harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\mathcal W}_{\mu_{1}\cdots \mu_{n}}^{( n )}( x_{1}, \cdots , x_{n}) \; = \; \left. \frac{\delta^{n}{\mathcal W}[ b_{\mu}]}{\delta b_{\mu_{1}}( x_{1}) \cdots \delta b_{\mu_{n}}( x_{n})}\right|_{b_{\mu}= 0}\; .
\frac{1}{m_{k+1}}\log \|Df^{m_{k+1}}|E^u_{z}\| \leq \lambda^u(\delta_Q)+\frac{1}{3}\log b\cdot\frac{1}{m_{k+1}}\sum_{i=1}^kr_i &= \lambda^u(\delta_Q)+\frac{1}{3}\log b\cdot\frac{1}{m_{k+1}}\#V_{m_{k+1}}.
\sigma_{\widetilde{i-1}}=L_{\widetilde{i-1}}\pi\stackrel{\eqref{piLx}}{=}\pi^{-1}L_{\widetilde 1}^{-1}L_{\widetilde i}\pi^2=\sigma_{\widetilde 1}^{-1}\sigma_{\widetilde i}\sigma_{\widetilde 0}=\sigma_{\widetilde 1}^{-1}\sigma^i_{\widetilde 1}\sigma_{\widetilde 0}^{1-i}\sigma_{\widetilde 0}=\sigma^{i-1}_{\widetilde 1}\sigma_{\widetilde 0}^{2-i}.
\mathcal E_{\alpha,g}(t):= \frac{2}{\alpha-1}t^2 (\Phi(x(t))- \inf_{\mathcal H}\Phi )+ (\alpha-1) \| x(t) - x^{*}+ \frac{t}{\alpha-1}\dot{x}(t) \|^2 + 2\int_t^{+\infty}\tau \langle x(\tau) - x^{*}+ \frac{\tau}{\alpha-1}\dot{x}(\tau), g(\tau) \rangle d\tau.
\tau_0 = 1, \tau_1 = x, \tau_2 = x^3 - 3t_3, \tau_3 = x^6 - 15t_3x^3 - 45t_3^2 + 45t_5x, \tau_4 = x^{10}-45t_3x^7 + 315t_5x^5 + 4725t_3t_5x^2 - 4725t_3^3x - 4725t_5^2 - 1475t_7x^3 + 4725t_3t_7.
\begin{dcases}-\int_0^T e^{\lambda_0(T-t)}p(t) \, dt = m_{2,0}, \ \ \ -\int_0^T p(t)\, dt = m_{2,0}, -\int_{0}^{T}\overline{e^{\lambda^h_k (T-t)}}p(t)\,dt = m_{2,k}, \forall |k|\geq k_0, -\int_{0}^{T}\overline{e^{\widehat \lambda_{n_l}(T-t)}}p(t)\,dt = m_{1,l}, \forall \, 1\leq l\leq l_0,\end{dcases}
\Pi_{\nu}^{\mu}={\frac{\rho}{1 2}}\left[ \begin{array}{c c c}{{- ( \rho + 2 \delta \rho )}}&{{}}&{{2 a ( \rho + P ) ( v + B )_{| j}}}\\{{}}&{{}}&{{}}\\{{- 2 a^{- 1}( \rho + P ) v^{| i}}}&{{}}&{{\left\{2 P + \rho + 2 ( 1 + P / \rho ) \delta \rho + 2 \delta P \right\}\delta_{j}^{i}- ( 1 + 3 P / \rho ) \delta \pi_{j}^{i}}}\\ \end{array}\right] \, .
X_i\varphi = \eta^2G(\eta |Tu|)^{\gamma+1}X_iX_lu + (\gamma+1)\eta^3 G(\eta |Tu|)^{\gamma}g(\eta |Tu|)X_lu\,X_i(|Tu|) + \Big(2\eta G(\eta |Tu|)^{\gamma+1}+(\gamma+1)\eta^2 G(\eta |Tu|)^{\gamma}g(\eta |Tu|)|Tu|\Big) X_lu X_i\eta
\left[q^d\right]\sum_{m=0}^N(-y)^m s_{(\ell,1^{N-m})}(z_1, \ldots, z_{N+1})= (-1)^N\left[t^\ell\right]\frac{yt^{N(d-1)+1}}{(1-t)^{Nd}(1-yt)^{d+1}}.
