harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
[ R , M ] = - \mathrm{\frac{i}{2}}\sum_{\rho \in \Delta_{+}}g_{| \rho |}\thinspace{\frac{| \rho |^{2}\cos ( \rho \cdot q )}{\sin^{2}( \rho \cdot q )}}\thinspace [ \mathrm{e}^{2 \mathrm{i}q \cdot \hat{H}}, \hat{s}_{\rho}] .
v(x,i) = \frac{1}{(2\alpha)^{\tilde{d}}}\int_{[-\alpha,\alpha]^{\tilde{d}}}\psi(\boldsymbol{u}(x-\boldsymbol{\omega}_{\tilde{d}},i))d\boldsymbol{\omega}_{\tilde{d}}, \;\; |x|\leq1-\alpha,
v \cdot p_{e,w_{00}}+ p_{0,w_{00}}= v^{\ell(w_0)}+ v^{\ell(w_0)-2}+ v^{\ell(w_0)-4}+ \cdots + v^{\ell(w_0)-2n}, v \cdot p_{e,u_{00}}+ p_{0,u_{00}}= \phantom{v^{\ell(w_0)}+}\ \v^{\ell(w_0)-2}+ v^{\ell(w_0)-4}+ \cdots + v^{\ell(w_0)-2n}.
\delta_{L}\xi_{\mu}= (\delta_{L}g_{\mu\nu}) \xi^{\nu}+g_{\mu\nu}\delta_{L}\xi^{\nu}= (\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}) \xi^{\nu}= (\nabla_{\mu}\xi_{\nu})\xi^{\nu}= -{\cal J}_{\mu},
\begin{aligned}\dot{\xi}^{x}_{n}(s)=\xi^{v}_{n}(s),\ \ \dot{\xi}^{v}_{n}(s)=\frac{1}{\epsilon}\tilde{B}(x_{n+\frac{1}{2}})\xi^{v}_{n}(s) +F(x(t_n+s))-F(\tilde{x}_{n}(s))+\varsigma^{0}_{n}(s).\end{aligned}
\nabla(v^{(r,s)}) = r v^{(r-1,s+1)}\otimes \xi, \nabla(v^{[r,s]}) = (r + 1) v^{[r+1,s-1]}\otimes \xi, \varphi(v^{(r,s)}) = p^r v^{(r,s)}, \varphi(v^{[r,s]}) = p^{-r}v^{[r,s]}.
\frac{1}{2 \pi i}\int_{\Gamma_j}\frac{\psi_j(n_j; \lambda_j)}{\lambda_j}\, d\lambda_j= \frac{1}{2 \pi i}\int_{\Gamma_j}\left(\prod_{k = 1}^{n_j}\frac{k p_j}{\lambda_j + k p_j}\right) \frac{d\lambda_j}{\lambda_j}= -1,
|\mathcal{F}|\leq 2^{n-1}+{n-1 \choose \lfloor \frac{n-1}{2}\rfloor}=\begin{cases}\displaystyle\sum_{i=n/2}^n{n \choose i}\mbox{if}n\mbox{is even,}{n-1 \choose \frac{n-1}{2}}+\displaystyle\sum_{i=(n+1)/2}^n{n \choose i}\mbox{if}n\mbox{is odd.}\end{cases}
\dot{y}_\alpha=\frac{1}{m_i}\left(\sum_{\xleftarrow[\alpha]{}i\xrightarrow[\beta]{}}e^{-y_\beta}-\sum_{\xleftarrow[\alpha]{}i\xleftarrow[\beta]{}}e^{-y_\beta}\right)-\frac{1}{m_j}\left(\sum_{\xrightarrow[\alpha]{}j\xrightarrow[\beta]{}}e^{-y_\beta}-\sum_{\xrightarrow[\alpha]{}j\xleftarrow[\beta]{}}e^{-y_\beta}\right).
\left( \begin{array}{c}{{\mathbf{E}^{\prime}}}\\{\mathbf{B}^{\prime}}\\ \end{array}\right) = \left( \begin{array}{c c}{a}&{c}\\{b}&{d}\\ \end{array}\right) \left( \begin{array}{c}{\mathbf{E}}\\{\mathbf{B}}\\ \end{array}\right)
f(\lambda_i(C)) = \f\Big(\sum_{j=1}^nD_{ij}\big(\tau \lambda_j(A)+(1-\tau) \lambda_j(B)\big)\Big) \geq \ \sum_{j=1}^nD_{ij}\big(\tau f(\lambda_j(A))+(1-\tau) f(\lambda_j(B))\big) = x_i,1\leq i\leq n,
\int_{\mathcal{G}_n}\frac{1}{\mathbf{N}_{k, \nu}\left(x\mathbf{N}_{K/E}(x)^{-n}\det y\right)}F\left(x, y^{-1}\right)\phi_\kappa\left(\mathbf{N}_{K/E}(x)y^{-1}\right)d\mu_\varphi = G_{k, \nu, F, \varphi, \kappa}.
