harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\hat{h}= \frac{\hbar^{2}}{2}\left( - \frac{1}{\mu ( q )}\Delta + \frac{\alpha}{\mu^{2}c^{2}}C_{1}\frac{{\bf q}}{q^{3}}\frac{\partial}{\partial{\bf q}}+ \frac{4 \pi \alpha \lambda}{\mu^{2}c^{2}}C_{1}\delta ({\bf q}) \right) + v ( q ) ,
T_n(t) =\frac{(-1)^n\sqrt{\pi}\sqrt{1-t^2}}{2^n(n-\frac{1}{2})!}\frac{d^n}{dt^n}\left(\left(1-t^2\right)^{n-\frac{1}{2}}\right), U_n(t) =\frac{(-1)^n(n+1)\sqrt{\pi}}{2^{n+1}(n+\frac{1}{2})!\sqrt{1-t^2}}\frac{d^n}{dt^n}\left(\left(1-t^2\right)^{n+\frac{1}{2}}\right).
\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)=\frac{1}{s-1}-\frac{1}{s}+\frac{1}{2}\int_{0}^{1}(\theta(it)-t^{-\frac{1}{2}})t^{\frac{s}{2}-1}dt+\frac{1}{2}\int_{1}^{\infty}(\theta(it)-1)t^{\frac{s}{2}-1}dt.
Z_{\; \; \alpha_{L}}^{\alpha_{L - 1}}Z_{\; \; \alpha_{L + 1}}^{\alpha_{L}}= C_{\alpha_{L + 1}}^{\alpha_{L - 1}\beta_{0}}\frac{\delta S_{0}}{\delta \Phi^{\beta_{0}}}, \; \alpha_{L + 1}= 1 , \cdots , M_{L + 1},
J(n,m):=\int_{1}^{\infty}\frac{arcoth^{n}x}{x^{m}}dx=\int_{1}^{\infty}log^{n/2}(\frac{x+1}{x-1})\frac{dx}{x^{m}}=\int_{0}^{\infty}z^{n}\frac{tanh^{m-2}z}{cosh^{2}z}dz.
\Omega_{D}^{k}({\cal A}) = \left( \begin{array}{c c}{\Lambda^{k}( X )}&{0}\\{0}&{\Lambda^{k}( X )}\\ \end{array}\right) + \left( \begin{array}{c c}{0}&{\Lambda^{k - 1}( X )}\\{\Lambda^{k - 1}( X )}&{0}\\ \end{array}\right) .
\|\sigma(u,t) \|_{\mathcal{C}_h}\leq C \sum_{c \in \mathcal{T}_h}\| (u - u_c)^2\|_{L^1(c)}= C \sum_{c \in \mathcal{T}_h}\| (u - u_c)\|^2_{L^2(c)}&\leq C \sum_{c \in \mathcal{T}_h}h^2 \, |u|^2_{H^1(c)}= C \, h^2 \, |u|^2_{H^1(\Omega)}.
N_A(s,t)= N\big[{\bf \Sigma^s}({\bf I^s_l}(A)),{\bf \Omega_{u}^s}({\bf I^s_l}(A)),{\bf \Omega_{v}^s}({\bf I^s_l}(A))\big](s) &\times N\big[{\bf \Sigma^t}({\bf I^t_l}(A)),{\bf \Omega_{u}^t}({\bf I^t_l}(A)),{\bf \Omega_{v}^t}({\bf I^t_l}(A))\big](t).
- \frac{1}{4}( \partial \phi )^{2}+ \frac{h^{2}}{4}( \partial C )^{2}e^{- \alpha \phi - 2 q A + 2 C}- \frac{k^{2}}{4}( \partial \chi )^{2}e^{2{\tilde{d}}B + \beta \phi + 2 \chi},
( - 1 )^{F}= ( - 1 )^{1 + \sum_{j = 2}^{n}\sum_{r > 0}( b_{- r}^{- j}b_{r , j}+ b_{- r}^{j}b_{r , - j}) + \sum_{j = n + 1}^{5}\sum_{n > 0}( b_{- n}^{- j}b_{n , j}+ b_{- n}^{j}b_{n , - j})}\cdot \! \! \! \prod_{k = n + 1}^{5}( b_{0}^{k}b_{0}^{- k}- b_{0}^{- k}b_{0}^{k})
\begin{array}{cc}\displaystyle{\langle V^{A}(z_{1})V^{B}(z_{2})\rangle =C_{V}\frac{I^{AB}(\hat{{\cal X}}_{12})}{(\det{\cal X}_{12})^{\frac{1}{2}\eta}}}~~~~ ~~~~I^{AB}(\hat{{\cal X}}_{12})=\textstyle{\frac{1}{4}}\mbox{tr}(\gamma^{A}\hat{{\cal X}}_{12}\gamma^{B}\hat{{\cal X}}_{21})\end{array}
\mathbf{\Sigma}_{\mu}:=\frac{\hbar}{i}\Sigma^{\alpha}\!_{\mu}(q)\frac{\partial}{\partial q^{\alpha}},\widehat{\mathbf{\Sigma}}_{\mu}:=\frac{\hbar}{i}\widehat{\Sigma}^{\alpha}\!_{\mu}(q)\frac{\partial}{\partial q^{\alpha}}.