M_n^{-1}\left\| g^{(1)}_n\right\|_1 = \left\| \chi_{[0, R]}(|\eta|)\phi_0 \mathfrak{X}_n \varphi_n^{(1)}\right\|_1 + M_n^{-1}\left\| \chi_{(R, \infty)}(|\eta|) \phi_0 |v_{12}|^{\frac 12}\left[H_0 + k_n^2 \right]^{-1}|v_{13}| \psi_n \right\|_1 ,
\left[ \frac{\partial^{2}}{\partial p_{0}^{2}}- \frac{\partial^{2}}{\partial \vec{p}^{2}}\right] \Psi ( p ) = \frac{4 \lambda_{F}\Psi ( p )}{( p_{0}^{2}- \vec{p}^{2}+ W^{2}- m^{2})^{2}- 4 p_{0}^{2}W^{2})}
K=F(\mathbb{D})\cap \pi_2^{-1}(\partial\bar{\mathbb{D}}_2)\ \ \ \ \K_{\lambda,\mu}=F_{\lambda,\mu}(\mathbb{D})\cap \pi_2^{-1}(\partial\bar{\mathbb{D}}_2)\ \ \ \ \ \hat{K}_{\lambda,\mu}=F_{\lambda,\mu}(\mathbb{D})\cap \pi_2^{-1}(\Gamma)
(\begin{matrix}a_{0}\\a_{2}\\ \end{matrix})_{j}=\frac{1}{70}(\begin{matrix}34&-10\\ -10&5\end{matrix})(\begin{matrix}1&1&1&1&1\\ 4&1&0&1&4\\ \end{matrix})(\begin{matrix}y_{j-2}\\y_{j-1}\\y_{j}\\y_{j+1}\\y_{j+2}\end{matrix})
\delta \left( p_{1}- q_{1}\right) \frac{e^{- i q_{1}^{'}x}}{\sqrt{2{\omega}_{1}\left( q_{1}^{'}\right)}}+ \delta \left( p_{1}- q_{1}^{'}\right) \frac{e^{- i q_{1}x}}{\sqrt{2{\omega}_{1}\left( q_{1}\right)}}
\begin{aligned}u_1''(t) + \lambda_1 f_1(u_1(t),u_2(t)) = 0, \t\in (0,1), u_2''(t) + \lambda_2 f_2(u_1(t),u_2(t)) = 0, \t\in (0,1), u_1'(0)=0,\u_1(1)+u_1'(1) =0, u_2'(0)=0,\u_2(1)-\xi u_2(\eta) =0,\ \eta \in (0,1), \ 0 < \xi<1,\end{aligned}
{\displaystyle\sum\limits_{b\in\mathbb{Z}_{W}^{\ast}}}f_{1}(b) >(\frac{5}{8}+\frac{3\delta}{8})\phi(W),{\displaystyle\sum\limits_{b\in\mathbb{Z}_{W}^{\ast}}}f_{i}(b) >(\frac{5}{8}-(\frac{5\eta}{4}+\frac{\delta}{8}))\phi(W)\text{}(i=2,3).
\langle\xi_{i_1}\psi_1^{k_1},\cdots,\xi_{i_n}\psi_n^{k_n}\rangle_{g}^{W,G}:=\int_{\overline{\mathcal{M}}_{g,n}}\Lambda_{g,n}^{W,G}(\xi_{i_1},\cdots,\xi_{i_n}) \prod_{m=1}^n\psi_m^{k_m}
\frac{v_{1}/l}{u_{1}/l}=\frac{v_{1}}{u_{1}}=\frac{t}{s}=\frac{q_{1}}{p_{1}}=\frac{q_{1}/l}{p_{1}/l}\|I_{\alpha}(|f|^{l})^{1/l}\|_{\mathcal{M}_{v_{1}}^{u_{1}}}=\|I_{\alpha}(|f|^{l})\|_{\mathcal{M}_{v_{1}/l}^{u_{1}/l}}^{1/l}.
\delta_{n}^{(2)}= \delta_{n}^{(2)}(v,A,b,\sigma,q,B,\eta,\gamma) = \frac{b K_n}{\gamma^{1 + 1/q}n^{1/2-1/q}}+ \frac{(b\sigma K_n^{3/2})^{1/2}}{\gamma^{1 + 1/q}n^{1/4}}.
\begin{aligned}a^1=A^1\,, \,\,\, b^1=B^1\,, a^2=g_{+,1}^{-1}A^2\,, \,\,\, b^2=B^2 g_{-,1}\,,\ldots\,, a^d=g_{+,d-1}^{-1}\cdots g_{+,1}^{-1}A^d\,, \,\,\, b^d=B^d g_{-,1}\cdots g_{-,d-1}\,,\end{aligned}
J_z^-\cong (J_z^-\cap T_i(U_z^-))\otimes\left(\bigoplus_{n=0}^\infty Kf_i^{(n)}\right), J_z^-\cong \left(\bigoplus_{n=0}^\infty K f_i^{(n)}\right)\otimes (J_z^-\cap T_i^{-1}(U_z^-)).
(\begin{matrix}x_{c}\\y_{c}\end{matrix})=(\begin{matrix}A&B/2\\B/2&C\end{matrix})^{-1}(\begin{matrix}-D/2\\ -E/2\end{matrix})=(\begin{matrix}(BE-2CD)/(4AC-B^{2})\\ (DB-2AE)/(4AC-B^{2})\end{matrix}).
\|\tilde c\| =\sup_k \|(\mathrm{ad}_k\otimes_{min}\mathrm{id})(\tilde c)\| &\leq\sup_{k}\limsup_{i}\|(\mathrm{ad}_k\otimes_{min}\mathrm{id}_{k_i})(c_i)\| &\leq\limsup_{i}\|(\mathrm{ad}_i\otimes_{min}\mathrm{id}_{k_i})(c_i)\| &=\frac{1}{\lambda}<\|c\|,
\mathrm{Var}g_n(x)= ~\mathrm{E}\left[g_n(x)-\hat{g}_n(x)-\mathrm{E}\left(g_n(x)-\hat{g}_n(x)\right)\right]^2 + \mathrm{Var}\hat{g}_n(x) &+2\mathrm{E}\left[\left(g_n(x)-\hat{g}_n(x)-\mathrm{E}\left(g_n(x)-\hat{g}_n(x)\right)\right)\left(\hat{g}_n(x)-\mathrm{E}\hat{g}_n(x)\right)\right].
s_{3B}(k)=\sum_{j=0}^{k}(\begin{matrix}2j\\j\end{matrix})(\begin{matrix}2j\\j\end{matrix})(\begin{matrix}3j\\j\end{matrix})(\begin{matrix}k+j\\k-j\end{matrix})(-27)^{k-j}=1,-15,297,-6495,149481,...
\sum_{\gamma\in\mathcal{U}_{[n,j]}}w(\gamma)=\sum_{k=j}^{n-1}\sum_{\gamma\in\mathcal{U}_{[n,j,k]}}w(\gamma)=\sum_{k=j}^{n-1}a_{-j}^{(j)}\,B_{[k-1,j-1]}^{(1)}W_{[n-k-1,0]}.
