harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S_{\mathrm{SCF}}(n) = k + 2h - 1 + \sum_{i=1}^k S_{\mathrm{SCF}}(p_i) + \sum_{j=1}^h (S_{\mathrm{SCF}}(q_j) + S_{\mathrm{SCF}}(b_j)) &\leq 6\sum_{i=1}^k t(p_i) + 6\sum_{j=1}^h (t(q_j) + t(b_j)) - 1 &= 6t(n) - 1 ,
\mu_M^\mathrm{el}\left(\mathbf{x}_1,\mathbf{x}_2\right) = P_1^\mathrm{el}\left(Q_{1,1},Q_{2,2},Q_{1,2}\right) \times \sqrt{\frac{4\pi}{3}}\tilde{\mathcal{Y}}_{1,M}\left(\mathbf{x}_1\right)+ P_2^\mathrm{el}\left(Q_{1,1},Q_{2,2},Q_{1,2}\right) \times \sqrt{\frac{4\pi}{3}}\tilde{\mathcal{Y}}_{1,M}\left(\mathbf{x}_2\right),
\rho_U(\epsilon(\gamma -1)) =\rho_U(\epsilon\gamma-1 - (\epsilon -1)) =p_J(\psi_U(\epsilon\gamma))-p_J(\psi_U(\epsilon)) =p_{\Gamma}(\psi_U(\epsilon))p_{J}(\psi_U(\gamma))+p_J(\psi_U(\epsilon))-p_J(\psi_U(\epsilon)) =\epsilon p_J(\psi_U(\gamma)) =\epsilon \rho_U(\gamma).
- iw^{(s)}_n = -i\frac{\lambda_s}{s!}n! \underbrace{\sum_{j=0}^{n-s}\sum_{k=0}^{n-s-j}\cdots \sum_{l=0}^{n-s-j-k-\ldots}}_{s-1}\underbrace{a_j a_k \cdots a_l a_{n-s-j-k-\ldots -l}}_{s a}.
\mathbf{S}_n^{(t)}=\mathbf{S}_n^{(t-1)}+\sum_{i\in[K_c]}\mathbf{\Omega}_{n,i}^{(t)}\mathbf{\Upsilon}_{n,i}^{(t)}\mathbf{P}_{n,i}^{(t,\theta_t)}\mathbf{Q}_{n,i}^{(t,\theta_t)}, n\in\overline{\mathcal{S}}_w^{(t)}.
\kappa^{a}f^{\lceil tp^a \rceil}\kappa^{a'}f^{\lceil tp^{a'}\rceil}= \kappa^{a+a'}f^{{\lceil tp^a \rceil}p^{a'}}f^{\lceil tp^{a'}\rceil}= \kappa^{a + a'}f^{\lceil t p^{a+a'}\rceil}f^{r}
\widetilde{\Theta}:=\sum_{\ell_1\in\mathcal{I}_{e_1}}m_{e_1,\ell_1}^*\otimes\Theta(m_{e_1,\ell_1})\in \bigl(\mathbf{M}^{\pi_{\mathcal{S}(v_1)}}\otimes L\bigr)^{K_{v_1}}
\sum_{\alpha\in \Phi^-(W)\setminus \Phi^-(W_J)}\!\!\!\!T_\alpha -\!\!\!\sum_{\alpha\in \Phi^-(W)\setminus \Phi^-(W_I)}\!\!\!\!T_{w\alpha}= \sum_{\alpha\in \Phi^-(W)}(T_\alpha-T_{w\alpha}) - \sum_{\alpha\in \Phi^-(W_J)}\!\!\!\!T_\alpha + \sum_{\alpha\in \Phi^-(W_I)}\!\!\!\!T_{w\alpha}.
\alpha|_{X_{ijkl}}= (G_{ijk}dF_i\wedge dF_j + G_{jki}dF_j\wedge dF_k + G_{ikj}dF_i\wedge dF_k)|_{X_{ijkl}}&=(G_{jkl}dF_j\wedge dF_k + G_{klj}dF_k\wedge dF_l + G_{jlk}dF_j\wedge dF_l)|_{X_{ijkl}}.
\aligned \langle \chi_0, \chi_k \rangle = \sum_\gamma \frac{(\cos(a\gamma)-1)}{\gamma^2}\cdot \frac{(-1)^k2a\sin(a\gamma)}{k\pi+a\gamma}& - \sum_\gamma \frac{a\gamma\cos(a\gamma)-\sin(a\gamma)}{a \gamma^3}\cdot \frac{(-1)^k2a\sin(a\gamma)}{k\pi+a\gamma}.\endaligned
\int_0^{2\pi}\frac{d\theta}{2\pi}e^{-|z|^2}\,\prod_\pm\left\{e^{xe^{\pm i\theta}|z^\pm_{2M+1}|^2}\,\prod_{\mu=1}^M e^{xe^{\pm i\theta}\left[e^{i\omega_\mu}|z^\pm_{\mu+}|^2+e^{-i\omega_\mu}|z^\pm_{\mu-}|^2\right]}\right\}\ \ \ ,
\textstyle \int_{\Omega}a(\nabla^{b}(\psi v),\nabla^{b}(\psi v)) = \int_{\Omega}(\lambda-c)|\psi v|^{2}- \int_{\Omega}\Re f \psi^{2}\overline{v}+ \int_{\Omega}a(\nabla \psi, \nabla \psi)|v|^{2}.
H_{p}={\frac{Q}{| \vec{z}- \vec{z}_{0}|^{7 - p}}}, H_{W}= 1 +{\frac{Q_{W}}{( x^{2}+{\frac{4 Q}{( p - 5 )^{2}}}| \vec{z}- \vec{z}_{0}|^{p - 5})^{\frac{p^{2}- 8 p + 1 9}{2 ( p - 5 )}}}}.