M^{- 1}( x ) = \frac{1}{2 \pi^{2}r}\left[ \frac{\pi}{2}\cos ( \frac{e^{2}r}{8}) + \mathrm{c i}( \frac{e^{2}r}{8}) \sin ( \frac{e^{2}r}{8}) - \mathrm{s i}( \frac{e^{2}r}{8}) \cos ( \frac{e^{2}r}{8}) \right]
E + \frac{1}{2 m}\left( c^{2}r^{2}- n c \right) = k \frac{\left( n - 1 \right) !}{\left( n - d \right) !}\left( \frac{N}{2 c}\right)^{d - 1}\left( d N r^{2}+ \frac{n N}{2 c}\right)
[\vec w]_{A_{\vec p, \vec r}}\le \Big[w^{\frac{\delta_{m+1}}{p}}\Big]_{A_{\frac{1-r}{r}\delta_{m+1}}}^{\frac1{\delta_{m+1}}}\prod_{i=1}^{m}\Big[w_i^{\frac{\theta_i}{p_i}}\Big]_{A_{\frac{1-r}{r}\theta_i}}^{\frac1{\theta_i}}
[\omega_1|\ldots|\omega_{k-1}|df|\omega_{k+1}|\ldots|\omega_n]=D_{x_0}([\omega_1|\ldots|\omega_{k-1}]\otimes f\otimes [\omega_{k+1}|\ldots|\omega_n]) +[\omega_1|\ldots|\omega_{k-1}|p^*(f)\omega_{k+1}|\ldots|\omega_n]-[\omega_1|\ldots|\omega_{k-1}p^*(f)|\omega_{k+1}|\ldots|\omega_n] \in \mathrm{im}(D_{x_0})+F_{n-1}\mathrm{Sh}(\Omega(C))
\hat\tau = \frac{z_--m}{z_--y_+}\delta_{y_+}+ \frac{m - y_+}{z_--y_+}\delta_{z_-},\;\;\;\;\;\; \check\tau = \frac{z_+-m}{z_+-y_-}\delta_{y_-}+ \frac{m - y_-}{z_+-y_-}\delta_{z_+}s =\pi_{x_-}(\{y_+\}) + \pi_{x_-}(\{z_-\})=\overline \pi_{x_-}^\star(\{y_-\}) - \pi_{x_-}(\{y_-\}) + \overline \pi_{x_-}^\star(\{z_+\}) - \pi_{x_-}(\{z_+\})>0.
\partial_{M}J^{M}( x , y ) = \frac{1}{9 6 \pi^{2}}\left[ \mathrm{T r}\left( Q_{L}( y ){\cal F}_{L}( x ) \tilde{\cal F}_{L}( x ) \right) - \mathrm{T r}\left( Q_{R}( y ){\cal F}_{R}( x ) \tilde{\cal F}_{R}( x ) \right) \right] \; ,
g_{q}(1) =q_{0}+q_{1}i+q_{2}j+q_{3}k, g_{q}(i) =-q_{1}+q_{0}i+q_{3}j-q_{2}k, g_{q}(j) =q_{2}+q_{3}i+q_{0}j+q_{1}k, g_{q}(k) =q_{3}-q_{2}i-q_{1}j+q_{0}k.
\tilde{F}(x_1,\cdots,x_{k-1},x_k)=\prod_{i=1}^{k-1}\left(\xi^{0}_{i+1}-\xi_i^0\right)\partial_2\cdots \partial_k \tilde{F}(\xi_1^0,\xi_2^1,\cdots,\xi^1_{k-1},\xi^{1}_k).
2 \sum_{i = 1}^{N}\delta \alpha_{i}\, \lambda_{i}= \delta \left[ \sum_{i = 1}^{N}( \alpha_{i}- 1 ) \sum_{j \neq i}2 \pi ( \alpha_{j}- 1 ) G ( \omega_{i}- \omega_{j}| \tau ) + \sum_{i = 1}^{N}( \alpha_{i}- 1 )^{2}\log | 2 \pi \eta^{2}| \right] \, ,
\phi{}_{j,k}-\phi{}_{j,k+\ell}=\frac{Q_{j,k+1}}{\prod_{i=j}^{k}\varsigma_{i}}\left(1-\frac{Q_{k+1,k+\ell+1}\mathbf{1}_{\mathbb{R}^{p}}}{\prod_{i=k+1}^{k+\ell}\varsigma_{i}}\right).
\Big( \hbar'(p - i - 1) - (a_s - a_{s'}) \Big) \times \dfrac{1}{\hbar'(p - i) - (a_s - a_{s'})}= \dfrac{\hbar'(p - l_{i}^{(s')}- 1) - (a_s - a_{s'})}{\hbar'(p - l_{i}^{(s')}) - (a_s - a_{s'})}.
\phi(H_{2k+1}) = \phi(T_k)\left(\phi(T_{k+1}) - \phi(T_{k-1})\right) begin{align*}.2em] = \phi(T_k)\left(\phi(P_2)\phi(P_{k-1}) - \lambda\phi(P_2)\phi(P_{k-4}) - \phi(P_2)\phi(P_{k-3}) + \lambda\phi(P_2)\phi(P_{k-6})\right) begin{align*}.2em] = \phi(P_2)\phi(T_k)q(\lambda).
f(B_1, B_2^0, B_2^1) = \frac{1}{2M}\left( -B_2^0 \log B_2^0 - B_2^1 \log B_2^1 \right. \left. - (B_1 - B_2^1) \log (B_1 - B_2^1) \right. & \left. - (M - B_1 -B_2^0) \log(M - B_1 -B_2^0) \right. \left. + B_1 \log B_1 + (M-B_1) \log(M-B_1)\right).
