\ldots,\,x_s ; & a_0,\,\ldots,\,a_{p}=t_0,\,\ldots,\,a_{q},\,t_1,\, a_{q+1},\,\ldots,\,a_s);
w^{\prime \prime}+ ( \frac{\nu^{\prime}}{\nu}+ \frac{R^{\prime}}{R}- 2 \Phi^{\prime}) w^{\prime}- \frac{( w^{2}- 1 ) ( 4 \kappa + ( 4 R^{2}- 3 ) w + w^{3})}{2 \nu R^{4}}= 0 ,
c^{2}d\tau^{2}=(g_{tt}-\frac{g_{t\phi}^{2}}{g_{\phi\phi}})dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}(d\phi+\frac{g_{t\phi}}{g_{\phi\phi}}dt)^{2}.
\sum_{l = 0}^{N - 1}\frac{w ( v_{2}, a - l )}{w ( v_{1}, - l ) \gamma ( b , l )}= \varphi_{1}( v_{1}, v_{2}) \frac{w ( v_{2}^{\prime}, - b ) w ( v_{2}/ ( \omega v_{1}) , a )}{w ( v_{1}^{\prime}, a - b )},
H(z,w)=\left( \frac{1}{2\pi i}\right)^{2}\int_{\sigma_{r}}\int_{\sigma_{s}}\sum_{m,n=0}^{\infty}\frac{z^{m}}{u^{m+1}}\frac{w^{n}}{v^{n+1}}H(u,v)dudv(z\in \Delta_{s},w\in \Delta_{r},v\in \sigma_{r},u\in \sigma_{s})
\mathbb{G}[\mathcal{A}]=(\mathbb{G}(A_{1}),\dots,\mathbb{G}(A_{m_{1}}))^{t}, \qquad\mathbb{E}[f|\mathcal{A}]=\left( \mathbb{E}(f|A_{1}),\dots,\mathbb{E}(f|A_{m_{1}})\right)^{t}, \mathbb{G}[\mathcal{B}]=(\mathbb{G}(B_{1}),\dots,\mathbb{G}(B_{m_{2}}))^{t}, \qquad\mathbb{E}[f|\mathcal{B}]=\left( \mathbb{E}(f|B_{1}),\dots,\mathbb{E}(f|B_{m_{2}})\right)^{t}.
P_0 = 1, \quad P_1 = -\tau, \quad P_2 = \frac{1}{2}\, \kappa^2, \quad P_3 = \frac{1}{2}\, \kappa^2\tau, \quad P_4 = \frac{1}{2}\, \kappa^{(1)\,2}+ \frac{1}{2}\,\kappa^2\tau^2 - \frac{1}{8}\,\kappa^4, \quad \ldots,
E\sup_{t_0 \leq t \leq t_1}|e_t^n|^q = E\sup_{t_0 \leq t \leq t_1}|e_t^n|^q I_{\Omega_1}+ E\sup_{t_0 \leq t \leq t_1}|e_t^n|^q I_{\Omega_2}=: D_1+D_2.
{\frac{1}{2 j + 1}}\; \left({\frac{D_{\mathrm{e f f}\s}^{j}}{D_{s}^{j}}}\right)^{2}\; = \; 4 \pi \sqrt{2}\; ( T_{D 2}\, \alpha^{\prime})^{2}\; ( k + 2 )^{3 / 2}\; \left[{\frac{\sin x_{j}}{x_{j}}}\right]
X^{t,x}_{s}=\ x+\xi(t,x)- \xi(s,X^{t,x}_{s})+(\lambda+1)\int_{t}^{s}\xi(r,X^{t,x}_{r})\mathrm{d}r +\int_{t}^{s}{( \nabla \xi(r,X^{t,x}_{r}) + \mathrm I_d)}\mathrm{d}W_{r},
\begin{array}{l l}{{}}&{{M^{2}\sinh^{2}( 2 \lambda \theta ) / ( P_{\mathrm{T}}( \theta ) P_{\mathrm{T}}( - \theta ) )}}\\{{}}&{{= k^{2}\sinh^{2}( 2 \lambda \theta ) [ \cos^{2}( \xi ) + \sinh^{2}( \lambda \theta ) + k^{2}\sinh^{2}( \lambda \theta ) \cosh^{2}( \lambda \theta ) ]^{- 1}.}}\\ \end{array}