\frac{d}{dt}F_t(O,1)= -F_t(O,1)+2F_t(O,2)+\frac{1}{\gamma}F_t(O,3), \frac{d}{dt}F_t(O,2)= -(1+\delta+\gamma)F_t(O,2)+(1+\delta+\gamma)F_t(e_1,1), \frac{d}{dt}F_t(O,3)= -(1+\delta+\gamma)F_t(O,3) &+2(1+\delta+\gamma)F_t(e_1,2)+\frac{(1+\delta+\gamma)^2}{2d\lambda}F_t(O,1).
B_+ \sim \mathcal{B}(2^{-1/2}), B_- \sim \mathcal{B}(1-2^{-1/2}) \mu_{B_+}= \begin{pmatrix}1- 2^{-1/2}2^{-1/2}\end{pmatrix}\mu_{B_-}= \begin{pmatrix}2^{-1/2}1-2^{-1/2}\end{pmatrix}.
\partial_i g(x) = \frac{1}{n}D_mG\bigg(\frac{1}{n}\sum_{k=1}^n\delta_{x_k},x_i\bigg), \partial_{ij}g(x) = \frac{1}{n^2}D_m^2G\bigg(\frac{1}{n}\sum_{k=1}^n\delta_{x_k},x_i,x_j\bigg), \ \i \neq j.
Z(t,\omega_2)= S_B(t-r)Z_0+\int_r^tS_B(t-q)\,d\omega_2(q) = S_B(t-r)Z_0+\int_0^{t-r}S_B(t-r-q)\,d\theta_r\omega_2(q) = S_B(t-r)Z_0+B\int_0^{t-r}S_B(t-r-q)\theta_r\omega_2(q)\,dq+\theta_r\omega_2(t-r).
(-\Delta_p)^s u(x) =2\int_{B_2}\frac{J_p(u_1(x)-u_1(2))-J_p(u_1(x)-u_1(y))}{|x-y|^{N+s\,p}}\, dy &\geq 2\int_{B_1}\frac{J_p(u_1(y)-u_1(x))-J_p(u_1(2)-u_1(x))}{|x-y|^{N+s\,p}}\, dy.
\left| \frac{S_1}{X_1}\right|^{K/2}\ll \frac{1}{H_1^{K/4}}+ \cdots + \frac{1}{H_{k-1}}+ \frac{1}{X_1 H_1 \cdots H_{k-1}}\sum_{h_1 = 1}^{H_1}\cdots \sum_{h_{k-1}=1}^{H_{k-1}}\left| \sum_n e( q p_1 (n)) \right|,
\begin{aligned}e^{-\frac{i}{h}b(x)}\int u(t,x) e^{\frac{i}{h}f(t,x)}\, dt \sim Ai(h^{-\frac{2}{3}}a(x)) h^{\frac{1}{3}}\sum_{0}^\infty u_{0,n}(x)h^{n}+ Ai'(h^{-\frac{2}{3}}a(x)) h^{\frac{2}{3}}\sum_{n = 0}^\infty u_{1,n}(x)h^n\end{aligned}
R=\frac{r(\theta_1\,|\,\phi_0,\phi_1)\,r(\theta_3\,|\,\phi_0,\phi_3)}{r(\theta_1+\theta_3\,|\,\phi_1,\phi_3)}\frac{P(\theta_1\,|\,\phi_2,\phi_3)\,P(\theta_3\,|\,\phi_1,\phi_2)}{P(\theta_1+\theta_3\,|\,\phi_0,\phi_2)}\,.
\Big|{\mathcal F_{(t,s)\to(\xi_0,\eta_0)}}\Big(\frac{\psi(t,s)}{\lambda_j}e^{i{\lambda_j(t-s)}}\Big)(\xi_0, \eta_0)\Big|^2 &=\frac{1}{\lambda_j^2}\left|{\mathcal F}(\psi)(\xi_0-\lambda_j, \eta_0+\lambda_j)\right|^2
\check{H}_{i}^{( 1 )}= \check{h}_{i}^{( 1 )}, \check{H}_{i}^{( p )}= \check{h}_{i}^{( p )}+ \sum_{k = 1}^{p - 1}{}_{p - 1}C_{k}\check{h}_{i}^{( k )}\check{H}_{i}^{( p - k )}, p = 2, 3, \dots .
G_{k}(\tau)=\left(\frac{\alpha_{0}(k+1)!}{2C_{k}}-\frac{2\alpha_{1}(k+1)ka_{k}^{2}}{(2\pi)^{2}}+\tilde{C_{k}}\right)(1+\tau)^{k}+\frac{b_{k}^{2}2\alpha_{1}(k+1)k}{(2\pi)^{2}}(1+\tau)^{k-2}.
\|u_\infty - \langle u^{in}\rangle \|_{(H^1)'}= \|u_\infty - u^{in}+ u^{in}- \langle u^{in}\rangle \|_{(H^1)'}& \ge \|u^{in}- \langle u^{in}\rangle \|_{(H^1)'}- \|u_\infty - u^{in}\rangle \|_{(H^1)'}& > \left[ \|u^{in}\|_\infty \|v^{in}\|_1 \|\gamma'\|_{L^\infty(0,V)}\right]^{1/2}- \|u_\infty - u^{in}\rangle \|_{(H^1)'}>0\,.
\begin{array}[c]{l}\mathring{R}ic_{\alpha\overline{\beta}},^{\overline{\beta}}=\left(Ric_{\alpha\overline{\beta}}-\frac{R}{n}h_{\alpha\overline{\beta}}\right),^{\overline{\beta}}=\frac{n-1}{n}(R_{\alpha}-inA_{\alpha\beta},^{\beta}).\end{array}
\lim_{p\to 0}\frac{R^{*(n)}_\lambda\big(p^{-1/4}\vec{x};t_0,p^{1/2}u_0;q,t;p\big)}{R^{*(n)}_\lambda\big(p^{-1/4}t^{n-1}v,\dots,p^{-1/4}v;t_0,p^{1/2}u_0;q,t;p\big)}=\frac{P_\lambda(\vec{x};q,t)}{P_\lambda\big(t^{n-1}v,\dots,v;q,t\big)}.
v_k^{(s,t)}=u_k^{(s,t)}(\kappa^{(t)}-u_{k-1}^{(s,t)}),k=1,2,\dots,2m, &V_k^{(s,t)}=-(\kappa^{(t)}-u_{k-1}^{(s,t)})(\kappa^{(t)}-u_{k}^{(s,t)}),k=1,2,\dots,2m.