\sup_{0<|h|<h_0}\left\|\frac{\delta_h \psi}{|h|^{\alpha}}\right\|_{L^p(B_R)}^p \le C\,(1-\alpha)\,[\psi]^p_{W^{\alpha,p}(B_{R+h_0})}&+C\,\left[\left(1+\frac{R}{h_0}\right)^p\,(R+h_0)^{-\alpha\,p}+\frac{h_0^{-\alpha\,p}}{\alpha}\,\right]\,\|\psi\|^p_{L^p(B_{R+h_0})},
\Big| \sum_{n=N}^{M}\frac{\varepsilon_{n}\lambda_{n}}{n^{\sigma}}\Big| \leq \Big\| \sum_{n=N}^{M}\frac{\varepsilon_{n}a_{n}}{n^{\sigma}}+ x_{0}\sum_{n=N}^{M}\frac{\lambda_{n}\varepsilon_{n}}{n^{\sigma}}\Big\| = \Big\| \sum_{n=N}^{M}\varepsilon_{n}\frac{a_{n}+ \lambda_{n}x_{0}}{n^{\sigma}}\Big\| \,.
y(x)\approx c_{1}Q^{-\frac{1}{4}}(x)exp[\frac{1}{\epsilon}\int_{x_{0}}^{x}\sqrt{Q(t)}dt]+c_{2}Q^{-\frac{1}{4}}(x)exp[-\frac{1}{\epsilon}\int_{x_{0}}^{x}\sqrt{Q(t)}dt].
|X_2| \ll 2^{2T}\sum_{\frac{ea^2}{L^2}< x_1 < \sqrt{a+dL}}2^{-\omega(x_1)}\cdot \frac{d^3L^3}{x_1a\phi(d)^3}(\log x_1)^{-\frac12}(\log(a+dL))^{-1}= \frac{d^3L^3}{a\phi(d)^3}2^{2T}(\log(a+dL))^{-1}\sum_{\frac{ea^2}{L^2}< x_1 < \sqrt{a+dL}}2^{-\omega(x_1)}\frac{(\log x_1)^{-\frac12}}{x_1}.
(\nabla_{X,Y,t}f, \nabla_{X,Y,t}^2 f)(\tilde X^k_j, \tilde Y^k_j, \tilde t^k_j) = (-\xi_k, -E_k), (\nabla_{X,Y,t}g, \nabla_{X,Y,t}^2 g)(X^k_j, Y^k_j, t^k_j) = (\eta_k, H_k).
Y\sim Pa(\alpha,x_{m})=\frac{\alpha x_{m}^{\alpha}}{y^{\alpha+1}}(y\ge x_{m})\Leftrightarrow X\sim iPa(\alpha,x_{m})=Power(x_{m}^{-1},\alpha)=\frac{\alpha x^{\alpha-1}}{(x_{m}^{-1})^{\alpha}}(0<x\le x_{m}^{-1})
\begin{array}{ll}\{c_{f,p},c_{g,v}\}_r(h_\lambda)=0, h_\lambda\in\frak h, \{c_{f,p},c_{g,v}\}_r(k_\lambda)=0, k_\lambda\in\frak k, \{c_{f,p},c_{g,v}\}_r(x_\alpha)=(\alpha|\rho)f(x_\alpha p)g(v)-(\alpha|\eta)f(p)g(x_\alpha v), \alpha\in\bf R.\end{array}
\sum_{\substack{i,j=1 begin{align*}1pt] i < j}}^n \left[\theta_0^i(\delta^i)^{t+\tau}\right]\left[\theta_0^j(\delta^j)^{t+\tau}\right]\left[(\delta^i)^{\Delta{t}}-(\delta^j)^{\Delta{t}}\right]\left[(\delta^i)^{\Delta\tau}-(\delta^j)^{\Delta\tau}\right].
\int_{-\infty}^\infty K_{t}(x,y)W(x)\mathrm{d}x = \sum_{k=0}^\infty t^{-k-1}\int_{\infty}^\infty h_k(x) \, W(x) dx \, h_k(y) &= \sum_{k=0}^\infty t^{-k-1}2^{-1/4}\langle h_k,h_0 \rangle h_k(y) = t^{-1}W(y)\, .
\phi(\bigoplus_{I}\sum_{p}(p + \iota_{E}(\omega)) \eta_{I,p}- \nabla_{\omega}(\iota_{E}(\bigoplus_{I}\eta_{I}))) = \bigoplus_{I}\sum_{p \neq -\iota_{E}(\omega)}\eta_{I,p}- \nabla_{\omega}(\phi((\iota_{E}(\bigoplus_{I}\eta_{I})))
\mathfrak{C}\,\,\equiv\,\,\left|A_{\Gamma_{\tau}}\right|_{\max}^{2}\left(\tau_{0}\right)\,+\,\max_{Z_{\tau}\in\Gamma_{\tau},\,\left|Z_{\tau}\right|=3\beta}\,\left|A_{\Gamma_{\tau}}\left(Z_{\tau}\right)\right|^{2}\,\,\leq\,\,C\left(n\right)
\mu^{\circ}= \mathcal{C}_{-1}(\alpha, \nu, 1, \varnothing, \lbrace 2, \ldots, \ell \rbrace) - \ell + 1 = \left \lceil \frac{\nu_1 + \sum_{i = 2}^{\ell}\min \lbrace \alpha_1, \alpha_i \rbrace}{\alpha_1}\right \rceil - \ell + 1 = \left \lceil \frac{\nu_1 + \alpha^*_1 + \cdots + \alpha^*_{\alpha_1}}{\alpha_1}\right \rceil - \ell.
{\overline{A}}_{\mu}( x , \beta ) = - \frac{i}{g}e^{i \theta_{a}T^{a}}\left[{\overline{\sigma}}_{\mu \nu}\frac{\rho^{2}( x - x_{0})_{\mu}}{( x - x_{0})^{2}[ ( x - x_{0})^{2}+ \rho^{2}]}\right] e^{- i \theta_{a}T^{a}},
\alpha_{i}= \sqrt{\frac{1}{2}\left( 1 - \eta_{i}\frac{\gamma}{\sqrt{\gamma^{2}+ 1}}\right)}, \hspace{0 . 5 c m}\beta_{i}= \eta_{i}\sqrt{\frac{1}{2}\left( 1 + \eta_{i}\frac{\gamma}{\sqrt{\gamma^{2}+ 1}}\right)}.