(\begin{matrix}1&0\\ \frac{n_{2}-n_{1}}{R_{2}n_{1}}&\frac{n_{2}}{n_{1}}\end{matrix})(\begin{matrix}1&t\\ 0&1\end{matrix})(\begin{matrix}1&0\\ \frac{n_{1}-n_{2}}{R_{1}n_{2}}&\frac{n_{1}}{n_{2}}\end{matrix})
T(z) =\exp\left(\frac{N}{2}(-1)^\nu\phi_A(z)\sigma_3\right)=\exp\left[ \frac{(-1)^{\nu}}{2}\left(\zeta(z)^2+2x\zeta(z)+\ell_x\right)\sigma_3\right], \ell_x = x^2 + N\phi_{A}(1/a) ,S=S(z)=\begin{bmatrix}0 1 1 0\end{bmatrix}\cdot\begin{bmatrix}0 1 -1 0\end{bmatrix}^{\nu},
\widehat{A}_{i,jk}=\sum^{ib_n}_{l=(i-1)b_n+1}\frac{x_{ij}x_{ik}-\widehat{\gamma}_{jk}(0)}{\sqrt{\widehat{\gamma}_{jj}(0)\widehat{\gamma}_{kk}(0)}},i=1,2,\dots,l_n,
\begin{array}{lcl}T_{\scriptscriptstyle}= \frac{1}{4}\left(T_{1}+T_{10}+T_{19}+T_{28}+T_{37}+T_{46}+T_{55}+T_{64}\right. & \left.+i\left(T_{65}+T_{74}+T_{83}+T_{92}+T_{101}+T_{110}+T_{119}-T_{128}\right)\right).\end{array}
0 \leq R_{2,n}(C,\epsilon) = \sum_{r=\epsilon n+1}^{\infty}\frac{T_rC^{r-1}e^{-Cr}}{(r-1)!}e^{-\delta_2 r}\leq \sum_{r=\epsilon n+1}^{\infty}\frac{T_rC^{r-1}e^{-Cr}}{(r-1)!}= R_{\epsilon n}(C).
e_0 = Z \otimes v_1 - Y \otimes v_2, e_1 = (uY - tZ) \otimes v_0 - (u^2X + (Q(t, u) + su) Z) \otimes v_2, e_2 = (u X - s Z) \otimes v_0 + L_1(X, Y, Z) \otimes v_1 + L_2(X, Y, Z) \otimes v_2
\sum\limits_{\tilde{s}'\in\mathcal{S}\cup\{\ast\}}p_{\tilde{S}^{(\epsilon)}_{GE}|S}(\tilde{s}'|s)p_{\tilde{S}|\tilde{S}^{(\epsilon)}_{GE}}(\tilde{s}|\tilde{s}') &=(1-\epsilon)p_{\tilde{S}|\tilde{S}^{(\epsilon)}_{GE}}(\tilde{s}|s)+\epsilon p_{\tilde{S}|\tilde{S}^{(\epsilon)}_{GE}}(\tilde{s}|\ast),s\in\mathcal{S}, \tilde{s}\in\tilde{\mathcal{S}}.
\|u_{\epsilon}\|_{4^*}^{4^*}=R^{4^*}\int_{\mathbb{R}^{N}}\varphi^{4^*}(x)(\frac{\epsilon}{\epsilon^{2}+|x|^{2}})^{N}dx &=R^{4^*}\int_{\mathbb{R}^{N}}\varphi^{4^*}(x)\frac{\epsilon^{N}}{(\epsilon^{2}+|x|^{2})^{N}}dx.
{g_{\tiny\rm UV}^{(a)}\over{g_{\tiny\rm IR}^{(a)}}}={\sin{\pi a\over{k+l+2}}\over{\sin{\pi\over{k+l+2}}}}{\sin{\pi\over{k+2}}\over{\sin{\pi a\over{k+2}}}}={g^{[l,k]}_{(1,a)}\over{g^{[l,k]}_{(a,1)}}},
0 \to H_{x}^{i}((\Omega^{m+i-1}_{X/\mathbb{Q}})^{\oplus j}) \to K^{M}_{m}(O_{X_{j},x_{j}}\ \mathrm{on}\x_{j}) \xrightarrow{t =0}K^{M}_{m}(O_{X,x}\ \mathrm{on}\x) \to 0.