\begin{array}{ccl}X = \varphi_2^vX_1^h+\psi_1^hY_2^v+cf_1^hf_2^v\left\{\psi_{2}^vY_1(f_1)^h\!-\!\varphi_{2}^vX_1(f_1)^h\right\}grad(f_{2}^v) & - cf_1^hf_2^v\left\{\psi_{1}^hY_2(f_2)^v\!-\!\varphi_{1}^hX_2(f_2)^v\right\}grad(f_{1}^h)\end{array}
h^{r^{k_{r}}, \ldots , a^{k_{a}-1}, \ldots , 1^{k_{1}}}_{\vec{w}}(x_{1}, \ldots , x_{k-1})=h^{r^{k_{r}}, \ldots , a^{k_{a}-1}, \ldots , 1^{k_{1}}}_{\vec{z}(\ell(t), \ldots , \ell(0))}(x_{1}, \ldots , x_{k-1}).
\sum_{k=1}^{\infty}\frac{T_\nu (\lambda_{\nu ,k})}{\lambda_{\nu ,k}^\sigma}J_{\sigma +\nu +1}(\lambda_{\nu ,k}) J_\nu (\lambda_{\nu ,k}t)=\frac{1}{2}\int_{0}^{\infty}{J_{\sigma +\nu +1}(z) J_\nu (zt)\frac{dz}{z^\sigma}}=\frac{\left( 1-t^2\right)^\sigma t^\nu}{2^{\sigma +1}\Gamma (\sigma +1)}
\Big\||x|^{-\frac s2}u^m\Big\|_2^2=\int_\Omega |x|^{-s}(u^m)^{2}dx\le C\Big(\int_\Omega |\nabla u^m|^{\frac{2N}{N+2-s}}dx\Big)^{\frac{N+2-s}{N}}.
\big\| \Xi^n \big\|_V = \big\| (P_h - I) u(t_n) \big\|_V \le \big\| P_h ( I - Q_h) u(t_n) \big\|_V + \big\| ( Q_h - I) u(t_n) \big\|_V \le \big( 1 + \| P_h \|_{\L(V)}\big) \| (Q_h - I) u(t_n)\| = \big( 1 + \| P_h \|_{\L(V)}\big) \mathrm{dist}_V(u(t_n), V_h),
s_1(B) := \sum_{k=1}^\infty \frac{1}{k!}\sum_{\substack{x_1,\ldots,x_k\in B, m_{i-1}\le d(x_1,\ldots,x_k)< m_{i}}}(\rho_+(x_1,\ldots,x_k)^2+\rho_-(x_1,\ldots, x_k)^2).
JO_{n+2}^{(3)}-4JO_{n}^{(3)}=\sum_{s=0}^{7}(J_{n+s+2}^{(3)}-4J_{n+s}^{(3)})e_{s}&=J_{n+2}^{(3)}-4J_{n}^{(3)}+(J_{n+3}^{(3)}-4J_{n+1}^{(3)})e_{1}+\cdots +(J_{n+9}^{(3)}-4J_{n+7}^{(3)})e_{7}&=e_{1}+e_{2}+e_{4}+e_{5}+e_{7}-2(1+e_{3}+e_{6})
\overline{{Z}}_{\Psi}= \int \prod D \Phi^{a}\prod D \vartheta^{b}\prod \delta ( d^{\gamma}) e x p \left({\frac{i}{\hbar}}W ( \Phi^{a},{\frac{\partial \overline{\Psi}}{\partial \Phi^{a}}}, \vartheta^{b},{\frac{\partial \overline{\Psi}}{\partial \vartheta^{b}}}\right)
\begin{array}{l}\vert 0,z\rangle_{-k}=\displaystyle\exp_{q^2}(-\frac{1}{2}|q^{-2}z|)\sum_{m=0}^\infty \frac{(-q^{-2}z)^m}{[m]_{q^2}!}({\cal D}_{-k})^m\vert 0\rangle_{-k}~,{\cal X}^{-k}\vert 0,z\rangle_{-k}=z\vert 0,z\rangle_{-k}~,\end{array}
2\sum_{i=1}^k\,s_it_i-\sum_{i=1}^k\,(s_i+t_i) =\sum_{i=1}^k\,\Big[s_i\,(t_i-1)+t_i\,(s_i-1)\Big] &\geq \sum_{i=1}^k\,(s_i+t_i-2)=n-2k.