\Psi_q(x) =\Psi_{q^2}(qx) \,\Psi_{q^2}(q^{-1}x) &=\Psi_{q^3}(q^2x)\, \Psi_{q^3}(x) \, \Psi_{q^3}(q^{-2}x) =\prod_{j=1}^k \Psi_{q^k}(q^{k-2j+1}x) \;.
\Lambda = \{(s_1,\dots,s_{n-1},p)\in \mathbb G_n \,: \; (s_1,\dots,s_{n-1}) \in \sigma_T(F_1^*+pF_{n-1}\,,\, F_2^*+pF_{n-2}\,,\,\dots\,, F_{n-1}^*+pF_1) \},
\widehat V_h (\sigma,\tau) \, = v_{00}+ h^\frac 23 \, v_{01}\tau + h v_{20}\sigma^2+ h^\frac 7 6 v_{11}\sigma \tau+ h^{\frac 43}v_{02}\tau^2 + h^{\frac 32}v_{30}\sigma^3+ \mathcal O (h^\frac 53)\,.
\frac{W_2^2(\mu_{t+s},\nu)}{2}\ge \int_X \varphi_t \, d\mu_{t+s}- \int_X \psi_t \, d \nu , \frac{W_2^2 ( \mu_t , \nu )}{2}= \int_X \varphi_t \, d \mu_t - \int_X \psi_t \, d \nu .
\tau_{\rm IS}(a(0,1)) =a(1,1)a(1,2)a(1,3),\; \tau_{\rm IS}(a(0,2))=a(2,1) \tau_{\rm IS}(a(1,1)) =a(1,1)a(1,2)a(1,3),\; \tau_{\rm IS}(a(1,2))=a(0,1)a(0,2),\;\tau_{\rm IS}(a(1,3))=a(2,1) \tau_{\rm IS}(a(2,1)) = a(0,1)a(0,2).
|h^{(st)}_{k-i,k-j}| \leq \left(\frac{2}{\pi}\right)^2\max_{x\in[-1,1],y\in[-1,1]}|H_{st}(x,y)| \int_{-1}^1 \int_{-1}^1 \frac{1}{\sqrt{(1-x^2)(1-y^2)}}dxdy =4\max_{x\in[-1,1],y\in[-1,1]}|H_{st}(x,y)| \leq 8(e-1)k^2 \|P(\cdot)\|\|v(\cdot)\|,
\begin{aligned}C_{k}: 2(3^{k-2}) \times 2(3^{k-2}) \mbox{square matrix (i.e., $2\times 2,6\times 6, 18\times 18,\dots$)}A_{k}: 2(3^{k-2}) \times 2(3^{k-1}) \mbox{rectangular wide matrix}B_{k}: 2(3^{k-1}) \times 2(3^{k-2}) \mbox{rectangular tall matrix.}\end{aligned}
\frac{16}{\lambda(t)}\int_{0}^{\infty}\frac{R^{3}}{(1+R^{2})^{3}}\int_{t+6 R \lambda(t)}^{\infty}ds \lambda''(s) \left(\frac{s-t}{1+(s-t)^{2}}-\frac{(s-t)}{\lambda(t)^{2-2\alpha}+(s-t)^{2}}\right) &=\frac{16}{\lambda(t)}\int_{t}^{\infty}K_{3}(s-t,\lambda(t)) \lambda''(s) ds
T_{7}=-\frac{2\pi i}{c\left( cz+d\right)}kB_{0}(B)\sum\limits_{v=0}^{k-1}f(-cv)P_{2}\left( \frac{cv+cR_{2}-mkR_{1}}{k}+m\left[ R_{1}\right]\right) & =-\frac{2\pi i}{c\left( cz+d\right)}B_{0}(B)P_{2}\left( cR_{2}-dR_{1},A\right) .
\Omega_{i}= \frac{l_{i}}{( l_{i}^{2}+ r_{H}^{2})}\frac{r_{0}^{3 / 2}}{( \pi N )^{1 / 2}}\sp J_{i}= \frac{4}{3}\frac{V_{5}( \pi N )^{1 / 2}}{( 2 \pi )^{6}}r_{0}^{3 / 2}l_{i}
\Gamma_{b}(w+b)=\sqrt{2\pi}\frac{b^{bw-\frac{1}{2}}}{\Gamma(bw)}\Gamma_{b}(w),\Gamma_{b}(w+b^{-1})=\sqrt{2\pi}\frac{b^{-b^{-1}w+\frac{1}{2}}}{\Gamma(b^{-1}w)}\Gamma_{b}(w).
\nabla_{\alpha}v \nabla_{\mu}v\: m_{\beta}m_{\nu}R^{\alpha\beta\mu\nu}\biggl|_{{\cal I}^+}=-\: \frac{8}{r}\:\Bigl(\frac{\partial^2}{{\partial u}^2}C_1 +i\frac{\partial^2}{{\partial u}^2}C_2 \Bigr) +O\left(\frac{1}{r^2}\right)
I_1(r)=C_{\nu}e^{i(\frac{\nu\pi}2+\frac{\pi}4)}e^{i\phi_r(N+\sqrt{N})}\int_0^\infty(r\rho)^{-\frac{n-1}2}e^{\frac{ir[\rho-(N+\sqrt{N})]^2}{2(N+\sqrt{N})}}\chi_{N}(\rho)\rho^{n-1}\mathrm{d}\rho.
d \hat{\mu}(e^{i\theta})= \frac{2^{b+\overline{b}+2}|\Gamma(b+2)|^2}{2\pi\, \Gamma(b+\overline{b}+3)}e^{(\pi-\theta)\mathcal{I}m(b+1)}[\sin^{2}(\theta/2)]^{\mathcal{R}e(b+1)}d\theta .