\mathbf{K}_{d,e}= \left[\begin{matrix}\Phi(k_{d,e,1}) \mathbf{0}\cdots \mathbf{0}\mathbf{0}\Phi(k_{d,e,2}) \cdots \cdots \cdots \cdots \ddots \mathbf{0}\mathbf{0}\mathbf{0}\mathbf{0}\Phi(k_{d,e,m})\end{matrix}\right],
S:=\left\{{\frac{{u}_{2}{u}_{6}-{u}_{4}{u}_{5}}{{u}_{6}}},{\frac{-{u}_{3}{u}_{5}+{u}_{5}{u}_{6}}{{u}_{6}}},{\frac{-{u}_{2}{u}_{5}+{u}_{4}{u}_{6}}{{u}_{6}}}, \right. \left.{\frac{2\,{u}_{3}{u}_{6}-{u}_{5}^2+2\,{u}_{6}^2}{{u}_{6}}}, -{\frac{{u}_{1}{u}_{5}}{{u}_{6}}}\right\}
(\begin{matrix}A^{\prime0}\\A^{\prime1}\\A^{\prime2}\\A^{\prime3}\end{matrix})=(\begin{matrix}cosh\phi&-sinh\phi&0&0\\ -sinh\phi&cosh\phi&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix})(\begin{matrix}A^{0}\\A^{1}\\A^{2}\\A^{3}\end{matrix})
C_{j,k}:= \left[(3 k + j) a - (6k+2), (3 k + j) a - 4 - 6 \left\lfloor \frac{j-1}{3}\right\rfloor \right]_2 & \cup \left[(3k+j) a - 2 - 6 \left\lfloor \frac{j-1}{3}\right\rfloor, (3 k + j) a \right]
\frac d{dt}\sum\limits_{\beta=0}^{1}\sum\limits_{|\alpha|=0}^{3-\beta}\left(\|\sqrt{G_1}\partial_\tau^\alpha\partial_y^\beta\tilde{u}\|_{L^2}^2+\|\sqrt{G_2}\partial_\tau^\alpha\partial_y^\beta\tilde{\theta}\|_{L^2}^2+\|\sqrt{G_3}\partial_\tau^\alpha\partial_y^\beta\tilde{q}\|_{L^2}^2\right)+\frac{C_0}{2}D(t) \le CE(t)^{\frac53}+Cf(t)E(t)^{\frac12}.
|D_{\rho}u(\ell)|^{k}\leq \bigg( \sum_{i = 1}^{N_{\rho}}|D_{\rho_{i}}u(\ell_{i})| \bigg)^{k}\leq |N_{\rho}|^{k-1}\sum_{i = 1}^{N_{\rho}}|D_{\rho_{i}}u(\ell_{i})|^{k}\leq C |\rho|^{k-1}\sum_{i = 1}^{N_{\rho}}|D_{\rho_{i}}u(\ell_{i})|^{k},
{{\bf{H}}_{{\rm{T}},k}}= \sum\nolimits_{l \in{\mathcal L}_{{\rm T}}}{{\alpha_{{\rm{T}},k,l}}{{\bf{a}}_{{\rm{S}},k}}\left({\Theta_{k,l}^{\rm{A}}}\right){\bf{a}}_{\rm{T}}^H\left({\Theta_{k,l}^{\rm{D}}}\right)},
0 = \epsilon_{i_{1}i_{2}\dots i_{n}i_{n + 1}}\epsilon_{j_{1}j_{2}\dots j_{n}j_{n + 1}}= \sum_{P \in S_{n + 1}}\ ( - 1 )^{\pi ( P )}\ \delta_{i_{1}P ( j_{1})}\dots \delta_{i_{n + 1}P ( j_{n + 1})},
\frac{\langle E(s)\rangle}{A}=-\frac{\hbar c^{1-s}\pi^{2-s}}{2a^{3-s}}\frac{1}{3-s}\sum_{n}|n|^{3-s}=-\frac{\hbar c^{1-s}\pi^{2-s}}{2a^{3-s}(3-s)}\sum_{n}\frac{1}{|n|^{s-3}}.
\int_Q |\nabla u(x) |^p \,\d x = \int_{(x_1,\dots, x_{n-1})}\int_{y_1}^{y_2}|\nabla u(x_1, \dots, x_{n-1},t) |^p\,\d t \ge \ell(Q)^{n-1}c^p \ell(Q)^{1-p}= c^p\ell(Q)^{n-p}.
P^{({\pm}N)(J)}_{\infty}(g) = \lim_{l{\rightarrow}{\infty}}P^{({\pm}N)(J)}(u) = \left(\begin{array}{cc}\frac{1+{\gamma}}{2}{\cal P}^{({\pm}N)}(g) 0 0 \frac{1-{\gamma}}{2}{\cal P}^{({\pm}N)}(g) \end{array}\right).
\int_{\mathcal{X}_n^c \times \mathcal{X}}f(x)f(y) \int_{l_x}^{v_x}(h_uF)(u,v_y) \,du \,dy \,dx = O \biggl( \frac{k^{-\frac{1}{2}+ \frac{2 \alpha}{\alpha+d}- \epsilon}}{n^{\frac{2 \alpha}{\alpha+d}- \epsilon}}\biggr).