{\rm e}^{\phi^i (x)}= \xi_1^{(i)}(x) \bar{\xi}_1^{(i)}(x),{}~~~~\psi_+^i(x) = \frac{\xi_1^{(i)}(x) \bar{\xi}_2^{(i)}(x)}{\xi_1^{(i)}(x)\bar{\xi}_1^{(i)}(x)},{}~~~~\psi_-^i(x) = \frac{\xi_2^{(i)}(x) \bar{\xi}_1^{(i)}(x)}{\xi_1^{(i)}(x)\bar{\xi}_1^{(i)}(x)}.
( - )^{{\bar{\alpha}}+{\bar{\beta}}+ \bar{\alpha}\bar{\beta}}\frac{{\bar{\vartheta}}[_{\bar{\beta}}^{\bar{\alpha}}]}{{\bar{\eta}}}\rightarrow \frac{{\bar{\vartheta}}[_{\bar{\beta}}^{\bar{\alpha}}]^{5}}{{\bar{\eta}}^{5}}{\frac{1}{2}}\sum_{\gamma , \delta}\frac{{\bar{\vartheta}[_{\delta}^{\gamma}]}^{8}}{{\bar{\eta}}^{8}}
\langle \rho ( k ) \rho ( - k ) \rangle = \left({\frac{e}{2 \pi}}\right)^{2}\left( k^{2}-{\frac{k^{4}}{M^{2}+ k^{2}}}\right) = \left({\frac{e}{2 \pi}}\right)^{2}{\frac{M^{2}k^{2}}{M^{2}+ k^{2}}}\ .
\{f,g\}_\kappa = i\left( \frac{\partial^R f}{\partial \bar z^A}g^{{\bar A}B}_\kappa \frac{\partial^L g}{\partial z^B}- (-1)^{(p(A)+\kappa)(p(B)+\kappa)}\frac{\partial^R f}{\partial z^A}g^{{\bar A}B}_\kappa \frac{\partial^L g}{\partial \bar z^B}\right),
P(E ; \{z= \bar{z}\}) = \mathcal{H}^{n-1}( \partial^{*}E \cap \{z = \bar{z}\}) = \mathcal{H}^{n-1}(( \partial^{*}E )_{\bar{z}}) & \leq \mathcal{H}^{n-1}\left(B^{n-1}\left( (0, r_{\ell}^{\vee}(\bar{z}) \right) \right) &= \mathcal{H}^{n-1}( \partial^{*}F_{\ell}\cap \{z = \bar{z}\}) &= P (F_{\ell}; \{z=\bar{z}\}) \leq P(E ; \{z= \bar{z}\}),
{\cal W}=\epsilon_{ijk}\Phi_i\Phi_j\Phi_k + \epsilon_{ijk}\Phi^l_i\Phi^l_j\Phi^l_k + \epsilon_{ijk}\Phi^i_k Q^{ij}Q^{ij}+ \Phi_i Q^i Q^i+ Q^{ij}Q^{jk}Q^{ki}+ Q^{ij}Q^i Q^j~.
\int_{0}^{\infty}(\gamma_{n}\ast\gamma_{n-1}\ast\cdots\gamma_{2}\ast H_{1})(t+s)dt \leq\int_{0}^{\infty}(\gamma_{n}\ast\gamma_{n-1}\ast\cdots\gamma_{2}\ast H_{1})(t)dt &=\Vert\gamma_{n}\Vert_{L^{1}}\cdots\Vert\gamma_{j+2}\Vert_{L^{1}}\Vert H_{1}\Vert_{L^{1}}<\infty,
\begin{pmatrix}-1 -1 \cdots -1 -1 0 0 \cdots 0 0 1 0 \cdots 0 0 1 1 \ddots 0 0 \vdots \vdots \ddots \vdots \vdots 1 1 \cdots 1 0 1 1 \cdots 1 1\end{pmatrix}.
Z [ A_{+}, \rho , \tilde{\rho}] = \int \! D \bar{\psi}D \psi D B_{+}^{\prime}D \theta e^{- \int \! d^{2}x [ L_{\mathrm{{C F}}}+ L_{\mathrm{B}}+ \rho e^{- i \theta}\psi_{-}^{\dag}\psi_{+}+ \tilde{\rho}e^{i \theta}\psi_{+}^{\dag}\psi_{-}]}\quad .
(g_1+g_2)^\infty(w) \le \lim_{\delta\searrow 0}\inf_{\lambda\in(0,\delta)}[\lambda g_1(w/\lambda) + \lambda g_2(w/\lambda)] &\le \sup_{\lambda>0}\lambda g_1(w/\lambda) + \lim_{\delta\searrow 0}\inf_{\lambda\in(0,\delta)}\lambda g_2(w/\lambda) &=g_1^\infty(w) + g_2^\infty(w),
\bar{b}(\theta) = \bar{b}_1 \theta + \bar{b}_2 \theta^2 + \bar{b}_3 \theta^3 = \bar{B}\begin{bmatrix}\theta \theta^2 \theta^3 \end{bmatrix}\bar{B}= \begin{bmatrix}\bar{b}_1 \bar{b}_2 \bar{b}_3 \end{bmatrix}
\{D\in K^{\times}: \mathcal{H}(D^{2}A,D^{3}B)\le X,~~~ D\cdot (A,B) \mathcal{P}(m',n')\}= \{D\in K^{\times}:\mathcal{H}(D^{2}A,D^{3}B)\le X\}= \{D\in K^{\times}: \mathcal{H}(D^{3}A,D^{3}B)\le X,~~~ D\cdot (A,B)\mathcal{P}(m,n)\},
\begin{cases}-\partial_{tt}F_{n}^\omega+\Delta F_{n}^\omega = 2 \nabla P_{\leq N^\gamma}u_{n-1}\cdot \nabla F_{n}^\omega F_{n}^\omega|_{t=0}= Q_N f_0^\omega~,~~ \partial_t F_{n}^\omega|_{t=0}= Q_N f_1^\omega~. \end{cases}
\frac{1}{1+t}\bigg[\binom{n}{k}_t-\binom{[\frac{k}{2}]+[\frac{n-k}{2}]}{[\frac{k}{2}]}_{t^4}\bigg] \frac{t}{1+t}\bigg[\binom{n}{k}_t-\binom{[\frac{k}{2}]+[\frac{n-k}{2}]}{[\frac{k}{2}]}_{t^4}\bigg]
e_{\bot}( x_{\bot}) = g \nabla_{\bot}\int d y^{-}d y_{\bot}d ( x_{\bot}- y_{\bot}) \left\{f^{3 a b}A_{\bot}^{a}( y_{\bot}, y^{-}) \Pi_{\bot}^{b}( y_{\bot}, y^{-}) + \rho_{m}^{3}( y_{\bot}, y^{-}) \right\}\frac{\tau^{3}}{2}\, .