G(\beta,L)=\frac{3\mu}{8\pi m}\left[(e^{m\beta}-1)^{-1}+(e^{mL}-1)^{-1}+2\sum_{n_{1},n_{2}=1}^{\infty}e^{-m\sqrt{\beta^{2}n_{1}^{2}+L^{2}n_{2}^{2}}}\right] ,
\left( \stackrel{[ 0 , 0 ]}{z}^{A}, \stackrel{[ 0 , 0 ]}{z}^{A_{1}}\right) , \left( \stackrel{[ 1 , 0 ]}{\cal P}_{1 a_{0}}, \stackrel{[ 0 , 1 ]}{\cal P}_{2 a_{0}}\right) , \left( \stackrel{[ 1 , 1 ]}{\lambda}_{a_{0}}\right) ,
Q(r_{1},r_{2},t)=-\frac{\hbar^{2}}{2m}(\frac{\nabla_{1}^{2}R_{A}(r_{1},t)}{R_{A}(r_{1},t)}+\frac{\nabla_{2}^{2}R_{B}(r_{2},t)}{R_{B}(r_{2},t)})=Q_{A}(r_{1},t)+Q_{B}(r_{2},t)
L_{0,0}*L_{0,0}=c^{-1}\phi(L_{0,0})*c^{-1}\phi(L_{0,0})=c^{-2}\phi(L_{0,0}\cdot L_{0,0})=c^{-2}\phi(cL_{0,0})=c^{-2}\cdot c^2 L_{0,0}=L_{0,0},
4 \int_{2}^{\infty}\frac{d y}{2 \pi}\, \frac{1}{\sqrt{y^{2}- 4}}\, \frac{y \left( \sqrt{1 + \sigma_{0}}\sqrt{1 + \sigma_{1}}- \coth 2 x_{0}\right) + ( \sigma_{0}+ \sigma_{1})}{y^{2}+ 2 y \sqrt{1 + \sigma_{0}}\sqrt{1 + \sigma_{1}}+ 2 ( \sigma_{0}+ \sigma_{1})}
\textstyle L_z = \textstyle \smash{\sum_v (-1)^v{z+A \choose v}\sum_{w\geqslant v}X^w (-1)^{w}{A+w-v-1 \choose w-v}= \sum_v X^v{z+A \choose v}\sum_{u}(-X)^{u}{A+u-1 \choose u}}& = \textstyle \smash{(1 + X)^{z+A}(1-(-X))^{-A}= (1+X)^z = \sum_{w}X^w{z \choose w}}.
\begin{aligned}\frac{|x_i^1u_{x_i}(P_1)-x_i^2u_{x_i}(P_2)|}{\rho^{\alpha}(P_1,P_2)}\leq C \eta^{1-\alpha}\|u\|_{C^{2+\alpha}_{WF}([0,T]\times\bar S_{n,m})},\end{aligned}
Q(x, y, z_{1}, z_{2}, z_{3}) =\tfrac{3\sqrt{3}}{2}\det \begin{pmatrix}-\tfrac{1}{\sqrt{3}}x + y z_{3}\bar{z}_{1}\bar{z}_{3}-\tfrac{1}{\sqrt{3}}x - y z_{2}z_{1}\bar{z}_{2}\tfrac{2}{\sqrt{3}}x\end{pmatrix}.
\frac{X}{V}\sum_{u_1,\ldots,u_k}\left(\prod_{i=1}^k\varphi(u_i)\right)\sum_{s_{1,2},\ldots,s_{k-1,k}}\biggl(\prod_{\substack{1\leq i,j\leq k i\neq j}}\mu(s_{i,j})\biggr)\sum_{\substack{d_1,\ldots,d_k e_1,\ldots,e_k u_i\mid d_i,e_i\forall i s_{i,j}\mid d_i,e_j\forall i\neq j}}\frac{\lambda_{d_1,\ldots,d_k}\lambda_{e_1,\ldots,e_k}}{\prod_{i=1}^kd_ie_i}.
\begin{aligned}(1+|y|)|\nabla_y \phi(y, \tau)|\chi_{B_{2R}(0)}(y)+ |\phi(y,\tau)| &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\lesssim \tau^{-\nu}(1+|y|)^{-a}\|h\|_{2s+a,\nu, \eta}, \tau\in (\tau_0,+\infty),y\in \mathbb{R}^n,\end{aligned}
(B^{-1})_0({\bf b}) = e^{-b_0}, (B^{-1})_1({\bf b}) = 2 e^{-\frac{3}{2}b_0}b_1, (B^{-1})_2({\bf b}) = e^{-2b_0}(-4 b_2 + 8 b_1^2), & (B^{-1})_3({\bf b}) = e^{-\frac{5}{2}b_0}\bigl(8 b_3-60 b_2 b_1+50 b_1^3\bigr).
A = \int \frac{d \tau}{\tau^{2}}\sqrt{{1 + \vec{\phi}_{\tau}^{\, 2}+ \vec{\phi}_{\sigma}^{\, 2}+ \vec{\phi}_{\tau}^{\, 2}\vec{\phi}_{\sigma}^{\, 2}- \left( \vec{\phi}_{\tau}\vec{\phi}_{\sigma}\right)^{2}}}.