\mathrm{I = \frac{2 \Gamma ( 4 - \o m e g a - \s i g m a )}{( 4 \p i )^{2}}\int_{0}^{1}d x x^{- \s i g m a}\int_{0}^{x}d G \int_{G}^{1 - x + G}d y [ H_{0}^{( 1 )}]^{\o m e g a + \s i g m a - 4}+ F . T .}
nB_2+lA_2= n U^n_l Str\left(\ldots \phi^{i_{2k}'}\ldots \underbrace{\phi^{i_{2k-1}'}\phi^{i_n}}\ldots \partial_{a_l}\phi^{j_l}\ldots \right) +l U^n_l Str\left(\ldots \phi^{i_{2k}'}\ldots \underbrace{\phi^{i_{2k-1}'}\partial_{a_l}\phi^{j_l}}\ldots \right)
W\in C^1((0,\infty); L^{3,\infty}(\mathbb R^3)\cap L^q_\sigma(\mathbb R^3)),\quad\forall q\in (3,\infty), &\nabla^2 W\in C((0,\infty); L^{3,\infty}(\mathbb R^3)\cap L^q(\mathbb R^3)),\quad\forall q\in (3,\infty).
\left[X_{\varepsilon}, \overline{L_{k}}\right]_{T}(p) = -e^{g_{\varepsilon}(p)}\left(\overline{L_{k}}g_{\varepsilon}(p)-\alpha(\overline{L_{k}})(p)\right) + b_{\varepsilon, k}(p)\left[L_{k}, \overline{L_{k}}\right]_{T}(p)\;
\begin{aligned}\delta(t) = - \frac{\beta}{m-1}\, (T-t)^{-\alpha-1}\, [-\log(T-t)]^{\frac{\beta}{m-1}-1}& \, + \frac{C^{m-1}m}{a^2(m-1)^2}\, (T-t)^{-\alpha m}\, [-\log(T-t)]^{\frac{\beta(2-m)}{m-1}}\geq \, \frac{C^{m-1}m}{2 a^2(m-1)^2}\, (T-t)^{-\alpha m}\, [-\log(T-t)]^{\frac{\beta(2-m)}{m-1}}\end{aligned}
f(x_m) =\frac{Np-2N-4s_1}{N(p-2)}x_m= \frac{Np-2N-4s_1}{N(p-2)}(\frac{4s_1}{KN(p-2)})^{\frac{4s_1}{N(p-2)-4s_1}}= \frac{Np-2N-4s_1}{2s_1p-Np+2N}M(\phi)^{\frac{s_1}{s_c}}M(u_0)^{\frac{s_c-s_1}{s_c}}>2E(u_0).
qz_i^2A_{i,2,0}=\frac{\lambda_1}{q}z_i^2A_{i,1,0}-\frac{\lambda_3(q+1)}{q\sqrt{q}(q^2+q+1)}z_iA_{i,0,0}+\frac{q+1}{q^2+q+1}A_{i,0,0}-\sqrt{q}z_i^3A_{i,1,0}.
c_{F|ideal}=\sqrt{\frac{2\gamma^{2}}{\gamma-1}(\frac{2}{\gamma+1})^{\frac{\gamma+1}{\gamma-1}}[1-(\frac{p_{e}}{p_{c}})^{\frac{\gamma-1}{\gamma}}]}+\frac{A_{e}}{A_{t}}(\frac{p_{e}-p_{a}}{p_{c}})
\begin{array}{l c l}{{( P \times V ) \times G}}&{{\longrightarrow}}&{{P \times V}}\\{{\big ( ( a , v ) \, , \, g \big )}}&{{\longmapsto}}&{{( a g \, , \, g^{- 1}\, v ) , \hspace{0 . 7 c m}a \in P , \quad v \in V , \quad g \in G .}}\\ \end{array}
\overline{g}_{\alpha\beta}=\overline{\varphi}_{\alpha}\circ\overline{\varphi}_{\beta}^{-1}:\overline{\varphi}_{\beta}(\overline{U}_{\alpha}\cap\overline{U}_{\beta})\rightarrow\overline{\varphi}_{\alpha}(\overline{U}_{\alpha}\cap\overline{U}_{\beta})
X_{i,j}(x,y) = -W_{i+1,j+1}(x)\left( (p+y-i)\frac{2y}{1+2y}A(y)-(p-y-1-i)\tilde{D}(y)\right) &\quad\;+(p+y-i-1) (p-y-2-i)W_{i+2,j+2}(x)C(y)\,.