= \sum\limits_{n=0}^{N-1}{f^-_n}(3t)^{-n/3}\int\limits_{-\infty}^{-\mu}\theta^{-n}\mathrm{Ai}(\eta-\theta)d\theta + \int\limits_{-\infty}^{-\mu}R_N(\theta{\root 3 \of{3t}}) \mathrm{Ai}(\eta-\theta) d\theta,
(Gf)(x-v_{i})=T_{w_{x}}^{-1}(T_{\sigma_{x}(i)}^{-1}\cdots T_{\sigma_{x}(i)+d_{i}^{+}(x)-1}^{-1}). \left( \rho(T_{\sigma_{x}(i)+d_{i}^{+}(x)-1}\cdots T_{\sigma_{x}(i)})f\right)\!(w_{y}y).
\left[ g_{\mu \nu}( u ) \right] = \left( \begin{array}{c c}{1}{u^{0}\vec{u}^{\mathrm{T}}}{\hline u^{0}\vec{u}}{- I + \vec{u}\otimes \vec{u}^{\mathrm{T}}( u^{0})^{2}}\end{array}\right) ,
\left[\begin{matrix}\mathbf{a}_1 \cdots \mathbf{a}_{2g}\end{matrix}\right]=\left[\begin{matrix}\mathbf{b}_1 \cdots \mathbf{b}_{2g}\end{matrix}\right]\alpha=\left[\begin{matrix}\mathbf{b}_1' \cdots \mathbf{b}_{2g}'\end{matrix}\right]\alpha'.
\partial_zA_2-A_{3}\partial_z\varphi_0+\bar{A}_{1}A_{3}+ \bar{A}_{2}A_{2}=\partial_{\bar{z}}\bar{A}_2-\bar{A}_{3}\partial_{\bar{z}}\varphi_0+A_{1}\bar{A}_{3}+A_{2}\bar{A}_{2}\,.
\Big((\gamma_j,\gamma'_j)\big|(\boldsymbol{\gamma}_{-j},\boldsymbol{\gamma}'_{-j})=(\boldsymbol{a},\boldsymbol{b})\Big)\sim\mathcal{N}(\boldsymbol{\eta}^{(j)},\sigma^2\,\mathbf{R}^{(j)})\ ,
||F_{6}(t,r)||_{L^{2}(r dr)}^{2}\leq C \int_{\frac{t}{4}}^{\frac{t}{2}}\frac{\lambda(t)^{4}r (v_{4}(t,r)^{2}+v_{5}(t,r)^{2})}{r^{8}}dr + C \int_{\frac{t}{2}}^{\infty}\frac{\lambda(t)^{4}(v_{4}^{2}+v_{5}^{2}) r dr}{r^{8}}&\leq \frac{C}{t^{8}\log^{10b+4N-2}(t)}
\mathbb{E}_{f_0,T}\|\hat{f}-\mathbb{E}_{f_0,T}\hat{f}\|_{\infty}\leq \sum_{j=0}^{j_n}\mathbb{E}_{f_0,T}\max_{k}(|Z_{jk}| +|W_{jk}|)\big\|\sum_{k=0}^{2^j-1}|\psi_{jk}|\big\|_{\infty}&\lesssim \sum_{j=0}^{j_n}2^{j/2}\Big(\sqrt{(\log_2 n)\eta/n}+\sqrt{1/n}\Big) \lesssim \sqrt{2^{j_n}\eta (\log_2 n)/n},
\sup_{x\in\mathbb R^2}(1+|x|)|u(x)| &\leq C(1+|a|^{-1})\left[\int_{\mathbb R^2}(1+|x|)|f(x)|\,dx+\sup_{x\in\mathbb R^2}\,(1+|x|^3)\big(\log\,(e+|x|)\big)|f(x)|\right].
\frac{\mathrm{d}^2\eta(\theta)}{\mathrm{d}\theta^2}=\frac{\mathrm{d}}{\mathrm{d}\theta}\Bigg(-\log(1+\theta)+\frac{1-\theta}{1+\theta}+\log\theta+1\Bigg) &=-\frac{1}{1+\theta}-\frac{2}{(1+\theta)^2}+\frac{1}{\theta}&=\frac{1-\theta}{\theta(1+\theta)^2}&>0,\quad\theta\in(0,1),
\displaystyle\sum\limits_{f=1}^{l_n}b^{st}_{f}p^{r}_{11}p^{n}_{\tilde{f}f}=\sum\limits_{e=1}^{l_{r-1}}\sum\limits_{f=1}^{l_n}b^{ef}_{\tilde{f}}p^{r-1}_{es}p^{n}_{ft}.
\psi_k=(\mathcal{A}^T-\mathrm{I}) e^{\mathrm{i}k x}=\varepsilon\partial_x (\beta_1 e^{\mathrm{i}k x})+O(\varepsilon^2)=\varepsilon(\beta_1)_x e^{\mathrm{i}k x}+\varepsilon\beta_1\,\partial_x e^{\mathrm{i}k x}+O(\varepsilon^2).
\bigg[\partial_{n+1}x_{n+1}^{1-2s}\partial_{n+1}+x_{n+1}^{1-2s}\sum_{j,k=1}^{n}a_{jk}\partial_{j}\partial_{k}\bigg]w =f\quad\mathbb{R}_{+}^{n+1}, w =0\quad\mathbb{R}^{n}\times\{0\}.
r^{*,\pi}_t(x_t|y^{t-1}_{t-M},y_t)=\Big(\frac{q_t(y_t|y^{t-1}_{t-M},x_t)}{\nu^{\pi}_{t}(y_t|y^{t-1}_{t-J})}\Big){\pi}_{t}(x_t|y^{t-1}_{t-J}),~\forall{x_t}\in{\cal X}_t,~t\in\mathbb{N}_0^{n-1}.