\nu_l = g_\lambda(\zeta_1(r)) - \sum_{j=1}^{k-1}Re\left[ G_\lambda(\zeta_1(r), \zeta_{j+1}(r)) e^{\frac{2\pi i}{k}j l}\right] & > g_\lambda(\zeta_1(r)) - \sum_{j=1}^{k-1}G_\lambda(\zeta_1(r), \zeta_{j+1}(r)) = \nu_0,
\begin{aligned}d(\phi (y), \psi (y)) \leq d(\phi (y), \psi (y)) + d(\psi (y), \eta (y)) \leq \tilde L \min\{d(\eta(\hat y), \eta (y)), d(\psi(\hat y), \phi (y))\}+M_\phi + M_\eta , = \tilde L \min\{d(\eta(\hat y), \eta (y)), d(\eta(\hat y), \phi (y))\}+M_\phi + M_\eta, \end{aligned}
\frac{(\frac{q^{2}}{4\pi\epsilon_{0}L_{1}^{2}})}{(\frac{\frac{q^{2}}{4}}{4\pi\epsilon_{0}L_{2}^{2}})}=\frac{mgtan\theta_{1}}{mgtan\theta_{2}}\Rightarrow4{(\frac{L_{2}}{L_{1}})}^{2}=\frac{tan\theta_{1}}{tan\theta_{2}}
\begin{cases}\displaystyle \frac{\partial x}{\partial u}(t,s)=\frac{\partial y}{\partial u}+\int_s^t \frac{\partial \xi}{\partial u}(\tau,s)\, d\tau \displaystyle\frac{\partial \xi}{\partial u}(t,s)=\frac{\partial \eta}{\partial u}-\int_s^t \frac{\partial^2}{\partial x\partial x}V(\tau,x(\tau,s)) \frac{\partial x}{\partial u}(\tau,s)\, d\tau,\end{cases}
\alpha \xi^{m}-\alpha \eta^{m}+q_{m}(\xi,\eta)=-\alpha \eta^{m}\left(\frac{\xi^{m}}{-\eta^{m}}+1+ \frac{q_{m}(\xi,\eta)}{-\alpha \eta^{m}}\right)\geq \frac{-\alpha}{2}\eta^{m}>c |\eta|.
O_{0}= \int_{0}^{1}{\frac{d x}{1 - x^{2}}}\bigl ( x^{- 1 / 6}- x^{1 / 6}\bigr ) ={\frac{1}{2}}\Bigl ( \psi \Bigl ({\frac{7}{1 2}}\Bigr ) - \psi \Bigl ({\frac{5}{1 2}}\Bigr ) \Bigr ) .
\left[\begin{array}{cc}\Psi_1(t) \Psi_2(t) \end{array}\right]=\left[\begin{array}{cc}\cos{\theta_v}-\sin{\theta_v}\sin{\theta_v}\cos{\theta_v}\end{array}\right]\left[\begin{array}{cc}\Psi_e(t) \Psi_{\mu}(t) \end{array}\right]\,,
Z_{\Lambda_{C}}= \int \d \tilde{\Phi}_{k}\e^{- N ({\frac{g}{3}}\makebox{t r}\tilde{\Phi}_{k}^{3}+{\frac{U}{2}}\makebox{t r}\tilde{\Phi}_{k}^{2}+ c \makebox{t r}\tilde{\Phi}_{k}\Lambda_{C})}
\big| D_2(wH) \big| = \big|(D_2 w)^{}\big| = \big| \big( -\tfrac{1}{2}(\sin{2x})(\sin{y}){L_w}_{*}e_1 + \tfrac{1}{2}(\sin{2x})(\cos{y}){L_w}_{*}e_2 - (\sin^2{x}){L_w}_{*}e_3 \big)^{}\big| = \tfrac{1}{2}\big| \sin{2x}\big|,
\begin{aligned}\int_{0}^{s}\Bigl(\int_{-\infty}^{\infty}P(xu)dF(x) \Bigr)du=\int_{0}^{s}\Bigl(\int_{0}^{\infty}P(xu)dF_1(x) \Bigr)du = \int_0^{\infty}\Bigl(\int_0^sP(xu)du \Bigr)dF_1(x).\end{aligned}
f(-\alpha)\overline{\mu}_Y-\alpha\underline{\mu}_X=\overline{h}, &f(-\alpha)\underline{\mu}_Y-\alpha\overline{\mu}_X=\underline{h}, &f(\alpha)\overline{\mu}_Y+\alpha\overline{\mu}_X=\overline{h}, &f(\alpha)\underline{\mu}_Y+\alpha\underline{\mu}_X=\underline{h}.
|\varphi_Z(\theta) - \varphi_{Z'}(\theta) | = |\mathbb{E}_{\mu_0}(e^{i\theta Z_t}) - \mathbb{E}_{\mu_0}(e^{i\theta Z'_t})| \leq \mathbb{E}_{\mu_0}(|e^{i\theta Z_t}-e^{i\theta Z'_t}|) \leq |\theta| \mathbb{E}_{\mu_0}(|Z_t-Z'_t|) \leq \pi \mathbb{E}_{\mu_0}(|Z_t-Z'_t|),
S = \int d t \left\{\, P_{a x}\dot{X}^{a x}- \left[ \frac{g^{2}}{2}\, X^{a x}X^{a x}+ \frac{1}{4 g^{2}}\, ( \partial_{[ a}P_{b ] x})^{2}\right] + \lambda_{x}X_{, a}^{a x}\right\}
\hat{I}_{>2,<\epsilon}(k):=\{\,\,\alpha_{k'}\,\,|\,\, k' \leq k, \,\, J(\alpha_{k'+1},\alpha_{k'})>2,\,\,\,\alpha_{k'+1},\,\,\alpha_{k'}\,\,\mathrm{satisfy\,\,2,3,4,5,6\,\, in\,\,Lemma\,\, \ref{mainlemma}}\}
\{x : 0 \le c_{k+1}(x) \le r_{k+1}\}= \bigcup_{i=1}^{q_k}\Big( I_{k+1}(i) \cup I_{k+1}(i + q_k) \cup \dots \cup I_{k+1}(i + r_{k+1}q_k) \Big) &= \bigcup_{i=1}^{q_k}\left( i\theta - \| q_k \theta\|, i\theta + r_{k+1}\|q_k \theta\| + \|q_{k+1}\theta\| \right).