{\rm{Tr}}_{L^{2}}\Bigr(f e^{-tP}\Bigr) \equiv \int_{M}dx \sqrt{{\rm{det}}g}\; f(x){\rm{Tr}}U(x,x;t) \sim (4\pi t)^{-m/2}\sum_{l=0}^{\infty}t^{l/2}A_{l/2}(f,P,{\cal B}).
f_1' \textstyle = \sum_{r - 2 > i(p-1) + 2 > 2}\Upsilon_i^{(1)'}x^{i(p-1)+2}y^{r-i(p-1)-2}, f_2' \textstyle = \sum_{r - 2 > i(p-1) + 2 > 2}\Upsilon_i^{(2)'}x^{i(p-1)+2}y^{r-i(p-1)-2},
0 \rightarrow E= \mathcal{O}_{\mathbb{P}^1}(-d_1) \oplus \dots \oplus \mathcal{O}_{\mathbb{P}^1}(-d_r) \rightarrow & \:\:\:\:\:\:\:\:\:\:\:\: F=\mathcal{O}_{\mathbb{P}^1}(-d_1) \oplus \dots \oplus \mathcal{O}_{\mathbb{P}^1}(-d_r) \oplus \mathcal{O}_{\mathbb{P}^1}(-d_{r+1}') \oplus \dots \oplus \mathcal{O}_{\mathbb{P}^1}(-d_{r+p}')
v(q) = v(o)+(X u)(p_o) x + (Y u)(p_o) y + (T u)(p_o) t & + \frac{1}{2}(X^2 u)(p_o) x^2 + \frac{1}{2}(Y^2 u)(p_o) y^2 + [2(Tu)(p_o)+(XYu)(p_o)]xy + O(r^3).
Z_{m}( \kappa ) ={\frac{\Gamma ( 1 / ( 2 m ) ) \, e^{- 1 / ( 4 \kappa )}}{2^{1 - 1 / m + ( - 1 )^{m}/ 2}\sqrt{2 \pi}\kappa^{1 / 2 - 1 / ( 2 m )}}}\,{}_{1}F_{1}( 1 / ( 2 m ) , 1 / 2 ; 1 / ( 4 \kappa ) )
S_1 =\sum_{k=0}^\infty\frac{1}{(2k)!}\bigg(\frac{2\beta}{\alpha}\bigg)^{2k}\Gamma\bigg(k+\frac{1}{2}\bigg)\Gamma\bigg(\nu+k+\frac{1}{2}\bigg), S_2 =-\sum_{k=0}^\infty\frac{1}{(2k+1)!}\bigg(\frac{2\beta}{\alpha}\bigg)^{2k+1}k!\Gamma(\nu+k+1).
(T_{i}-1)(T_{i}+q)=0 (1 \le i<k), T_{i}T_{i+1}T_{i}=T_{i}T_{i+1}T_{i}(1 \le i \le k-2), T_{i}T_{j}=T_{j}T_{i}(|i-j|>1), X_{i}X_{j}=X_{j}X_{i}(i, j=1, \ldots , k), X_{i+1}T_{i}-T_{i}X_{i}=T_{i}X_{i+1}-X_{i}T_{i}=(\alpha+\beta X_{i})(\gamma+\delta X_{i+1}) (1\le i<k), X_{i}T_{j}=T_{j}X_{i}(i\not=j, j+1).
{}^{w^\vee}(\chi^\vee\cdot(\varphi_\sigma,\varrho_\sigma))={}^{w^\vee}(\varphi_{\chi\otimes\sigma},\varrho_{\chi\otimes\sigma})=(\varphi_{{}^w(\chi\otimes\sigma)},\varrho_{{}^w(\chi\otimes\sigma)}=(\varphi_{\chi\otimes\sigma},\varrho_{\chi\otimes\sigma}).
B(fg) -f Bg = \sum_j ( B(f_{<j-2}g_j) - f_{<j-2}B g_j) & + \sum_j ( B(f_j g_{<j-2}) - f_j Bg_{<j-2}) & + \sum_j (B(f_j \tilde g_j) - f_j B\tilde g_j),
\alpha_{\pm}\bigl(\mathcal{E}(n)\bigr) =\mathcal{E}(n\pm 1)-\mathcal{E}(n), R_{-1}\bigl(\mathcal{E}(n)\bigr)R_0\bigl(\mathcal{E}(n)\bigr)^{-1}=A_n+C_n, a^{(+)}\phi_n(x)=A_n\phi_{n+1}(x), a^{(-)}\phi_n(x)=C_n\phi_{n-1}(x).