\sum_{s_1\geq\dots\geq s_{r-1}\geq0}\frac{q^{s_1^2+\dots+s_{r-1}^2-s_1-\cdots- s_i+s_{r-1}}}{(q)_{s_1-s_2}\dots(q)_{s_{r-2}-s_{r-1}}(q^2;q^2)_{s_{r-1}}}=\sum_{k=0}^{i}\frac{(q^{2r},q^{r-i+2k-1},q^{r+i-2k+1};q^{2r})_\infty}{(q)_\infty},
\begin{aligned}0 =\prod_{\substack{Nv_j=v_{\ell}i\leq\ell\leq h(i-1)}}\left( \prod_{j=j_{\ell}}^{i-1}{d_{(1,\ldots, n),(v_1, v_2,\ldots,v_{h(j)},Nv_j , \ldots v_n)}}\right).\end{aligned}
\mathcal{L}_\omega: 0 \to \underset{|\alpha| = 0}{\oplus}\mathbb{C}_{V_\omega^{(\alpha)}\times K}\to\underset{|\alpha| = 1}{\oplus}\mathbb{C}_{V_\omega^{(\alpha)}\times K}\to\cdots \to\underset{|\alpha| = n}{\oplus}\mathbb{C}_{V_\omega^{(\alpha)}\times K}\to 0,
\ll \Big(\frac{1}{p}\Big)^{- \frac{\dim X}{2}+ M + 1}\sup_{(x,y) \in X \times X}\big|\Delta^{\widetilde{X}}_{\widetilde{y}}v_M(\widetilde{y}, e^{i \theta + 2u}\widetilde{x})\big| \int_0^u \int_X s^{M}G((u - s)/p, x, y)|G(s/p, y, e^{-i \theta + 2u}x) | dv (y)ds.
-m\lambda\log(\tfrac{\lambda}{2}) =\frac{\log(L\log(m))}{L\log(m)}\log\left(\frac{2Lm\log(m)}{\log(L\log(m))}\right) begin{align*}3pt] \leq\frac{(\log(2L\log(m))^2}{L\log(m)}+\frac{\log(L\log(m))}{L}\leq C_4+\frac{\log(L\log(m))}{L}
\begin{gathered}z_0 = 1; z_n = Y^{-s}(1-q^s)^{-1}\left(g_n + \sum_{\stackrel{i + j = n}{i,j>0}}z_j\alpha^j(g_i)\right) \textrm{for}n\geq 1; z: = \sum_{i \geq 0}z_i X^i. \end{gathered}
\left|-\left\langle ABh\partial_x\partial_\tau^3\tilde{h},\partial_\tau^3u\right\rangle-\left\langle ABh\partial_x\partial_\tau^3u,\partial_\tau^3\tilde{h}\right\rangle\right| = \left|\left\langle\partial_x(ABh)\partial_\tau^3\tilde{h},\partial_\tau^3u\right\rangle\right| \le CD(t)^{\frac14}E(t)^{\frac54}.
y_p^u =\mathbf{h}_{pp}^H\mathbf{w}_1{s}_1+\mathbf{h}_{sp}^H\mathbf{w}_2{s}_2+n_p, y_s^u =\mathbf{h}_{ss}^H\mathbf{w}_2{s}_2+\mathbf{h}_{ps}^H\mathbf{w}_1{s}_1+n_s,
\Delta \left(T \curvearrowleft S \right) (p_1, \ldots, p_{n+k}) = n \times T(p_1, \ldots, p_n) \otimes S(p_{n+1}, \ldots, p_{n+k}) & + (T \curvearrowleft S_1) \otimes S_2 (p_1, \ldots, p_{n+k}) + (T_1 \curvearrowleft S) \otimes T_2 (p_1, \ldots, p_{n+k}) &+ T_1 \otimes (T_2 \curvearrowleft S) (p_1, \ldots, p_{n+k}),
\mathcal{R}\;=\;\mathcal{F}(\Delta^{\mathsf{L}}_1+ s)\mathcal{F}^*\;,\mathcal{P}\;=\;\mathcal{F}\Big(\sum_{j=1}^{\mathsf{L}}\omega_j\,|j\rangle\langle j|\Big)\mathcal{F}^*\;\in\;\mathbb{C}^{\mathsf{L}\times\mathsf{L}}\;,
\mu_{Mas}(E^u_\cdot(0),E^s_\cdot(0)) =\mu_{Mas}(\gamma_1)+\mu_{Mas}(\gamma_2)+\mu_{Mas}(\gamma_3) &=\mu_{Mas}(\gamma_2)=\mu_{Mas}(E^u_\cdot(t_0),E^s_\cdot(-t_0)),
\psi_{\beta}^{(2)}\left( 0,x;u_{\beta}^{(2)}\right) =k_{\beta}\left\{-p_{1}8u_{\beta}^{(2)}\left( 1+\frac{1}{2}\beta^{2}u_{\beta}^{(2)2}\right) +p_{2}u_{\beta}^{(2)}-u_{\beta}^{(2)}+O\left( \beta^{4}\right) \right\}
Rf_{!}u:Rf_{!}\mathcal{F}\to Rf_{!}Rc_{1\ast}c_{1}^{\ast}\mathcal{F}=Rf_{!}Rc_{1!}c_{1}^{\ast}\mathcal{F}\;\;\;\cong Rf_{!}Rc_{2!}c_{1}^{\ast}\mathcal{F}\overset{u}{\to}Rf_{!}\mathcal{F}
C_{3.1}= \frac{\mathfrak{Q}_{1}\mathfrak{Q}_{2}}{\mathfrak{R}}\sup_{m \in \mathbb{R}}(1+|m|)^{\mu - \mathrm{deg}(R)}\int_{-\infty}^{+\infty}\frac{1}{(1+|m-m_{1}|)^{\mu - \mathrm{deg}(Q_{1})}(1+|m_{1}|)^{\mu - \mathrm{deg}(Q_{2})}}dm_{1}
m_{n , \pm}^{2}= -{\cal M}_{0}^{2}( \alpha_{P}, R ) + \left( \frac{n}{R}\right)^{2}\pm \sqrt{4 \left( \frac{n \alpha_{P}}{R^{2}}\right)^{2}+{\cal M}_{0}^{4}( \alpha_{P}, R )}.