\int\limits_0^\pi Cf(u,\beta,\psi)\sin\psi d\psi = \int\limits_0^\pi C(M_Af)(x,e_n,\psi) \sin\psi d\psi &= \frac{\pi}{|S^{n-1}|}\int\limits_{S^{n-1}}R(M_Af)(\omega,\omega\cdot x)|\omega\cdot e_n|d\omega.
\left[ \widehat{{\mathcal J}}_{\ell / r}^{0}\left( t , \underline{{x}}\right) , \widehat{{\mathcal J}}_{\ell / r}^{0}\left( t , \underline{{y}}\right) \right] = \pm i \frac{q^{2}}{4 \pi^{2}}\left( \underline{{B}}\left( \underline{{x}}, t \right) \cdot \underline{{\nabla}}\right) \delta \left( \underline{{x}}- \underline{{y}}\right) ,
u(0)=\frac{1}{|B_{r/2}|}\int_{B_{r/2}}u \geq \frac{1}{|B_{r/2}|}\int_{B_{r/2}\cap\{u \geq N\}}u \geq \frac{N|B_{r/2}\cap\{u \geq N\}|}{|B_{r/2}|}
- 1 2 R_{\mu \nu}R^{\nu \rho}R_{\rho}^{\mu}+{\frac{9 n}{n - 1}}R R_{\mu \nu}R^{\mu \nu}-{\frac{3 ( n^{2}+ 4 n - 4 )}{4 ( n - 1 )^{2}}}R^{3}+{\cal O}( W ) ,
\sum(\Theta_{j\bar k}^{T_B}\partial/\partial t^k, \partial/\partial t^j)_H= -||\sum [\theta_k^*, \theta_k]||^2- \left( P^{\bot}(D^{{\rm End}(H)}_j \theta_k), P^{\bot}(D^{{\rm End}(H)}_k \theta_j) \right).
{\mathcal S}(X_2, \tau_2, v_2)-{\mathcal S}(X_1, \tau_1, v_1) = \left(\int_1^2{\mathcal X'}_{\epsilon}d\epsilon, \int_1^2\pi_{\epsilon}d\epsilon, \int_{1}^2{\mathcal V'}_{\epsilon}d\epsilon\right)
\Vert (B_\varepsilon (\vartheta)-\widehat{\zeta}Q_0^\varepsilon)^{-1}(B_\varepsilon (\vartheta)+\lambda_0 Q_0^\varepsilon )\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}&\leqslant 2(1+\lambda_0)\Vert f\Vert_{L_\infty}\Vert f^{-1}\Vert_{L_\infty}c(\phi).
A(M) = \frac{2M^2}{\log^2M}\bigg(1 + O\bigg( \frac{\log\log M}{\log M}\bigg) \bigg) \bigg( \zeta(2) + O\bigg( \frac1{\log^3M}\bigg) \bigg) + O\bigg( \frac{M^2\log\log M}{\log^3M}\bigg) &= \frac{\pi^2M^2}{3\log^2M}+ O\bigg( \frac{M^2\log\log M}{\log^3M}\bigg),
\{\theta_{1}, \theta_{2}\}_{\mathrm{c . b .}}= W_{~ 1 2}^{0}+ \left\{\theta_{1}[ \theta_{2}, r ] \right\}+ \mathrm{T r}_{3 4}( W_{~ 1 2 3 4}^{2}\theta_{3}\theta_{4}) + \mathrm{T r}_{3 4 5 6}( W_{~ 1 2 3 4 5 6}^{4}\theta_{3}\theta_{4}\theta_{5}\theta_{6}) \ldots ,
M^{(s)}_{k_{1},k_{2},\ldots,k_{n}}=\prod_{j=1}^{n}\left(\prod_{i_{j}=0}^{k_{j}-1}j_{s+2\sum^{j-1}_{p=1}{k_p}+2i_{j}+(j-1)}\right), \widetilde{M}^{(s)}_{k_{1},k_{2},\ldots,k_{n}}=\prod_{j=1}^{n}\left(\prod_{i_{j}=0}^{k_{j}-1}j_{-s-2\sum^{j-1}_{p=1}{k_p}-2i_{j}-(j-1)}\right),
\pi_{\beta,\rho}(g) = \pi^{\beta,\rho}\left(l(\overline T)(1+\varpi^lS)_r\right) = \tau\left(\varpi^{-(l-1)}B(S,\beta) +\varpi^{-l}B(\lambda(\overline T),\beta)\right)\cdot \rho(Y)\cdot\pi^{\beta}(v,1) = \tau\left(\varpi^{-l}B(T,\beta)-2^{-1}\varpi^{-1}B(T^2,\beta) \right)\cdot\rho(Y)\cdot\pi^{\beta}(v,1).
Z(s, f^{(Q_\tau)}, \phi)(g) = \frac{L^N((2n+1)s+1/2,\pi \boxtimes \chi, \varrho_{2n+1})}{L^N((2n+1)(s+1/2), \chi)\prod_{j=1}^nL^N((2n+1)(2s+1)-2j, \chi^2)}\times i^{nk}\,\chi(\tau)^{-n}\, \pi^{n(n+1)/2}\bigg(\prod\limits_{p|N}{\rm vol}(\Gamma_{2n}(p^{m_p}))\bigg) A_{\mathbf{k}}((2n+1)s- 1/2)\phi(g),
d\lambda^{(n)}_i(t) = 2 \sqrt{\lambda^{(n)}_i(t)}dB^{(n)}_i(t) + 2 \Big(|n - p| + 1\Big) dt + \sum_{j \neq i}\frac{4 \lambda^{(n)}_i(t)}{\lambda^{(n)}_i(t) - \lambda^{(n)}_j(t)}dt,
\psi_\mathrm{trial}:=\sum_{s=0}^N \sum_{|d|\le \sigma_N^2}c_{d,s}u_1^{\otimes(N-s+d)/2}\otimes_{\mathrm{sym}}u_2^{\otimes (N-s-d)/2}\otimes_{\mathrm{sym}}\Phi_{\mathrm{trial},s}.