G^1(x,\mathbf{y}) = \left( FP^1_1 \right) y^1+\left( FP^1_2 \right) y^2 + \left( L^1_1 \right) \left(y^1 \right)^2 + (L^1_2) \left(y^2 \right)^2+(Q_0^1) y^1 y^2 &+ \left( \frac{A^1_0}{F}\right) \left(y^1 \right)^3+ \left( \frac{B^1_0}{F}\right) \left(y^1 \right)^2 y^2 + \left( \frac{D^1_0}{F}\right) y^1 \left(y^2 \right)^2 +F^2R_0^1.
{\rm (T3b)}\hskip 10pt{\lambda_r+i\choose i}y(s,t)_j= \sum_{h=0}^j{\lambda_s-i+j\choose j-h}{\lambda_r+i\choose h}y(r,s)_{i-h},\cr \phantom{}\hskip 120pt \hbox{for}1\leq j\leq \lambda_t, j<i\leq \lambda_s+j.
g^{-1}=\tau^{-1}(g^{-1}_{\tau^{-1}(\boldsymbol{0})}, \dots, g^{-1}_{\tau^{-1}(\boldsymbol{p-1})}) gh =\tau\sigma\sigma^{-1}(g_{\boldsymbol{0}},\dots, g_{\boldsymbol{p-1}})\sigma (h_{\boldsymbol{0}},\dots, h_{\boldsymbol{p-1}}) = \tau\sigma(g_{\sigma(\boldsymbol{0})}h_{\boldsymbol{0}}, \dots, g_{\sigma(\boldsymbol{p-1})}h_{\boldsymbol{p-1}})
\epsilon_j(\xi,t) = \frac{(-1)^{p+1}}{2}[[U_j]] R_{p+1}^- + \alpha_{p+2}(t) R_{p+1}^{-,(-1)}+ \ldots + \alpha_{2p+1}(t)R_{p+1}^{-,(-p)}+ \mathcal{O}(h^{2p+2}),
\left[{\frac{\partial}{\partial t}}+ \int d x \; \left( \Pi ( x ) \;{\frac{\delta}{\delta \Phi ( x )}}- \big ( m^{2}\Phi ( x ) - \nabla^{2}\Phi ( x ) \big ){\frac{\delta}{\delta \Pi ( x )}}+{\cal K}_{I}( x ) \right) \right] W [ \Phi , \Pi ; t ] = 0 \; ,
\langle \, X\,[\,\phi\,,\,\phi^\ast\,]\,\rangle_{_J}\,\,=\, \,\int\,{\cal D}\phi \,X\,\,\,exp{i\over \hbar}\,\{S[\phi\,,\,\phi^\ast \,] \,+\,J^A\,\phi^A \,\,\}
\sum_{n=0}^{\frac{m-1}{2}}F(n,0)\equiv \sum_{n=0}^{\frac{m-1}{2}}F(n,1) \equiv \sum_{n=0}^{\frac{m-1}{2}}F(n,2)\equiv\cdots\equiv \sum_{n=0}^{\frac{m-1}{2}}F\left(n,\frac{m-1}{2}\right) \pmod{[m]\Phi_m(q)^2}.
L:= \ 2\int_0^{+\infty}\lambda(t)\beta(t)\left[\sup\limits_{u \in C}\varphi_B\left(u,\frac{p}{\beta(t)}\right)-\sigma_C\left(\frac{p}{\beta(t)}\right)\right]dt & \ +3\int_0^{+\infty}\lambda^2(t)\beta^2(t)\|Bx(t)\|^2dt+3\|Dz+v\|^2\int_0^{+\infty}\lambda^2(t)dt<+\infty.
\delta_{\epsilon}A_{( \sigma )}^{\alpha \beta}= \varepsilon^{\alpha \beta \gamma \delta}\partial_{\gamma}\epsilon_{\delta ( \sigma )}\equiv \stackrel{( 0 )}{R}_{( \sigma )}^{\prime \alpha \beta \delta ( \rho )}\epsilon_{\delta ( \rho )}, \; \delta_{\epsilon}B_{\alpha}^{\; ( \sigma )}= 0 ,
=\prod \limits_{k=-(N_0-1)}^{N_0-1}\exp\{\xi_{0k}a_{0k}(t)\}\prod\limits_{j=0}^{N-1}\prod\limits_{l=-(M_j-1)}^{M_j-1}\exp\{\eta_{jl}b_{jl}(t)\}= \exp\{\hat X(t)\}.