\begin{aligned}J_1= \int_0^\infty [(h+h_e\phi(y))\partial_x+g\partial_y]D^\alpha h\cdot\langle y\rangle^{2(l+k)}D^\alpha u\mathrm{d}y &+\int_0^\infty [(h+h_e\phi(y))\partial_x+g\partial_y]D^\alpha u\cdot \langle y\rangle^{2(l+k)}D^\alpha h\mathrm{d}y.\end{aligned}
\quad\quad\quad\frac{\partial}{\partial c}M(c,x,1)= (4-c^2)[32x(1-x)(1+x)^2-40c^2x(1-x)(1+x)^2 &-6c^3x^2(2-13x+2x^2)+8c(-32+30x^2-31x^3+6x^4)]=0
\hat{\varrho}_{\rm fld}=\hat{\varrho}_m\left[1+\zeta\left({\hat{c}\over c}\right)^2\right]+\hat{\varrho}_r\left({\hat{c}\over c}\right)^2,\ \ \ \ \ \ \ \hat{\wp}_{\rm fld}={1\over 3}\hat{\varrho}_r\left({\hat{c}\over c}\right)^2,
\sum_{i}q_{i}^{2}\Bigl ( \sum_{a_{i}+^{*}\Gamma_{2}}4 e^{-{\pi t}p^{2}/ 2}- \sum_{2 a_{i}+^{*}\Gamma_{2}}e^{-{\pi t}p^{2}/ 2}\Bigr ) \Biggr ] + o ({\cal B}^{4})
p_1(x+y) \leq p_1(x) + p_1(y) - 2 \delta_1(p_1(y) p_1(u - v)) = p_1(x) + p_1(y) - 2 \delta_1( p_1(\tau x - y)) & \leq p_1(x) + p_1(y) - 2 \delta_1( \|\tau x - y\| / 2) \leq p_1(x) + p_1(y) - 2 \delta_1( \|x - y\| / 4) & = p_1(x) + p_1(y) - 2 \delta_1( t/ 4).
Q(z, z')\lesssim \begin{cases}r'^{-n}r^{s}(r/r')^{1-\frac n2+\nu_0-s},r<\frac{r'}2; d(z,z')^{-(n-s)},r\sim r'; r^{-n+s}(r'/r)^{1-\frac n2+\nu_0},r>2r';\end{cases}
(L(x,\xi)\eta)_i\eta^i=\vert\xi\vert^2_g\vert\eta\vert^2_g+\left(1-\frac{2}{n}\right)\vert\langle\xi,\eta\rangle\vert^2_g\geq\vert\xi\vert^2_g\vert\eta\vert_g^2.
J^i(\gamma)=J^i(\gamma^1,\ldots,\gamma^n)=\mathbb{E}\bigg[\int_0^T f^i(X_t,\gamma_t^i)\,dN^i_t+g^i(X_T)\bigg] = \mathbb{E}\bigg[\int_0^T \lambda f^i(X_t,\gamma_t^i)\,dt+g^i(X_T)\bigg]\,,
\mathbb{P}(\alpha(G^+) = |\sigma| + 1 \big| k = k_0) \le \mathbb{P}(Y_I(\sigma) < \lambda_I - \xi \big | k = k_0) + \sum_v \mathbb{P}(A_v \big| (Y_I(\sigma) \ge \lambda_I - \xi) \cap (k = k_0)) =O(\exp{(-n^\delta)}).
\beta x+ (\beta+\alpha(\tau-1))y=0, &(\beta+\alpha-c\alpha)x + ((\beta+\alpha)\tau-c\alpha)y=0, &\beta^2-\alpha^2 + \tau \alpha^2 = 1, &\beta^2-\alpha^2 + \tau \alpha^2 + \alpha\beta(\tau-c)=1, &\tau \beta^2 + 2(\tau-c)\alpha\beta + (c^2-2c+\tau)\alpha^2=\tau', &\mu+x^2+2xy+\tau y^2=\mu'
S_{1}+S_{2}+\displaystyle\sum_{l=1}^{p-q}\left(\varphi_{il}^{\star\star}\left(Z\right)\right)^{(2)}\overline{\left(\varphi_{jl}^{\star\star}\left(Z\right)\right)^{(2)}}=0, \quad\quad\mbox{for all $i,j=1,\dots,q$,}
\mathbb{E}[{|\sqrt{\sum a_{ii}X_i^2}- \mathbb{E}[\sqrt{\sum a_{ii}X_i^2}]|}^p] = \int_{0}^{\infty}t^{p-1}\mathbb{P}(|\sqrt{\sum a_{ii}X_i^2}- \mathbb{E}[\sqrt{\sum a_{ii}X_i^2}]| \ge t) dt \ \le 2 \int_{0}^{\infty}t^{p-1}e^{\frac{-t^2}{2 \|A\|_{\infty}C}}dt \le C_p \|A\|_{\infty}^{\frac{p}{2}}.
g_i^j(x_i) = 0; \min_{x\in\partial B(x_i, \frac{\delta_1r_{x_i}}{3})}\sum\limits_{i=1}^N|g^j_i(x)|^2> 2\max_{x\in B(x_i, \delta_2r_{x_i})}\sum\limits_{i=1}^N|g^j_i(x)|^2;
\phi_{\kappa}(\theta, x) \ = \ \inf_{v_1 \in{\mathcal M}_1}E^{v_1, \hat v_2}_x \Big[e^{\frac{\kappa \|r_2\|_{\infty}}{\alpha}}e^{\theta \int^{T_\kappa}_0 e^{-\alpha t}r_1(X(t), v_1( t,X(t)), \hat v_2(\theta e^{- \alpha t}, X(t)))dt}\Big].