\big(\nabla{\theta}^{k}_{\phi}, \,\nabla u\big) = \big(\nabla(\phi^{k}- I_{h}\phi^{k}), \nabla u\big) +\big(\vert\Psi^{k}_{h}\vert^{2}-\vert\Psi^{k}\vert^{2},\, u \big), \forall u \in X_{h}^{r}.
B_\mathbb{C}(H_{\alpha}- H_{\beta}, A) = \frac{2\alpha(A)}{\langle \alpha, \alpha \rangle}- \frac{2\beta(A)}{\langle \beta, \beta \rangle}&= 0 ( \alpha|_\mathfrak{a}= \beta|_\mathfrak{a}|\alpha| = |\beta|).
\phi \psi = (\sigma \tau,(\phi_{\tau^{-1}1}\psi_1, \phi_{\tau^{-1}2}, \phi_{\tau^{-1}3}\psi_3)) &= (\sigma \tau,(\phi_{\tau^{-1}1}\psi_1, \varepsilon, \phi_{\tau^{-1}3}\psi_3)) \in \prescript{\sigma \tau}{}N_l^A
I_{1}(x,n)=\int_{x}^{\infty}d\tau\frac{1}{e^{\tau}-1}\tau^{n}=\sum_{q=1}^{\infty}e^{-qx}(\frac{x^{n}}{q}+\frac{nx^{n}}{q^{2}}+...\frac{n!}{q^{n+1}}),
\quad\partial u_{1\dots k}= \sum_{1\leq p< \lceil \frac{k}{2}\rceil}\sum_{\sigma \in S(p, k-p)}\Bigr[ u_{\sigma (1)\dots \sigma (p)},u_{\sigma (p+1)\dots \sigma (k)}\Bigr] &+\frac{1}{2}\sum_{\sigma \in S(\frac{k}{2},\frac{k}{2})}\Bigl[ u_{\sigma (1)\dots \sigma (\frac{k}{2})}, u_{\sigma (\frac{k}{2}+1) \dots \sigma (k)}\Bigr]
I(s,\Omega) = \frac{1}{2\pi i}\int_{\Omega}\frac{\sigma_{2u}(m) \sigma_{2u}(n) \cos(\pi u)}{(mn)^{u}\zeta(1 + 2u)\zeta(1-2u)}\Gamma (s -\tfrac 12 + u) \Gamma(s - \tfrac 12 - u) \d u ,
\left[ J^{\alpha \beta}( x ) , S_{-}^{\alpha^{\prime}\beta^{\prime}}( y ) \right] = - \delta ( x - y ) \left( \delta^{\alpha \alpha^{\prime}}S_{-}^{\beta \beta^{\prime}}( x ) + \delta^{\alpha \beta^{\prime}}S_{+}^{\beta \alpha^{\prime}}( x ) \right) ,
\lambda = \log\frac{\theta''-\theta'}{\theta''-\theta}+n log \left\lvert\frac{t-\theta}{t-\theta'}\right\rvert+ m \log \left\lvert\frac{\theta'}{\theta}\right\rvert &= \log\left(1+\frac{(\theta-\theta')(x-\theta'' y)}{(\theta''-\theta)(x-\theta' y)}\right)
\begin{array}{ccll}$$X_if_\epsilon = X_if \hskip 1.1cm i\in\{1,3,4\},$$ $$X_2f_\epsilon = X_2f+\alpha+\beta x_2+\gamma x_3,$$ $$X_{ij}f_\epsilon = X_{ij}f \hskip 1cm (i,j)\in\{1,4\}\times\{1,4\},(i,j)\not=(2,2),(3,2),$$ $$X_{22}f_\epsilon = X_{22}f+\beta+\gamma x_1.$$\end{array}
\Big\|\phi_i(g_j)\,(w_{in}\circ g_j^{-1})-\sum_{s=j+1}^{m}\phi_i(g_s)\,(w_{i}\circ g_s^{-1})(\,\cdot\,-g_s\xi_{rn}+g_j\xi_{rn})\Big\|^2 &=\Big\|\phi_i(g_j)\,(w_{in}\circ g_j^{-1})-\sum_{s=j}^{m}\phi_i(g_s)\,(w_{i}\circ g_s^{-1})(\,\cdot\,-g_s\xi_{rn}+g_j\xi_{rn})\Big\|^2+\Big\|\phi_i(g_j)\,(w_i\circ g_j)\Big\|^2+o_n(1).