D_{mm'}^j(\psi)= \frac{1}{\sqrt{(j+m')!}}~\frac{1}{\sqrt{(j-m')!}}~ (\frac{\partial}{\partial \xi_1^*})^{j+m}~(\frac{\partial}{\partial \xi_2^*})^{j-m}~ <\xi\mid U(\psi)\mid j,m'>
\alpha_N^0= \alpha_N(0,\lambda) = -i \sum_{k=1}^N\frac{1}{(2i\lambda)^k}r_k(0)= O\Big(\frac{1}{\lambda}\Big) \beta_N^0= \beta_N(0,\lambda) = -i \sum_{k=1}^N\frac{1}{(2i\lambda)^k}s_k(0)= O\Big(\frac{1}{\lambda}\Big)
\begin{array}{c}\displaystyle \frac{1}{\tau}( a^\circ(z -z_0), \psi )_H +( b^\circ z, \psi )_H +\bigl( A^\circ \partial_x z +\nu^2 \partial_x z +p \omega^\circ, \partial_x \psi \bigr)_H begin{align*}1ex] = \langle k^\circ, \psi \rangle_{V_0}, \mbox{for any $ \psi \in V_0 $.}\end{array}
abt\left[w_{1}w_{2}\right]_{q}\sum_{i=0}^{w_{3}-1}q^{2w_{1}w_{2}i}e^{\left[w_{1}w_{2}i\right]_{q}abt}+\left(q-1\right)\left[w_{1}w_{2}\right]_{q}\sum_{i=0}^{w_{3}-1}q^{w_{1}w_{2}i}e^{\left[w_{1}w_{2}i\right]_{q}abt}.
( \mathrm{M a s s})^{2}={\frac{1}{4 \lambda_{2}^{( 0 )}}}\left[{\frac{1}{2}}( \hat{\alpha}_{L})^{2}+ N_{L}- 1 \right] ={\frac{1}{4 \lambda_{2}^{( 0 )}}}\left[{\frac{1}{2}}( \hat{\alpha}_{R})^{2}+ N_{R}- \delta \right] .
\langle a \rangle(x,\xi) = \lim_{T\to\infty}\frac{1}{T}\int_0^T a\circ\Phi_{t\omega}^H(x,\xi) dt=\int_{\mathbb{T}_\omega}a\circ\Phi_\tau^H(x,\xi)\nu_\omega(d\tau)\in\mathcal{C}^\infty(\mathbb{R}^{2d}).
\lim_{n \rightarrow \infty}T^{-1}T \left( \sum_{k=0}^{\infty}\alpha_{k}e_{k}\otimes \bar{e}_{k}\right) T^{-1}\xi_{n}=\lim_{n \rightarrow \infty}\left( \sum_{k=0}^{\infty}\alpha_{k}e_{k}\otimes \bar{e}_{k}\right) T^{-1}\xi_{n}= T^{-1}\eta,
k_{1,\lambda}(y',y_d) = \int_{\mathbb R}\frac{1}{2\omega_\lambda(\xi)}e^{-\omega_\lambda(\xi)|y_d|}d\xi &= \int_{\mathbb R}e^{iy'\cdot \xi}\frac{1}{2\omega_\lambda(\xi)}e^{-\omega_\lambda(\xi)|y_d|}\chi_R(\xi)d\xi & + \int_{\mathbb R}e^{iy'\cdot \xi}\frac{1}{2\omega_\lambda(\xi)}e^{-\omega_\lambda(\xi)|y_d|}(1-\chi_R(\xi))d\xi &=I+II.
\sum_{1\leq i<j\leq n}\left| \begin{array}{cc}\alpha d_i (1-\alpha)a_{i,j}(1-\alpha)a_{j,i}\alpha d_j\end{array}\right|=\sum_{1\leq i<j\leq n}\left| \begin{array}{cc}\alpha d_i (1-\alpha)a_{i,j}(1-\alpha)a_{j,i}\alpha d_j\end{array}\right|,
A_{\cal F}=-\tilde{A}^TK\tilde{A}-\frac{1}{4}\tilde{A}^TZ\tilde{A}- \tilde{A}^TJ\tilde{A}+4{\cal M}^{-1\AB}_{\mu\nu}\partial_\rho{\cal H}_B^{\rho\nu}\partial_\sigma{\cal H}_A^{\sigma\mu}
\left. \frac{\partial U_{4}}{\partial v}\right|_{v = v_{s}}= - \frac{1}{v_{s}^{3}}+ v_{s}+ \frac{3 u v_{s}^{3}}{1 6 \pi^{2}}[ \mathrm{l n}\frac{v_{s}^{2}u}{M^{2}}- \frac{1}{2}] = 0 .
\left[\int_{\mathbb{R}}\frac{1+t^{2}}{|1-tG_{\mu_{1}}(F_{\rho_{1}}(x_{0}))|^{2}}\,d\sigma(t)\right]\left[\int_{\mathbb{R}}\frac{d\mu_{1}(t)}{(F_{\rho_{1}}(x_{0})-t)^{2}}\right]=1.