S_0=\begin{pmatrix}Id 0 0 0 -Id 0 0 0 0\end{pmatrix},S_a=\begin{pmatrix}0 A_a B_a A_a^{tr}0 C_a B_a^{tr}C_a^{tr}0\end{pmatrix},1\leq a\leq m_{+},
v v_{k}v_{k}^{*}(1-v^{*}v)v_{n-j}=v (v^{*}v v_{k}v_{k}^{*})(1-v^{*}v)v_{n-j}=vv_{k}v_{k}^{*}v^{*}v(1-v^{*}v)v_{n-j}& = v_{k+1}v_{k+1}^{*}(v-vv^{*}v)v_{n-j}=0,
\Gamma(\pi) =\lim_{t\to\infty}\frac{1}{N}\sum_{\ell=1}^{N}\frac{\sum_{i=1}^{R_\ell^\pi(t)}(\rho_\ell (\frac{(T_{\ell i}^\pi)^2}{2}+T_{\ell i}^\pi Z_{\ell i}^\pi)+c_\ell)}{t},
\left( 1 +{\frac{1 6}{9}}{\frac{z_{0}^{2}}{l^{2}}}\right) + \left({\frac{3 2}{9}}\omega z_{0}^{2}- l^{2}\right) \varepsilon^{2}+ \left({\frac{1 6}{9}}l^{2}z_{0}^{2}\omega^{2}\right) \varepsilon^{4}> 0 ,
\langle k_1,k_2,k_3|V|p\rangle= (16 \pi^3) \delta^3(p-k_1-k_2-k_3) \;\Theta\bigl(p^--k_1^--k_2^--k_3^-\bigr)\; u_4(-k_1,-k_2,-k_3,p) \;.
\Vert \mathcal{B}- \mathcal{B}_{n}\Vert_{\infty}=\Vert \mathcal{B}- \mathcal{P}_{n}\mathcal{B}+ \mathcal{P}_{n}\mathcal{B}- \mathcal{B}_{n}\Vert_{\infty}\leq \Vert \mathcal{B}- \mathcal{P}_{n}\mathcal{B}\Vert_{\infty}+ \Vert \mathcal{P}_{n}\mathcal{B}-\mathcal{B}_{n}\Vert_{\infty},
z_{h,k+l}= \binom{n}{h}\frac{(\alpha+2l+1)_{n-l-h}(\beta+2k+1)_{h-k}}{(2k+2l+\sigma+1)_{n-k-l}}, begin{align*}1ex] z_{hi}= \frac{(2i+\sigma)(i+k+l+\alpha+\beta)(i-n-1)}{(\alpha+l+i-k)(i+n+\sigma)(2i+\alpha+\beta-1)}z_{h,i-1}(i = k+l+1,k+l+2,\ldots,n);
\begin{aligned}|N_{G}(v)\cap (V(T')\backslash \{u_{1},\cdots,u_{s}\})| =|N_{G}(v)\cap (Y\backslash \{u_{1},\cdots,u_{s}\})| &\leq |Y|-\lfloor\frac{s}{2}\rfloor &\leq t-\lfloor\frac{s}{2}\rfloor\end{aligned}
\frac{\partial^{2}F}{\partial t^{2}}(x,t)=\alpha(\frac{\partial^{2}F}{\partial x_{1}^{2}}(x,t)+\frac{\partial^{2}F}{\partial x_{2}^{2}}(x,t)+\frac{\partial^{2}F}{\partial x_{3}^{2}}(x,t))+\beta P(x,t)
a \sharp b \,(x,\xi) := \frac{1}{(\pi h)^2}\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}}\int_{\mathbb{R}}e^{\frac{2i}{h}\sigma (y, \eta; \, z, \zeta)}a(x+z, \xi + \zeta) b(x+y, \xi +\eta) \; dy d\eta dz d\zeta
R_j^h(\ell):= \sum_{\ell_1+\ell_2=\ell \atop \pi(\ell_2) \nparallel v(j)}\frac{-\left(2\pi(\ell_1)\cdot\pi(\ell_2)\right) V(\ell_1) V(\ell_2)}{(\omega\cdot \ell_2 + 2 j \cdot \pi(\ell_2) - |\pi(\ell_2)|^2)(\omega\cdot \ell_2 + 2 (j-\pi(\ell_1)) \cdot \pi(\ell_2) - |\pi(\ell_2)|^2)}\,,
\frac{dw}{dx}=-\frac{\frac{f^{\prime}(k)k}{1-\tau}}{\frac{f^{\prime}(k)}{1-\tau}[\tau\frac{f^{\prime}(k)}{f^{\prime\prime}(k)}+k]}=-\frac{1}{\tau\frac{f^{\prime}(k)}{kf^{\prime\prime}(k)}+1}
S^{\mathrm{soft}}(\kappa, \pi;u,v) =S^{(\mathrm{soft})}(\kappa;u,v) +\sum_{p=1}^{m}\Big( \phi_{1,p}(\kappa, \pi;u)\phi_{2,p-1}(\kappa, \pi;v) - \phi_{1,p-1}(\kappa, \pi;u) \phi_{2,p}(\kappa, \pi;v)\Big),
F_N\left( \left[ \frac1n\sum_{i=1}^{n}\frac{\frac1N\Vert z_i\Vert^2}{\check{d}_i}\right]^{-1}\right) \geq F_N\left( \check{d}_n \left[ \frac1n\sum_{i=1}^{n}\frac1N\Vert z_i\Vert^2 \right]^{-1}\right).