G_{01^s0^{n-s-1}}= G_{1^s0^{n-s}}- \theta_1 G_{1^{s-1}0^{n-s+1}}=F_{1^s}(\theta_1,\theta_2 ,\ldots,\theta_n) - \theta_1 F_{1^{s-1}}(\theta_2 ,\ldots,\theta_n) = F_{1^s}(\theta_2,\theta_3 ,\ldots,\theta_n)= G_{\alpha}(\theta_2,\theta_3 ,\ldots,\theta_n)\,.
{\cal M}= \left( \begin{array}{c c c}{{G^{- 1}}}&{{G^{- 1}( B + C )}}&{{G^{- 1}A}}\\{{( - B + C ) G^{- 1}}}&{{( G - B + C ) G^{- 1}( G + B + C )}}&{{( G - B + C ) G^{- 1}A}}\\{{A^{T}G^{- 1}}}&{{A^{T}G^{- 1}( G + B + C )}}&{{I_{1 6}+ A^{T}G^{- 1}A}}\\ \end{array}\right) .
\sum_{j = 0}^k{\sum_{s = 0}^j{\binom kj\binom js\left({\frac{{f_1}}{{f_2}}}\right)^j\left({-\frac{{1}}{{f_1}}}\right)^s X_{m - (b-c)k + (b - a)j + as}}}= \left(-\frac{f_3}{f_2}\right)^kX_m\,.
A=\begin{bmatrix}a_{1,1}a_{1,2}\dots a_{1,n-1}a_{1,n}a_{1,2}a_{2,2}\dots a_{2,n-1}a_{2,n}\vdots \vdots \ddots \vdots \vdots a_{1,n-1}a_{2,n-1}\dots a_{n-1,n-1}a_{n-1,n}a_{1,n}a_{2,n}\dots a_{n-1,n}a_{n,n}\end{bmatrix}\in \mathsf{S}^2(\mathbb{C}^n)
W^{(1)}-W^{(3)}= b_0 y+( 78 b_0 + b_1 )y^5+( 1688 b_0 + 32 b_1 + b_2 )y^9 &+(- 32382 b_0 - 553 b_1 - 14 b_2 )y^{13}&+(- 2525349 b_0 - 37184 b_1 - 678 b_2 )y^{17}+ \cdots.
\frac{qx_ix_{i+1}-q^{-1}x_{i+1}x_i}{q-q^{-1}}=1, x_i^3x_{i+2}-[3]_qx_i^2x_{i+2}x_i+[3]_qx_ix_{i+2}x_i^2-x_{i+2}x_i^3=0,
\frac{1}{2}\mathfrak{c}_\omega(a_0\wedge a_1\wedge a_2\wedge a_3)= \mathrm{Tr}_\omega(Pa_0(1-P)a_1Pa_2(1-P)a_3P)- &-\mathrm{Tr}_\omega(Pa_0(1-P)a_3Pa_2(1-P)a_1P)= = \lim_{N\to \omega}\frac{1}{\log(N+2)}\sum_{k=0}^N \sum_{m=k}^\infty k a_{0,k}a_{2,m}\left(a_{1,-m}a_{3,-k}-a_{1,-k}a_{3,-m}\right),
\begin{array}{lll}dx_t=\Big[\frac{\partial b_t}{\partial x}x_t + \frac{\partial b_t}{\partial u}\beta_t \Big] dt+ \Big[\frac{\partial \sigma_t}{\partial x}x_t + \frac{\partial \sigma_t}{\partial u}\beta_t\Big] dW_t + \Big[\frac{\partial \gamma_t}{\partial x}x_{t^{-}}+ \frac{\partial \gamma_t}{\partial u}\beta_t\Big] dM_t; x_0=0.\end{array}
\dot{v}(s) = A(t)e^{-(t-s)A(t)}u(s) + e^{-(t-s)A(t)}\dot{u}(s) & = A(t)e^{-(t-s)A(t)}u(s) - e^{-(t-s)A(t)}A(s)u(s) & = A(t)e^{-(t-s)A(t)}(A(s)^{-1}- A(t)^{-1}) A(s) u(s)
u\leq v \Leftrightarrow \ell(v)=\ell(u)+\ell(vu^{-1}) &\Leftrightarrow \ell(v\xi^{-1})=\ell(u\xi^{-1})+\ell(vu^{-1}) &\Leftrightarrow \ell(v\xi^{-1})=\ell(u\xi^{-1})+\ell(v\xi^{-1}(u\xi^{-1})^{-1}) &\Leftrightarrow u\xi^{-1}\leq v\xi^{-1}&\Leftrightarrow \rho_\xi(u)\leq \rho_\xi(v).