E(Az) &=\left(\begin{array}{ccccccc}\Delta_0E(\lambda_{1},z)v_{1,l_{1,1}}^1 \!\!\!\cdots\!\!\! \sum_{h=1}^{l_{1,1}}\Delta_{l_{1,1}-h}E(\lambda_{1},z)v_{1,l_{1,1}}^{l_{1,1}}\!\!\!\cdots\!\!\! \Delta_0E(\lambda_{k},z)v_{k,l_{k,q_k}}^1 \!\!\!\cdots\!\!\! \sum_{h=1}^{l_{k,q_k}}\Delta_{l_{k,q_k}-h}E(\lambda_{k},z)v_{k,l_{k,q_k}}^{l_{k,q_k}}\end{array}\right)P^{-1},
\left(1 - q^{3n}\right) U_\S(n) = q^{3n-2}\left(1 + q + q^2\right) U_\S(n-1) &- q^{6n-6}\left(1 + q + q^2 + q^{3n-3}\right) U_\S(n-2) + q^{9n-12}U_\S(n-3),
V( X^n_t , \alpha^n_t)- V( X^n_0, \alpha^n_0) = \int_{0}^{t}L^nV( X^n_s , \alpha^n_s)ds + \int_{0}^{t}\nabla V \sigma^n(s, X^n_{s}, \mathcal{L}_{X^n_{s}}, \alpha^n_s)dW_s & +\int_{0}^{t}\int_{\mathbb R}V( X^n_s, \alpha_0+ h( X^n_s, \alpha^n_s, z)) - V( X^n_s, \alpha^n_s)\mu(ds,dz),
=\left( \begin{array}{cc}(\int_M dU_{\beta_k}\wedge*dU_{\beta_j}) (- \int_M dU_{\beta_k}\wedge*dU_{\alpha_j}) ( -\int_M dU_{\alpha_k}\wedge*dU_{\beta_j}) ( \int_M dU_{\alpha_k}\wedge*dU_{\alpha_j}) \end{array}\right)
\lim_{\left(\frac{1}{2}\epsilon\xi^{2}k_{X}+ \epsilon\xi^{2}R_{\psi}(X-\epsilon\xi,T),\epsilon\xi\right)\rightarrow (0,0)}R_{i}\left(\theta - k\xi + \frac{1}{2}\epsilon\xi^{2}k_{X}+ \epsilon\xi^{2}R_{\psi}(X-\epsilon\xi,T),X - \epsilon\xi,T\right) = 0
\ell\left( t\right) =\frac{M^{\left( \alpha\right)}\left( t;\mathbf{F}\right) -U^{\left( \alpha\right)}}{\mathbf{A}_{0}\left( t\right)}-\mu_{\alpha}^{\left( 1\right)}\frac{\left( M^{\left( 1\right)}\left(t;\mathbf{F}\right) \right)^{\alpha}-\left( U^{\left( 1\right)}\right)^{\alpha}}{\mathbf{A}_{0}\left( t\right)}.
q_{2,\infty}^*(\mathcal F_2^c)\triangleq \max_{\mathbf{T}}q_{2,\infty}\left(\mathcal F_2^c,\mathbf{T}\right) s.t. 0\leq T_n\leq 1,\n\in \mathcal F_2^c, &\quad\sum_{n\in \mathcal F_2^c}T_n=K_2^c,
\Re(f(\zeta))-\Re(\zeta) \ge \Re(\beta) -\frac{1}{\Re\left(\zeta\right)\cdot\log\left(\Re\left(\zeta\right)\right)\cdots(\log^{\circ k}(\Re(\zeta))){}^{1+\varepsilon}}& =\rho_{\beta ,\varepsilon,k}^{-}\left(\Re\left(\zeta\right)\right) ,
\Delta_{\Sigma}g(z) = \gamma \Sigma_{00}(z) \left( \frac{2b'(x)^2}{b(x)^3}- \frac{b''(x)}{b(x)^2}\right) \| y \|^2_d + \frac{2\gamma}{b(x)}\sum_{i=1}^d \Sigma_{ii}(z) - 4\frac{b'(x)}{b(x)^2}\sum_{i=1}^d y_i \Sigma_{0i}(z).
\frac{(n-1)(n-3)}{4}\frac{62}{315}\int_{-\pi/2}^{\pi/2}x^8\sin^2(2x) = \frac{(n-1)(n-3)}{4}\frac{31\pi(2835-1890\pi^2+378\pi^4-36\pi^6+2\pi^8)}{1451520}&\approx \frac{(n-1)(n-3)}{4}0.36024.
\begin{array}{l}{{\mu_{+}x^{+}= q^{2}x^{+}\mu_{+}~ , ~ ~ \mu_{+}\partial_{+}= q^{- 2}\partial_{+}\mu_{+}~ ,}}\\{{\Lambda x^{k}= q^{2}x^{k}\Lambda ~ , ~ ~ \Lambda \partial_{k}= q^{- 2}\partial_{k}\Lambda ~ , \mathrm{f o r}~ k = 0 , \pm ~ .}}\\ \end{array}
|W_c(\mu, \nu) - W_c(\tilde \mu, \tilde \nu)| = \left( W_c( \mu, \nu_t)^{\frac{1}{2}}+ W_c(\tilde \mu, \tilde \nu)^{\frac{1}{2}}\right)\left| W_c(\mu, \nu)^{\frac{1}{2}}- W_c(\tilde \mu,\tilde \nu)^{\frac{1}{2}}\right|.
e^{R}( f ) ( g ) = \frac d{d s}{\Big |}_{s = 0}f ( \exp ( s e ) \cdot g ) = \frac d{d s}{\Big |}_{s = 0}\breve{f}( g^{- 1}\cdot \exp ( - s e ) ) = e^{L}( \breve{f}) ( g^{- 1}) \ ,
k_1=C_1=C\otimes 1\otimes 1,k_2=C_2=1\otimes C\otimes 1, k_3=C_3=1\otimes 1\otimes C,k_4=C_{123}=\delta^{(3)}(C), begin{align*}0.5em] X=C_{12}=\delta^{(2)}(C)\otimes 1,Y=C_{23}=1\otimes \delta^{(2)}(C)\ .
\mathbb{P}\Bigg(\max_{m\in\mathcal{M}_n}\Big|\Big|Q^{(\mathsf{B}_n)}_{I,\mathbf{U}}Q^{(\mathsf{B}_n)}_{\mathbf{Z}|M=m,I,\mathbf{U}}-Q^{(\mathsf{B}_n)}_{I,\mathbf{U}}Q^n_{Z|U}\Big|\Big|_{\mathsf{TV}}>e^{-n\gamma_1}\Bigg)\leq e^{-e^{n\gamma_2}}.
\begin{bmatrix}R_{s,\mathsf{k}}+\omega_0 j L_{s,\mathsf{k}}(\theta_\mathsf{k}) \omega_0 j \mathrm{r}(\theta_\mathsf{k}) l_{sf,\mathsf{k}}0 R_{f,\mathsf{k}}\end{bmatrix}\begin{bmatrix}i_{s,\mathsf{k}}i_{f,\mathsf{k}}\end{bmatrix}= \begin{bmatrix}v_\mathsf{k}v_{f,\mathsf{k}}\end{bmatrix}.