\int_{\{\Psi_1<0\}}|\tilde F-f_0F^{1+\delta}|^2e^{-\varphi}=\int_{\{\Psi_1<0\}}|\frac{\tilde F}{F^{1+\delta}}-f_0|^2=\int_{\{\Psi_1<0\}}|\frac{\tilde F}{F^{1+\delta}}|^2-\int_{\{\Psi_1<0\}}|f_0|^2.
\mathcal{A}(g_K, \phi, \sigma, m) = -2\nabla^2\sigma + \nabla^2\phi + 4d\sigma\otimes d\sigma - d\sigma\otimes d\phi - d\phi\otimes d\sigma -\frac{1}{m}d\phi\otimes d\phi + (g_K(\nabla\sigma, \nabla\phi) - 2|\nabla\sigma|^2 - \Delta\sigma - e^{2\sigma})g_K.
\Bar{\delta}^{\Psi\Psi}_{p(p-1)qqj}= \frac{\lambda_{pq(j-1)}^2}{\lambda_{(p-1)q(j-1)}^2}\Bar{\delta}^{\Phi\Phi}_{p(p-1)qq(j-1)}, \delta^{\Phi\Phi}_{ppq(q-1)j}= \frac{\lambda_{pqj}^2}{\lambda_{p(q-1)j}^2}\delta^{\Psi\Psi}_{ppq(q-1)(j+1)},
\beta_{00}=\sum_{\rho=0}^N\frac{\left(\eta_{0\rho}\right)^2}{4}\left[\left(\frac{\tilde{\omega}}{\Omega_\rho}-\frac{\Omega_\rho}{\omega}\right){\rm e}^{i\Omega_\rho t}+\left(\frac{\Omega_\rho}{\tilde{\omega}}-\frac{\tilde{\omega}}{\Omega_\rho}\right){\rm e}^{-i\Omega_\rho t}\right]\;.
(\sigma_\mu)_{\alpha \dot\alpha}J^{\mu\alpha}=J^\alpha_{\alpha \dot\alpha}= i\,\frac{T_G}{4\pi^2}\, \mbox{Tr}\left[ \bar G_{\dot\alpha\dot \beta}\bar \lambda^{\dot\beta}\right]\,,\qquad\vartheta^\mu_{\mu}=\frac{T_G}{16\pi^2}\,\mbox{Tr}\,\left[G_{\rho\sigma}G^{\rho\sigma}\right]\,.
\mu_1|u_1|_4^4\sum_{i=2}^d\mu_i|u_i|_4^4+2b\mu_1|u_1|_4^4\sum_{1<i<j}|u_iu_j|_2^2 \leq b^2\Big(\sum_{j=2}^d |u_1u_j|_2^2\Big)^2 &\leq b^2\Big(\sum_{j=2}^d |u_1|_4^2|u_j|_4^2\Big)^2.
Cov(X,Y)=Cov(X-k_{x},Y-k_{y})=\frac{\sum_{i=1}^{n}(x_{i}-k_{x})(y_{i}-k_{y})-(\sum_{i=1}^{n}(x_{i}-k_{x}))(\sum_{i=1}^{n}(y_{i}-k_{y}))/n}{n}.
\bar z^N+(-\bar z-\xi)^N=\lambda_N\xi^N+\sum_{j<N}p_{ijk}z^i\xi^jw^k, \sum_i\bar z^i(-\bar z-\xi)^{N-1-i}=\lambda_{N-1}'\xi^{N-1}+\sum_{j<N-1}q_{ijk}z^i\xi^jw^k.
||\tilde{\mathcal{A}}\mathcal{A}_j\partial_j{\mathbf V}||^2_{s-1,\ast,t}\lesssim ||\tilde{\mathcal{A}}\mathcal{A}_j{\mathbf V}||^2_{s,\ast,t}&\leq C(K)\Big(||{\mathbf V}||^2_{s,\ast,t}+||\dot{{\mathbf U}}||^2_{W^{1,\infty}_{\ast}(\Omega_t)}||\hat{W}||^2_{s,\ast,t}\Big),
f=\sum_{j=1}^{\infty}\hat a_j\hat u_{j,\lambda}=\sum_{j=1}^{\infty}\hat a_j\sqrt{\mu_j(\lambda)+b}\cdot\frac{\hat u_{j,\lambda}}{\sqrt{\mu_j(\lambda)+b}}=\sum_{j=1}^{\infty}\hat a_j\sqrt{\mu_j(\lambda)+b}\cdot \gamma_0(u_{j,\lambda}),
b ( g ) = B_{1}e^{\chi_{2}{\cal X}_{2}}e^{\chi_{3}{\cal X}_{3}}+ e^{- \chi_{1}{{\cal X}_{1}}^{t}}B_{2}e^{\chi_{3}{\cal X}_{3}}+ e^{- \chi_{1}{{\cal X}_{1}}^{t}}e^{- \chi_{2}{{\cal X}_{2}}^{t}}B_{3}.