F_{\mu \nu}^{( 0 )}: = i \, ( k_{\mu}\zeta_{\nu}- k_{\nu}\zeta_{\mu}) e^{i \, k \cdot x}+ i \, ({k^{\prime}}_{\mu}{\zeta^{\prime}}_{\nu}-{k^{\prime}}_{\nu}{\zeta^{\prime}}_{\mu}) e^{i \, k^{\prime}\cdot x}
{{\boldsymbol{d}}_k}={\boldsymbol{\Theta}}_k^{- 1}\left[{\begin{array}{*{20}{c}}{{{{\boldsymbol{\hat x}}}_{k|k - 1}}}{{{\boldsymbol{y}}_k}-{{{\boldsymbol{\hat y}}}_{k|k - 1}}+{{\boldsymbol{H}}_k}{{{\boldsymbol{\hat x}}}_{k|k - 1}}}\end{array}}\right],
\begin{array}{l}{{\lambda \left( \kappa_{a d}^{S}\epsilon_{d b c}\alpha_{c}+ \kappa_{b d}^{S}\epsilon_{d a c}\alpha_{c}\right) + R \left( \beta_{a}k_{b}+ \beta_{b}k_{a}\right) = 0}}\\{{\ \ \lambda R \left( \alpha_{a}k_{b}- \alpha_{b}k_{a}\right) + \kappa_{a d}^{S}\epsilon_{d b c}\beta_{c}- \kappa_{b d}^{S}\epsilon_{d a c}\beta_{c}= 0 .}}\\ \end{array}
\sup_{\{R\leq \epsilon^{-1}r_\epsilon\}}w^-_{\gamma+2}|(\Delta_{\omega^-}-\epsilon^2X) v_i| =\sup_{\{r\leq r_\epsilon\}}w_{\epsilon,\gamma+2,\delta}|(\Delta_{\omega_\epsilon}-X) u_i|\longrightarrow 0.
( \hat{\nu}_{\epsilon}, a ) = \sum_{i , k}\hat{\nu}_{\epsilon i}( \delta_{i j}- \frac{1}{2}{\cal C}_{i j}) a_{j}= \sum_{i \in \Gamma_{\epsilon}^{( 0 )}}\hat{\nu}_{\epsilon i}a_{i}- c o s \frac{\pi}{h}\sum_{i \in \Gamma_{\epsilon + 1}^{( 0 )}}\hat{\nu}_{\epsilon + 1 i}a_{i}
a_{2 n - 1}( l ) = -{\frac{1}{2}}- ( 2 n - 1 ) -{\frac{2 \mathrm{e}^{-{\frac{1}{l}}}}{{l}{}^{( 2 n -{\frac{1}{2}})}\ ( n - 1 ) ! \ \Gamma ( n +{\frac{1}{2}})}}.
f_{r_t|r_m}(y)= \sum^{t-m-1}_{n=1}\frac{2y(\lambda_c \pi)^{n}(y^2-x^2)^{n-1}}{n!}e^{-\lambda_c \pi (y^2-x^2)}\times \left[\lambda_c \pi(y^2-x^2)-n\right] + 2\lambda_c \pi ye^{-\lambda_c \pi (y^2-x^2)}.
\frac{D(z)}{A'(z)}=\frac{\sum\limits_{j=1}^n\phi_j\prod\limits_{\substack{\ell=1 (\ell\neq j)}}^n(z-\lambda_\ell)}{\sum\limits_{j=1}^n\prod\limits_{\substack{\ell=1 (\ell\neq j)}}^n(z-\lambda_\ell)}.
V(\mathbf{j}) = \sum_{k=1}^{n-1}[(n-k+1)j_k - s(j_k)] - c(j_n,-\alpha(n)-1) \leq \sum_{k=1}^{n-1}[(n-k+1)j_k - s(j_k)]] = v(\mathbf{j}) < v(\mathbf{j}') = V(\mathbf{j}')
\|tu_{tt}\|_{L^p(\mathbb{R}^{2}_+)}+\|t^\frac{1}{2}u_{st}\|_{L^p(\mathbb{R}^2_+)}+\|u_{ss}\|_{L^p(\mathbb{R}^2_+)}+\|u_t\|_{L^p(\mathbb{R}^2_+)}+\|u\|_{L^p(\mathbb{R}^2_+)}&\qquad\le C_{pT}\|f\|_{L^p(\mathbb{R}^2_+)},
\frac{p^{(\alpha,\beta)}_{n+1}(0)}{p^{(\alpha,\beta)}_{n}(0)}=(n+1+\alpha)\rho^{(\alpha,\beta)}_n, \frac{p^{(\alpha,\beta)}_{n+1}(1)}{p^{(\alpha,\beta)}_{n}(1)}=(n+1+\beta)\rho^{(\alpha,\beta)}_n,
\nu_{LEQ}(\overrightarrow{\eta})= \nu_{EQ}(\overrightarrow{\eta})\exp \left( \varepsilon \sum_{i=1}^N \frac{m \rho_i^{(1)}}{\rho_i^{(0)}(m+\rho_i^{(0)})}\eta_i \right) + O(\varepsilon^2).