\sup_{x \geq t}\left(x^{3/2}|e'''(x)|\right) = y(t)^{3/2}|e'''(y(t))| \leq \frac{C}{y(t)^{3/2}\log^{b+1}(y(t))}+ \frac{C \sup_{x \geq y(t)}\left(x^{3/2}|e'''(x)|\right)}{\sqrt{\log(\log(y(t))}}&\leq \frac{C}{t^{3/2}\log^{b+1}(t)}+ \frac{C \sup_{x \geq t}\left(x^{3/2}|e'''(x)|\right)}{\sqrt{\log(\log(t))}}
P(S_n'(cn)\leq \lfloor n-p_n^{-1}\rfloor) =P(n-S_n'(cn)\geq n-\lfloor n-p_n^{-1}\rfloor) &=P(\mathrm{Bin}(n,1-\pi_n(cn))\geq n-\lfloor n-p_n^{-1}\rfloor) &=P(\mathrm{Bin}(n,1-\pi_n(cn))\geq\lceil p_n^{-1}\rceil) &\leq\mathrm{e}^{-\frac{\lceil p_n^{-1}\rceil}{2}\log\left(\frac{\lceil p_n^{-1}\rceil}{n(1-\pi_n(cn))}\right)}.
\|u_{\varphi(n)}(\delta)-u_{\varphi(n)}(0)\|^{2}_{L^{2}}=\|u(\delta)\|^{2}_{L^{2}}+\|u(0)\|^{2}_{L^{2}}-2Re \langle u_{\varphi(n)}(\xi)-u_{\varphi(n)}(0) \rangle_{L^2}.
\begin{cases}\Delta^2u=\mu\rho u,{\rm in\}\Omega, \frac{\partial^2u}{\partial\nu^2}=0,{\rm on\}\partial\Omega,{\rm div}_{\partial\Omega}(D^2u\cdot\nu)+\frac{\partial\Delta u}{\partial\nu}=0,{\rm on\}\partial\Omega,\end{cases}
PV(y) = \sum_{x \in \mathcal{X}}P(x) V(y|x) = \sum_{x \in \mathcal{X}}\frac{P(x) W(y|x)^{1-s}PV(y)^s}{\displaystyle \sum_{\tilde{y}\in \mathcal{Y}}W(\tilde{y}|x)^{1-s}PV(\tilde{y})^s},
X^{\mu}( \sigma , \tau ) = q^{\mu}+ 2 \alpha^{\prime}p^{\mu}\tau + i \sqrt{\alpha^{\prime}}\sum_{n \neq 0}\frac{1}{n}[ \alpha_{n}^{\mu}e^{- i n ( \tau - \sigma )}+ \tilde{\alpha}_{n}^{\mu}e^{- i n ( \tau + \sigma )}]
\mathbb{P}(X_{r}>T_{(k)}) =\sum_{j=0}^{r-1}(-1)^{j}\frac{s^{j}}{j!}f^{(j)}_{n,k}(s) &=\sum_{j=0}^{r-1}(-1)^{j}\frac{s^{j}}{j!}g^{(k)}_{n,k}(s),
\begin{aligned}\nabla_k U_{ij}\equiv \, \nabla U (e_i, e_j, e_k) = \nabla_{kij}u + \nabla_k A^{ij}(x, u, \nabla u) \equiv \, \nabla_{kij}u + \nabla'_k A^{ij}(x, u, \nabla u) + A^{ij}_z (x, u, \nabla u) \nabla_k u + A^{ij}_{p_l}(x, u, \nabla u) \nabla_{kl}u \end{aligned}
c_2 = (1+e^{h\sigma})\left( \zeta(1-e^{h\sigma}) + 1+e^{h\sigma}\right)-2\zeta h\sigma e^{h\sigma}\;, c_1 = 2(1+e^{h\sigma})\left( \zeta(1+e^{h\sigma}) + 1-e^{h\sigma}\right)a_\sigma-4\zeta a e^{h\sigma}\;, c_0 = \left( (1-e^{h\sigma})\left( \zeta(1+e^{h\sigma}) + 1-e^{h\sigma}\right) + 2\zeta h\sigma e^{h\sigma}\right)a_{\sigma}^2 \;,
(\frac{\Gamma_{e}}{\delta x_{PE}}A_{e}+\frac{\Gamma_{w}}{\delta x_{WP}}A_{w}-S_{p})\phi_{P}=(\frac{\Gamma_{w}}{\delta x_{WP}}A_{w})\phi_{W}+(\frac{\Gamma_{e}}{\delta x_{WP}}A_{e})\phi_{E}+S_{u}
|\vec{q}_1|= \frac{|\left<\vec{q}_1,A\vec{q}\right>|}{|A\vec{q}|}= \frac{|\left<\tilde{\vec{q}}-\sqrt{N}a_{\infty}\vec{q},A\vec{q}\right>|}{|A\vec{q}|}\leq \frac{m^2}{|A\vec{q}|}\ll_{\Omega}\frac{m^2}{|\vec{q}|}.
\int_I|g\cdot b-f\cdot a| \leq \int_I|g\cdot b-f\cdot b|+\int_I|f\cdot b-f\cdot a| &= \int_I|g-f||b|+\int_I|f||b-a| &\leq \|b\|\int_I|g-f|+\|b-a\|\int_I|f| &\leq (\|a\|+\delta)\delta+\delta\|f\|_{L^1}<\epsilon.