\begin{array}{rcl}\dot{x}(t) \triangleq{{f}_{E}}\left(x(t),{{u}_{0}}(t),{{u}_{1}}(t),v(t) \right) & = \left[ 1-v(t) \right]{{f}_{0}}\left( x(t),{{u}_{0}}(t)\right) & +v(t){{f}_{1}}\left( x(t),{{u}_{1}}(t) \right) \end{array}
\int_{0}^{\infty}\frac{d k}{\pi}k^{4}\frac{d}{d k}\left( \delta_{m}^{\ell}( k ) - \delta_{m}^{( 1 ) \ell}( k ) - \delta_{m}^{( 2 ) \ell}( k ) \right) + \sum_{j}( \kappa_{j , m}^{\ell})^{4}= 0
\begin{aligned}\left| \partial_\xi^\beta\left( H_{+}(a,c;2i\phi(t)|\xi|) \right) \right| \lesssim (\phi(t)|\xi|)^{a-c}(1+ |\xi|)^{-|\beta|}\phi(t)|\xi|\geq 1, \left|\partial_\xi^\beta \left( H_{-}(a,c;2i\phi(t)|\xi|)\right)\right| \lesssim (\phi(t)|\xi|)^{-a}(1+|\xi|)^{-|\beta|}\phi(t)|\xi|\geq 1.\end{aligned}
u=\frac{J_{3}P_{3}^{0}(sin\theta)}{r^{4}}=J_{3}\frac{1}{r^{4}}\frac{1}{2}sin\theta(5sin^{2}\theta-3)=J_{3}\frac{1}{r^{7}}\frac{1}{2}z(5z^{2}-3r^{2})
\liminf_{k \to \infty}{\cal H}^1(J_{y_k}) \ge \kappa_1^{-1}\int_{S^1}\hat{\mu}^{0,\xi}_{\tilde{v}}(\Omega) \, d{\cal H}^1(\xi) & = \kappa_1^{-1}\int_{S^1}\int_{J^\xi_{\tilde{v}}}|\nu_{\tilde{v}}(x) \cdot \xi| \, d{\cal H}^1(x)\, d{\cal H}^1(\xi) ={\cal H}^1( J_{\tilde{v}}).
L_{3}=2F\frac{\partial F}{\partial \lambda_{i}}\lambda_{p}^{-2}\lambda_{q}^{-1}h_{pqi}^{2}+F\lambda_{p}^{-2}\frac{\partial^{2}F}{\partial \lambda_{i}\partial \lambda_{j}}h_{iip}h_{jjp}&~+F\lambda_{p}^{-2}\sum_{i\neq j}(\frac{\partial F}{\partial \lambda_{i}}-\frac{\partial F}{\partial \lambda_{j}})(\lambda_{i}-\lambda_{j})^{-1}h_{ijp}^{2}.
\tilde A(y,s)=A(R_0^{k_0}y,s) &\tilde f(y,s)=R_0^{(2k_0(1-\alpha)+k_0\alpha)}f(R_0^{k_0}y,s) &\tilde q (y)=R_0^{2k_0(1-\alpha)}q (R_0^{k_0}y)
\mathcal{BL}_{p,q}^{\mathcal{T}}(F,f)=\left( \frac{p^{\prime}}{q^{\prime}}\right)^{q}\omega_{q}\left( \left( \frac{q^{\prime}}{p^{\prime}}\right)^{q-1}\frac{f^{q}}{F}\right)^{q}F
\left|(1 - \eta_{j, R})S_{\mu, \xi, j}\right|\lesssim \left(\frac{1}{R^{n-2s-a}}+\frac{1}{R^{4s-a}}\right)\frac{1}{R^{a-2s}}\sum_{j=1}^k\frac{\mu_j^{-2s}\mu_0^{\frac{n-2s}{2}+\sigma}}{1+|y_j|^{a}}.
g(n,\beta,\gamma) =\int_0^x e^{-\beta t}t^nt & =\frac{n!}{\beta^{n+1}}\left[ 1-e^{-\beta x}\sum_{m=0}^n \frac{(\beta x)^m}{m!}\right] , &[\gamma>0,Re\,\beta >0,n=0,1,2,...].