\textstyle \sum_{r - m > i(p-1)+\alpha > m}E_i \left[1,x^{i(p-1)+\alpha}y^{r-i(p-1)-\alpha}\right] \textstyle + \sum_{j(p-1)+\alpha \in [\alpha,m] \cup [r-m,r-\alpha]}F_j \left[1,x^{j(p-1)+\alpha}y^{r-j(p-1)-\alpha}\right] \textstyle \qquad = \sum_{i}E_i' \left[1,\theta^\alpha x^{i(p-1)}y^{r-\alpha(p+1)-i(p-1)}\right],
\frac{\tilde\lambda_{j}}{\tilde\lambda_{j-1}-\tilde\lambda_j}\leq \frac{\lambda_{j-1}}{\tilde\lambda_{j-1}-\lambda_{j-1}}=\frac{1}{1-A}\frac{\lambda_{j-1}}{\lambda_{j-2}-\lambda_{j-1}}\leq 2\frac{\lambda_{j-1}}{\lambda_{j-2}-\lambda_{j-1}},
a ={\partial_{\nu_{1}}}{\partial_{\bar{\nu_{1}}}}\log{\langle \nu \vert \nu \rangle}, b ={\partial_{\nu_{1}}}{\partial_{\bar{\nu_{2}}}}\log{\langle \nu \vert \nu \rangle}, c ={\partial_{\nu_{2}}}{\partial_{\bar{\nu_{2}}}}\log{\langle \nu \vert \nu \rangle}
(\sigma_\Delta^\leqslant-\sigma_0^\leqslant)^2 =\frac12\left((\sigma_\Delta^\leqslant-\sigma_0^\leqslant)^2+(t_\Delta^\leqslant-t_0^\leqslant)^2\right)+\frac12\left((\sigma_\Delta^\leqslant-\sigma_0^\leqslant)^2-(t_\Delta^\leqslant-t_0^\leqslant)^2\right) &=\frac12\bigl|s_\Delta^\leqslant-s_0^\leqslant\bigr|^2+\frac12\Re(s_\Delta^\leqslant-s_0^\leqslant)^2.
\sum_{w \in W_0}w \Big( \prod_{\alpha \in \widetilde{\Sigma}_0^+}\frac{1}{1 - e^{-\alpha^\vee}}\, \Big) \cdot e^{w\bar{\lambda}^\flat}= \sum_{\bar{\nu}^\flat \in{\mathcal Wt}(\bar{\lambda}^\flat)^\tau}P_{w_{\bar{\nu}^\flat}, w_{\bar{\lambda}^\flat}}(1) \cdot e^{\bar{\nu}^\flat}.
\bigl(\{x\in X \mid \forall i, |f_{i}(x)|< r_{i}\,|f'_{i}(x)|,\ \forall j, |g_{j}(x)|\le s_{j}\,|g'_{j}(x)|,\ \forall k, |h_{k}(x)|\le t_{k}\,|h'_{k}(x)|\}\bigr)_{H}= \{x\in X \mid \forall i, |f_{i}(x)|< r_{i}\,|f'_{i}(x)|,\ \forall j, |g_{j}(x)|\le s_{j}\,|g'_{j}(x)|,\ \forall k, |h_{k}(x)|\prec t_{k}\,|h'_{k}(x)|\}.
X^2_0(t)= \sum_{k+\ell\leq s}\sum_{1\leq j\leq J}|z^{(k,\ell)}_j(t)|^2,\X^2_-(t)=\sum_{k+\ell\leq s}\sum_{i\in -\mathbb{N}}|\xi^{(k,\ell)}_i(t)|^2,
\pi^0_l(h): \: \phi(Z) \mapsto \bigl( \pi^0_l(h)\phi \bigr)(Z) =\frac 1{N(cZ+d)}\cdot \phi \bigl( (aZ+b)(cZ+d)^{-1}\bigr), \pi^0_r(h): \: \phi(Z) \mapsto \bigl( \pi^0_r(h)\phi \bigr)(Z) =\frac 1{N(a'-Zc')}\cdot \phi \bigl( (a'-Zc')^{-1}(-b'+Zd') \bigr),
|{}_2F_1\left(k-\tfrac{1}{2},k+\tfrac{z-1}{2};2k-1;x\right)| = \left|\tfrac{\Gamma(2k-1)}{\Gamma(k-1/2)^2}\int_{0}^{1}t^{k-3/2}(1-t)^{k-3/2}(1-xt)^{-k-(z-1)/2}\d t\right| & \leq \tfrac{\Gamma(2k-1)}{\Gamma(k-1/2)^2}\int_{0}^{1}t^{k-3/2}(1-xt)^{-1}\left(\tfrac{1-t}{1-tx}\right)^{k-3/2}\d t.
|\rho(ad-\rho(d)a)|^2 =\lim_\lambda |\rho(ad-\rho(d)au_\lambda)|^2=\lim_\lambda |\rho(a(d-\rho(d)u_\lambda))|^2 & \leq \lim_\lambda \rho(aa^*)\rho((d-\rho(d)u_\lambda)^*(d-\rho(d)u_\lambda))= 0.
\phi^{\alpha}_{J_1;J_2+\bold{1}_k}=S_k\phi^{\alpha}_{J_1;J_2}+D_{J_1}\left(\left(S_k\xi^i-\xi^i\right)u^{\alpha}_{\bold{1}_i;J_2+\bold{1}_k}\right)-\left(S_k\xi^i-\xi^i\right)u^{\alpha}_{J_1+\bold{1}_i;J_2+\bold{1}_k}.
\langle R^N(\bar\nabla_{e_j}\big(R^N(V,d\phi(e_k))d\phi(e_k)\big),\tau(\phi))d\phi(e_j),V\rangle =\langle R^N(V,d\phi(e_j))\tau(\phi),\bar\nabla_{e_j}\big(R^N(V,d\phi(e_k))d\phi(e_k)\big)\rangle.