=\sqrt{\frac{\|f_{1}-Mu^{(k)}_{1}-Eu^{(k)}_{2}\|^{2}_{2}+\|f_{2}+E^*u^{(k)}_{1}\|^{2}_{2}}{\|f_{1}\|^{2}_{2}+\|f_{2}\|^{2}_{2}}}< 10^{-6}.
\{a_{1}, a_{2}, \cdots a_{r-1}, a_{r}a_{r+1}\}= \{a_{1}, a_{2}, \cdots a_{r-1}, a_{r}\}a_{r+1}\phantom{{}=}+ (-1)^{\chi}a_{r}\{a_{1}, a_{2}, \cdots a_{r-1},a_{r+1}\},
\langle\overline{\psi}(k_{1})\overline{\psi}(k_{2})\cdot\cdot\cdot\overline{\psi}(k_{n})\psi(k_{1}^{\prime})\cdot\cdot\cdot\psi(k_{n})\rangle=\sum_{pairings}(-1)^{S}\prod_{pairsi,j}\delta(k_{i}-k_{j})\frac{1}{\gamma\cdot k_{i}-m}
\eta_{2,2}(x_1,t)= \frac{h_{2,3}\, \partial_{x_2}v^{(2,3)}_0(x_1,-(l_1+l_2),t)}{h_{2,3}\, \partial_{x_2}v^{(2,3)}_0(x_1,-(l_1+l_2),t) + h_{2,4}\, \partial_{x_2}v^{(2,4)}_0(x_1,-(l_1+l_2),t)}.
\Im m_{\mu_\alpha\boxplus\mu_\beta}(z)\sim \Im \omega_\alpha(z)\sim \Im \omega_\beta(z)\sim \begin{cases}\sqrt{\kappa+\eta}\,, \qquad \textrm{if}E\ge E_-\,, \frac{\eta}{\sqrt{\kappa+\eta}}\,, \textrm{if}E<E_-\,,\end{cases}
[\begin{matrix}A^{\prime}(x)\\B^{\prime}(x)\end{matrix}]=[\begin{matrix}u_{1}(x)&u_{2}(x)\\u_{1}^{\prime}(x)&u_{2}^{\prime}(x)\end{matrix}]^{-1}[\begin{matrix}0\\f\end{matrix}]=\frac{1}{W}[\begin{matrix}u_{2}^{\prime}(x)&-u_{2}(x)\\ -u_{1}^{\prime}(x)&u_{1}(x)\end{matrix}][\begin{matrix}0\\f\end{matrix}],
Z ( \omega , \theta ) = \sum_{n = 0}^{N}\frac{Z_{n}( N , \omega ) e^{\theta n}}{n !}= \sum_{\sigma_{m}= 0 , 1}\exp \left( \sum_{0 \le m < k \le N - 1}A_{m k}\sigma_{m}\sigma_{k}+ ( \theta + \eta ) \sum_{m = 0}^{N - 1}\sigma_{m}\right) ,
\lambda(g,b) =\inf \left\{\int_M (R-\frac{1}{12}|H|^2+|\nabla f|^2)e^{-f}dV_g: \int_M e^{-f}dV_g=1\right\}&=\inf\left\{\int_M (R-\frac{1}{12}|H|^2)\omega^2+4|\nabla \omega|^2dV_g: \|\omega\|_{L^2}=1\right\}.
\frac{1}{g^{2}}\left( \begin{array}{l}{{\Gamma_{1}}}\\{{\Gamma_{2}}}\\{{\Gamma_{3}}}\\ \end{array}\right) = \left( \begin{array}{l l l}{{B_{1 1}}}&{{B_{1 2}}}&{{B_{1 3}}}\\{{B_{2 1}}}&{{B_{2 2}}}&{{B_{2 3}}}\\{{B_{3 1}}}&{{B_{3 2}}}&{{B_{3 3}}}\\ \end{array}\right) \left( \begin{array}{l}{{\Gamma_{1}}}\\{{\Gamma_{2}}}\\{{\Gamma_{3}}}\\ \end{array}\right) .
\begin{cases}\partial_tV^{(\alpha, a)}-\mu\Delta\sum\limits_{l=0}^\alpha C_\alpha^l (-1)^{\alpha-l}V^{(l, a)}-\nabla\cdot H^{(\alpha, a)}=f^1_{\alpha a}, \partial_t H^{(\alpha, a)}-\nabla V^{(\alpha, a)}=f^2_{\alpha a}\end{cases}
\|\log f(z;\boldsymbol{\xi})-\log f\left( 1;\boldsymbol{\xi}\right) \|_{H^{2}}^{2}=\left\Vert \log \left\vert h(e^{iw};\boldsymbol{\xi})\right\vert^{2}-\log \left\vert h(1;\boldsymbol{\xi})\right\vert^{2}\right\Vert^2_{L^{2}\left( \mathbb{T}\right)}<\infty
\d B_{\mu\nu}= \eta_{11}\theta^1 \partial_{[\mu}V_{\nu]}+ \eta_{AB}\theta^A \partial_{[\mu}W_{\nu]}^B \rightarrow F=\eta_{11}\frac{(X^1)^3}{X^0}+ \eta_{AB}\frac{X^1}{X^0}X^A X^B
K_{x}=[\begin{matrix}0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{matrix}],K_{y}=[\begin{matrix}0&0&1&0\\ 0&0&0&0\\ 1&0&0&0\\ 0&0&0&0\end{matrix}],K_{z}=[\begin{matrix}0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\end{matrix}]