K(\varepsilon ;\vartheta ;\zeta):=\left( [\Lambda^\varepsilon ]b(\mathbf{D})+\vartheta [\widetilde{\Lambda}^\varepsilon ]\right) S_\varepsilon (B^0(\vartheta )-\zeta \overline{Q_0})^{-1},\zeta \in \mathbb{C}\setminus \mathbb{R}_+.
\gamma_{k}^{2}\wedge\omega^{k-1}=-k.(k-1)!(1+f'(s))^{k-2}f''(s)\omega_{\Sigma_{1}}\wedge\dots\wedge\omega_{\Sigma_{k}}\wedge\frac{\sqrt{-1}dw\wedge d\bar{w}}{\lvert w\rvert^{2}}&=-k!(1+f'(s))^{k-2}f''(s)\omega_{\Sigma_{1}}\wedge\dots\wedge\omega_{\Sigma_{k}}\wedge\frac{\sqrt{-1}dw\wedge d\bar{w}}{\lvert w\rvert^{2}}
\int_{(j\!-\!1)h}^{\tau}\! \int_{\mathbb{T}^3}\!\!\biggl(\frac{1}{h}D^j \!\biggr) \, dx \, dt\!\geqslant \! a(\!\tau\!) C_D \!\!\int_{\mathbb{T}^3}\!\!|{\delta\Theta}^j|^2 \!+\!\bigl(|F^{j-1}|^{p-2}\!+\!|F^j|^{p-2}\bigr)|\delta F^j|^2 \,dx \! \geqslant \! 0
U=\left( \begin{array}{cccc}m_{1}1 m_{2}1 \ddots \ddots\end{array}\right) ,\ \L=\left( \begin{array}{ccccc}1 l_{11}1 \vdots \ddots \multicolumn{1}{l}{\ddots}\multicolumn{1}{l}{}l_{d1}\cdots l_{dd}1 0 \ddots \ddots\end{array}\right) .
\theta^{\prime}(x,\overline{y})\equiv \bigvee_{i=0}^{N_{\theta}}\bigvee_{j=0}^{N_{\theta}}[\dim(\theta(\cdot, \overline{y}))=i \; \& \; E(\theta(\cdot, \overline{y}))= j]\rightarrow x = \langle i,j\rangle
\Phi_\Xi (\zeta ,\zeta^{*})=\left\langle \exp (\zeta +\zeta^{*})\mid \exp(\varsigma +\tau^{*})\right\rangle e^{-\left\langle \tau^{*}\mid \varsigma\right\rangle}=e^{\left\langle \zeta \mid \tau^{*}\right\rangle+\left\langle \zeta^{*}\mid \varsigma \right\rangle -\left\langle \tau^{*}\mid \varsigma \right\rangle}
\lim_{k_3\rightarrow0}k_4=\frac{(1-k_1)+\sqrt{(1-k_1)^2-4C}}{2},\ \ \ \ \ \lim_{k_3\rightarrow0}k_5=-\frac{k_2}{2}+\sqrt{\left(\frac{k_2}{2}\right)^2-A}.
u_0 = Q v_0 = (\underbrace{k'_1,..,k'_1}_{k_1},\underbrace{k'_2,\ldots,k'_2}_{k_2-1},0,\underbrace{k'_3,\ldots,k'_3}_{k_3-1},0,\ldots,\underbrace{k'_{\nu},\ldots,k'_{\nu}}_{k_{\nu}-1},0),
\mathcal{W}=\frac{L_{0,0}}{L_{1,1}}+ L_{a,b}+ \sum_{i=1}^{a-1}\sum_{j=1}^b\frac{L_{i,j}}{L_{i+1, j}}+ \sum_{j=1}^{b-1}\sum_{i=1}^a \frac{L_{i,j}}{L_{i,j+1}}.
E_{\xi_{i}}= -T_{i}(E_{i + 1}) E_{\xi_{i + 2}}+ q^{-1}E_{\xi_{i + 2}}T_{i}(E_{i + 1}) = -(-E_{i}E_{i + 1}+ q^{-1}E_{i + 1}E_{i}) E_{\xi_{i + 2}}+ q^{-1}E_{\xi_{i + 2}}(-E_{i}E_{i + 1}+ q^{-1}E_{i + 1}E_{i}).
\widehat P(p)=\left( \begin{array}{ccc}4\,p_1 6\,p_2 8\,p_3 6\,p_2 4\,p_1^2+8\,p_3 2\,p_1p_2 8\,p_3 2\,p_1p_2 3\,p_2^2-8\,p_1p_3 \end{array}\right),\qquad\lambda^{(\det(\widehat P))}(p) = (24,\,0,\,- 8\,p_{1})\,.
\frac{n-(n_1+2n_2+3n_3)}{6}=\frac{n-(\mathrm{fix}_\Lambda(h^2)+\mathrm{fix}_\Lambda(h^3)-\mathrm{fix}_\Lambda(h))}{6}> \frac{n}{6}-\frac{2n}{6(q^2-q+1)}\ge \frac{q^3+1}{6}-\frac{q+1}{3}\ge 7,
S_{\mathrm{h y p}}( \epsilon_{\kappa}) \stackrel{0 \leftarrow \tau / \sigma}{\longleftarrow}S_{\mathrm{c a t e n a r y}}( \epsilon_{\sigma}) \stackrel{\tau / \sigma \rightarrow \infty}{\longrightarrow}S_{\mathrm{3 / 2 - p a r a b}}( \epsilon_{\kappa}) .
\begin{gathered}\tilde{\eta}^{h,x}(t)=\int_{0}^{t}\Pi(t,s)\sigma'(X(s,x)).e^{sA}h dW(s), \zeta^{h,x}(t)= \int_0^t \Pi(t,s)\sigma''(X(s,x))\cdot (\eta^{h,x}(s),\eta^{h,x}(s)) dW(s),\end{gathered}