[H_{i,r},E_j(z)]= a_{i,j}(r)C^{-(r+|r|)/2}\,z^r E_j(z)\,, &[H_{i,r},F_j(z)]=-a_{i,j}(r)C^{-(r-|r|)/2}\,z^r F_j(z)\,, &[H_{i,r},H_{j,s}]=\delta_{r+s,0}\cdot a_{i,j}(r)\eta_r\,,\eta_r=\frac{C^r-C^{-r}}{q-q^{-1}}\,.
\lim_{R\to\infty}\frac{\sigma_R^2(t)}{R}=\lim_{R\to\infty}4\pi\sum_{n\geq 2}C_H^n \int_{T_n(t)}\big(\ell_R*g_{\pmb{t}_n}^{(n)}\big)(0) d\pmb{t}_n = 4\pi\sum_{n\geq 2}C_H^n\int_{T_n(t)}g_{\pmb{t}_n}^{(n)}(0) d\pmb{t}_n.
\begin{cases}\begin{aligned}\sum_{i,j,k,p}(\bar h^{p^{\ast}}_{ijk})^2=0, \ \ \bar\lambda=0, \ \ \bar\lambda_1\bar\lambda_2\neq 0, & S<\sup H^2=\bar H^2 \leq 3S-2 \ \ \ \S<\sup H^2<\dfrac43S,\end{aligned}\end{cases}
|F(w)|=|F(\zeta_0+\mathrm{i}\tau)| \leqslant \int_{\Gamma}|F(\zeta)| \frac{C\tau|\mathrm{d}\zeta|}{|\zeta-\zeta_0|^2+\tau^2}\leqslant \int_{\mathbb{R}}\big|F\big(u+\mathrm{i}a(u)\big)\big| \frac{C\tau\sqrt{1+M^2}\,\mathrm{d}u}{|u-u_0|^2+\tau^2},
\left( \bar{\lambda}\chi \right)^{*}= \left( \bar{\lambda}^{i}\chi_{i}\right)^{*}= \chi_{i}^{\dagger}\left( \bar{\lambda}^{i}\right)^{\dagger}= \bar{\chi}^{i}\lambda_{i}= \bar{\lambda}^{i}\chi_{i}= \bar{\chi}\lambda = \bar{\lambda}\chi \ .
{}_{}{\Phi}_{p}(b;c;z)= \frac{1}{B(b, c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}exp\left[zt-\frac{p}{t(1-t)}\right] dt. (p>0 ;\p=0\and \Re(c)>Re(b)>0)
{\Gamma_{\theta}^{i j}}^{- 1}( \nu ) = \frac{\nu \sigma}{\omega^{2}}\left( - \delta_{i j}\cot \left( \pi \frac{\nu}{\omega}\right) + \left( 1 - \delta_{i j}\right) \csc \left( \pi \frac{\nu}{\omega}\right) \right) \, ,
\frac{\partial U^{- 1}}{\partial z_{\alpha}}+{\frac{1}{2 \pi \kappa}}\sum_{\beta \neq \alpha}\frac{\frac{1}{2}( \hat{Q}^{+}( \xi_{\alpha}) \hat{Q}^{-}( \xi_{\beta}) + \hat{Q}^{-}( \xi_{\alpha}) \hat{Q}^{+}( \xi_{\beta}) ) + \hat{Q}^{3}( \xi_{\alpha}) \hat{Q}^{3}( \xi_{\beta})}{z_{\alpha}- z_{\beta}}U^{- 1}= 0
\sigma(c_na_n, c_nb_n) = \sigma(a_n, b_n) &= \sigma(A_n(a,b), B_n(a,b)) = \frac{H_n(A_n(a,b), B_n(a,b))^{2d^n}}{\rho_d(a,b)^{Nd_n}}& = \frac{H_n(a_n, b_n)^{2d^n}}{\rho_d(a,b)^{Nd_n}}
\| M_{\chi}f \|_{L_2}^{2}= \int_{t_1}^{t_2}|f(e^{i\theta})|^2 \frac{d\theta}{2\pi}= \int_{\alpha_1}^{\alpha_2}|f(\psi(e^{it}))|^2 |\psi'(e^{it})| \frac{dt}{2\pi}&\leq m \int_{\alpha_1}^{\alpha_2}|f(\psi(e^{it}))|^2 \frac{dt}{2\pi}\leq m \| C_\psi f \|^2_{L^2},
\frac{1}{3}\int p_{1}= -\frac{1}{48\pi^{2}}\int d\tau \int d\vec{\sigma}\cdot \vec{\nabla}(\frac{2\vec{\nabla}V\cdot \vec{\nabla}V}{V^{3}})=-\frac{1}{6\pi^{2}}\int d\vec{\sigma}\cdot \vec{\nabla}(\frac{\vec{\nabla}V\cdot \vec{\nabla}V}{V^{3}})~~,
\left( \frac{1}{G_{4}( \mu )}- \frac{1}{G_{4}^{*}( g )}\right) = \left( \frac{g^{2}( \Lambda )}{g^{2}( \mu )}\right)^{2 - c g^{2}}\left( \frac{1}{G_{4}( \Lambda )}- \frac{1}{G_{4}^{*}( g )}\right) ,
I_\Delta \!=\! (x_1 x_2 x_3,\! x_1 x_2 x_4, x_1 x_3 x_5, x_1 x_4 x_6, x_1 x_5 x_6, x_2 x_3 x_6, x_2 x_4 x_5, x_2 x_5 x_6, x_3 x_4 x_5, x_3 x_4 x_6)
\bar{\varepsilon}_{i}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\cdots \gamma_{\mu_{n}}\psi_{m}^{i}= - \bar{\psi}_{m i}\gamma_{\mu_{n}}\cdots \gamma_{\mu_{2}}\gamma_{\mu_{1}}\varepsilon^{i}= - ( \bar{\varepsilon}_{i}\gamma_{\mu_{1}}\gamma_{\mu_{2}}\cdots \gamma_{\mu_{n}}\psi_{- m}^{i})^{*}.