\delta{\cal L}= \frac{-1}{16{\pi}^2}\sum_{\alpha}\int d^2 \theta\frac{W^{\alpha}W_{\alpha}}{4}\left([c(G_{\alpha}) -\sum_{r_{\omega}}T(r_{\omega})]g(\phi)+2\sum_{r_{\omega}}T(r_{\omega})\log det h_{\alpha\beta}(\phi,{\bar\phi})\right)+h.c
\mathbb{E}\left[U\left(L_T^{{\phi}^*}\right)|X_0=z\right]=-\frac{1}{\sqrt{\gamma_\beta(T)}}e^{f\left(\hat{\phi}_1^T(z),z\right)}\quad\textrm{by \eqref{matrixformofexp}, Lemma \ref{determinantwithgamma}and \eqref{f}}.
{\cal H}={\frac{1}{2}}P^{2}+ m_{L}^{2}\left( \sum_{\alpha_{i}\in \Pi \cap \Delta_{L}}e^{\alpha_{i}\cdot Q}+ e^{\alpha_{0}\cdot Q}\right) + m_{S}^{2}\sum_{\alpha_{i}\in \Pi \cap \Delta_{S}}e^{2 \alpha_{i}\cdot Q},
\{\tau_l < \epsilon \}\subset \{|X_{n_l}|_{\mathcal{E}(\tau_l \wedge \epsilon)}= |X_{n_l}(0)|_1 + ( M - 1 + 2^{-l}) \}\subset \{|X_{n_l}|_{\mathcal{E}( \tau_l \wedge \epsilon )}> |X_{n_l}(0)|_1 + (M-1) \}.
(l-z \frac{d}{d z}) \widetilde C_l^{\alpha}(z) = -2 \widetilde C_{l-2}^{\alpha+1}(z), \frac d{dz}\widetilde C_l^{\alpha}(z)= 2 \gamma (\alpha,l)\widetilde C_{l-1}^{\alpha+1}(z),
\begin{cases}E_{\text i}^+(\kappa_x) + E_{\text r}^-(\kappa_x) = E_{\text t}^+(\kappa_x) E_{\text i}^+(\kappa_x) - E_{\text r}^-(\kappa_x) = \alpha(\kappa_x) E_{\text t}^+(\kappa_x) \end{cases}
\begin{array}{l l}{{\nabla \cdot \mathbf{E}^{\prime}= \rho_{e}^{\prime},}}&{{\nabla \times \mathbf{B}^{\prime}= \mathbf{j}_{e}^{\prime}+ \frac{\partial \mathbf{E^{\prime}}}{\partial t},}}\\{{\nabla \cdot \mathbf{B}^{\prime}= \rho_{m}^{\prime}= 0 ,}}&{{\nabla \times \mathbf{E}^{\prime}= - \frac{\partial \mathbf{B^{\prime}}}{\partial t}.}}\\ \end{array}
S_{f, \tau, \Lambda}x=\sum_{\lambda\in \Lambda}f(\pi(\lambda)^{-1}x) \pi(\lambda)\tau =\frac{o(\Lambda)}{o(G)}\sum_{\mu\in \Lambda^0}f(\pi(\mu)^{-1}\tau) \pi(\mu)x=\frac{o(\Lambda)}{o(G)}S_{f, x, \Lambda^0}\tau, \forall x \in \mathbb{C}^{o(G)}.
\langle \hat{t}, \stackrel{\vee}{\phi}\rangle \propto \int d^{3 r - 1}p^{'}d^{3 v}q^{'}\hat{t}_{1}^{[ \mu ]}( p^{'}) \hat{t}_{2}^{[ \nu ]}( q^{'}) \psi_{[ \mu \nu ]}( p^{'}, q^{'}) ,
\prod_{j<p}|\lambda(D_{pp})-\lambda(D_{jj})|=\left|\Delta\left(\{\lambda_l\}_{l=1}^k\cup\{x_j\pm\mathrm{i}y_j\}_{j=1}^{(N-k)/2}\right)\right|\prod_{j=1}^{(N-k)/2}\frac{1}{2y_j}
\left[\varphi^*_{(d,-d)}{\mathcal{Z}}^{ct}_2\right] = \frac{1}{2}\left(\frac{d^2}{2}\left( (\psi_1-\delta_2)+(\psi_2-\delta_2) \right)+d^2\delta_2-\frac{d^2}{2}\delta_{1,1}\right)^2.
\tau+E_{(-1)}=\begin{pmatrix}\tau+(e_{11})_{-1}(e_{12})_{(-1)}\dots (e_{1n})_{(-1)}(e_{21})_{-1}\tau+(e_{22})_{(-1)}\dots (e_{2n})_{(-1)}\vdots \cdots \ddots \vdots (e_{n1})_{-1}(e_{n2})_{(-1)}\dots \tau+(e_{nn})_{(-1)}\end{pmatrix}.