\begin{array}{l l}{{\phi_{1}= 0}}\\{{\phi_{2}= \phi = \varphi ( r ) e^{i \theta}}}\\{{H_{\mu}= \frac{\sqrt{\beta}}{q}\frac{H ( r )}{r}\delta_{\mu}^{\theta}}}\\{{\tilde{D}= \frac{1}{2 \beta}\left( q ( \varphi^{2}- \eta^{2}) + \alpha e \sigma^{2}\right)}}\\{{G = 0 ,}}\\ \end{array}
(R_{\phi^{-1}}^*\tilde\eta)_p(Y_1,\dots,Y_k) = (p\phi^{-1},(R_{\phi^{-1}}^*\alpha)_p(Y_1,\dots,Y_k)) = (p\phi^{-1},\phi \circ \alpha_p(Y_1,\dots,Y_k)) = \mu_\phi(\tilde\eta_p(Y_1,\dots,Y_k))
H_{F}\ = \ \left( \vec{q}_{1}+ i \vec{q}_{2}\right) \cdot \left( \vec{\lambda}\wedge \vec{\lambda}\right) - \left( \vec{q}_{1}- i \vec{q}_{2}\right) \cdot \left( \frac{\partial}{\partial \vec{\lambda}}\wedge \frac{\partial}{\partial \vec{\lambda}}\right) \ .
\cfrac{1}{{a}_0}-\cfrac{1}{\pi}+\sum_{n=1}^\infty \Big(\cfrac{1}{{a}_n}-\cfrac{2}{\pi}\Big)=\cot\alpha, \cfrac{1}{{b}_0}-\cfrac{1}{\pi}+\sum_{n=1}^\infty \Big(\cfrac{1}{{b}_n}-\cfrac{2}{\pi}\Big)=-\cot\beta.
J(a,c) \equiv \det \begin{pmatrix}1 1 s (\overline{f}^{n-1}_{a,c}(0))^{t p}s (\overline{f}^{n-1}_{a,c}(1))^{t p}\end{pmatrix}\equiv s(\overline{f}^{m-1}_{a,c}(1)^{t p}- \overline{f}^{m-1}_{a,c}(0)^{t p}) \pmod{p}.
\begin{array}{c}{D^{\left[ A , B \right]}( \Lambda ) u ( \Lambda^{- 1}p ) = u ( p ) D^{\left( s \right)}( R^{- 1}( \Lambda , p ) )}\\{D^{\left[ A , B \right]}( \Lambda ) v ( \Lambda^{- 1}p ) = v ( p ) D^{\left( s \right) *}( R^{- 1}( \Lambda , p ) ) , \quad v ( p ) = u ( p ) D^{\left( s \right) *}( i \sigma_{2})}\\ \end{array}\,
F(\theta_{(\ell+1)q_n+m_n})=\left(\begin{matrix}\cos(\pi\theta_{(\ell+1)q_n+m_n})E-\lambda \sin(\pi\theta_{(\ell+1)q_n+m_n})\ \ -\cos(\pi\theta_{(\ell+1)q_n+m_n}) \cos(\pi\theta_{(\ell+1)q_n+m_n}) 0\end{matrix}\right)
\xi_{G^{\prime}}-\xi_G =(e_{v^{\prime}_1}-f_{v^{\prime}_1})+(e_{v^{\prime}_2}-f_{v^{\prime}_2})-(e_{v_1}-f_{v_1})-(e_{v_2}-f_{v_2}) &=(b-(-b))+(c-(-b-c))-(b+c-(-b-c))-(b-c) &=c.
\int_{\Gamma}\nu \varphi \cdot \hat{\mathrm{rot}}' \psi \mathrm{d}x= \int_{\Omega}\nabla \varphi \cdot \mathrm{rot}' \hat{\psi}\mathrm{d}x = 0, \int_{\Gamma}\nu^{\perp}\psi \cdot \nabla \hat{\varphi}\mathrm{d}\Gamma =\int_{\Gamma}\mathrm{rot}' \psi \cdot \nabla \hat{\varphi}\mathrm{d}x = 0
\sigma_{x_1}=\sigma^{-3}\bigl((x_1^2-y_1^2)x_1+y_1^2(x_1-y_1)\bigr)=\sigma^{-3}(x_1^3-y_1^3) &\sigma_{x_2}=2\sigma^{-3}(x_2-y_2) &\sigma_{x_1x_1}=-3\sigma^{-7}\bigl(x_1^3-y_1^3\bigr)^2+3\sigma^{-3}x_1^2 &\sigma_{x_1x_2}=-3\sigma^{-7}\bigl(x_1^3-y_1^3\bigr)(x_2-y_2) &\sigma_{x_2x_2}=-3\sigma^{-7}(x_2-y_2)^2+2\sigma^{-3}.
\sum_{j\leq i}s_{ij}^{a_i}\frac{\prod_{k\leq i-1}(1-s_{ij}/s_{i-1,k})}{\prod_{k\leq i}^{k\ne j}(1-s_{ij}/s_{ik})}=(t_i^2v^{2d_{i-1}-2d_i})^{a_i}.
\quad\, \sum_{K \geq 1}\sum_{V \geq AK}\exp\left( K \log(m^2) -V((c/2) \log m) \right) = \sum_{K \geq 1}\exp\left( 2K \log m - AK(c/2) \log m \right) (1 + o(1)) = m^{2 - Ac/2}(1 + o(1)),
\bar{c}_{4}= \sqrt{c^{2}+ R_{\tau}^{2}}, \ \ \bar{G}_{4}={\frac{m_{0}e^{2 m_{0}R_{0}}G_{5}( 1 + R_{\tau}^{2}/ c^{2})^{3 / 2}}{1 - \Omega_{R}^{2}}},
|A_1(x+h,y+h)-A_1(x,y)| = |A(2^{-N-1}R x+x_0 +2^{-N-1}R h,2^{-N-1}R y+x_0 +2^{-N-1}R h)-A(2^{-N-1}R x+x_0,2^{-N-1}R y+x_0)| \leq \Gamma (2^{-N-1}R)^\alpha |h|^\alpha .
\left|\mathcal{I}_s\left(z_k, \frac{1}{(t+|x|^2)^{\frac{\theta}{2}}}\right)(x)\right| =\frac{c_{N,s}}{2}\left|\int_{\Omega_{2/k}}\frac{-\rho(k\delta_\Omega(x))\big((t^2+|x|^2)^{-\frac{\theta}{2}}-(t^2+|y|^2)^{-\frac{\theta}{2}}\big)}{|x-y|^{N+2s}}dy\right| \leq \bar{c}(\Omega^{\prime},\Omega^{\prime\prime}, N, s,t)\,\textrm{vol}(\Omega_{2/k}),
\left|\mathbb{E}e^{\iota \sum_{i=1}^np_i(X^H_{s_i}-X^H_{r_i})}\right|=\prod_{j=1}^\infty\frac{e^{-\iota\lambda_j}}{\sqrt{1-2\iota\lambda_j}}=\prod_{j=1}^\infty(1+4\lambda_j^2)^{-\frac14}.
\|\overline w_{M_n,n}-{u}_T\|^2_{L^2(\Omega)}\le \Big[ \dfrac{1}{n}\sum_{k=1}^n \sigma_k \epsilon_k - \widetilde{G}_{n0}\Big]^2 +\sum_{p=1}^{M_n}\Bigg[ \dfrac{\pi}{n}\sum_{k=1}^n \sigma_k \epsilon_k \phi_p(x_k) - \widetilde{G}_{np}\Bigg]^2 \&+ \sum_{p=M_n+1}^\infty \Big<{u}_T, \phi_p\Big>^2.
\left( Q_{2}^{\dagger}\right)^{3 1}Q_{2}^{1 3}\u_{A}({r}) ={\frac{1}{m}}\left[ -{\frac{\partial^{2}}{\partial r^{2}}}-{\frac{1}{r}}{\frac{\partial}{\partial r}}+{\frac{\left( 2 \ell + 1 - \lambda \right)^{2}}{r^{2}}}+ \lambda{\frac{\delta ( r )}{r}}\right] u_{A}= E u_{A}({r})
\tau = \underbrace{\min\!\big( \mathcal{H}_2 - \rho(h_{j,1}) + 3q(n)+4C \big)}_{\ge\, -q(n)-C+3q(n)+4C \, = \, 2q(n)+3C}- \underbrace{\max\!\big( \mathcal{H}_1 \big)}_{\le\, q(n)+C}\ge q(n) + 2C,
\begin{array}[c]{lll}X_{0}:=L^{2}( 0,T;H^{-2}( 0,L) ) X_{1}:=L^{2}( 0,T;H_{0}^{2}( 0,L) ) \tilde{X}_{0}:=L^{1}( 0,T;H^{-1}( 0,L) ) \tilde{X}_{1}:=L^{1}( 0,T;( H^{3}\cap H_{0}^{2}) (0,L) ) \end{array}
\varepsilon D_{\vec{z}}\vec{F}\left({\vec{z},\varepsilon}\right)\vec{g}\left({\vec{F}\left({\vec{z},\varepsilon}\right),\vec{z},\varepsilon}\right)=\vec{f}\left({\vec{F}\left({\vec{z},\varepsilon}\right),\vec{z},\varepsilon}\right)