\iint_{\underline a, \underline b : \underline a \wedge \underline b = \phi}\sum_{n=0}^{\infty}\sum_{c_1\cdots c_n}\frac{\mu[c_1\cdots c_n]^q}{d( c_1\cdots c_n\underline a, c_1\cdots c_n\underline b)^{s(q-1)}}d\mu(\underline a)d\mu(\underline b)<\infty
J_{N}(x,y)= p^{N}{\textstyle\sum\limits_{J\in G_{N}^{0}}}{\textstyle\sum\limits_{K\in G_{N}^{0}}}{\textstyle\sum\limits_{J_{j}\in G_{J}^{M}}}{\textstyle\sum\limits_{K_{j}\in G_{K}^{M}}}A_{JK}\Omega\left( p^{M}\left\vert x-J_{j}\right\vert_{p}\right)\Omega\left( p^{M}\left\vert y-K_{j}\right\vert_{p}\right)
\vartheta_{j}\epsilon_{j+1}= s_j^{-1}\epsilon_j d_j^{9}t_j^{-18}C_{t_j}^2 \PT^{-1}_j \epsilon_j \le s_j^{-\frac{1}{2}}\epsilon_j^{\frac{3}{2}}\hat{c}^{-2}T^{-3}_j \le \hat{c}^{-2}\epsilon_0 T^{4(j+1)^2}T^{-3}_j \le \hat{c}^{-2}\epsilon_0 T_{j}
N(x;k):=\begin{cases}e^{ic_+(x)\sigma_3}\left(\frac{\psi_1^-(x;k)}{a(k)}, \psi_2^+(x;k)\right), \; k\in D^+, begin{align*}5pt] e^{ic_+(x)\sigma_3}\left(\psi_1^+(x;k),\frac{\psi_2^-(x;k)}{\overline{a(\bar{k})}}\right), \; k\in D^-, \end{cases}
\nabla (e^{-\tau \Phi}) =-\tau \nabla \Phi e^{-\tau \Phi}\Delta (e^{-\tau \Phi}) =\div (\nabla (e^{-\tau \Phi}))=-\tau \div ( \nabla \Phi e^{-\tau \Phi})=-\tau (\Delta \Phi) e^{-\tau \Phi}-\tau \nabla \Phi \cdot \nabla(e^{-\tau \Phi}) &=-\tau (\Delta \Phi) e^{-\tau \Phi}+ \tau^2 |\nabla \Phi|^2 e^{-\tau \Phi}.
Z = \int{\cal D}[ f ( \xi ) , g ( \xi ) , X ] \, \delta \left( g_{z z}^{f}\right) \delta \left( g_{\bar{z}\bar{z}}^{f}\right) \, \mathrm{d e t}\left( \frac{g_{z z}^{f}}{\delta f}\right) \mathrm{d e t}\left( \frac{g_{\bar{z}\bar{z}}^{f}}{\delta f}\right) \, e^{- S [ g , X ]}.
t r \langle z | \exp ( - i \tau H ) | y \rangle = N^{- 1}\int{\cal D}[ x ( t ) , \psi ( t ) ] \exp \left\{i \int_{0}^{\tau}d t \left[ L_{b o s}( x ( t ) ) + L_{f e r}( \psi ( t ) , x ( t ) ) \right] \right\},
|\mathrm{H}_{u}(u_k,v_k)\varphi|+|\mathrm{H}_{v}(u_k,v_k)\psi| \leqslant a|u_{k}|^{\theta(x)-1}(|\varphi|+|\psi|)+ b|v_{k}|^{\sigma(x)-1}(|\varphi|+|\psi|)\leqslant \mathsf{B}\in L^{1}(B_R).
\sqrt{n}\left\langle f,\Omega_{n}\right\rangle =\left\langle f,a^{\dag}\Omega_{n-1}\right\rangle =\frac{1}{\sqrt{2}}\left\langle f,\left(M_{x}-\partial_{x}\right) \Omega_{n-1}\right\rangle & =\frac{1}{\sqrt{2}}\left\langle \left( M_{x}+\partial_{x}\right)f,\Omega_{n-1}\right\rangle
\begin{aligned}\int_0^1\big|X_N(L,\alpha)\big|^2\,d\alpha \ll\frac{L}{N^3}\sum_{\substack{1\leq r\leq N^2 1\leq s\leq N^2}}W_N(w_r,\mathcal{A})W_N(w_s,\mathcal{A})\frac{\gcd(w_r,w_s)}{\sqrt{|w_rw_s|}}\ll \frac{L}{N^3}\sum_{1\leq r\leq N^2}W_N(w_r,\mathcal{A})^2\exp\left(\frac{10\log r}{\log\log (r+1)}\right).\end{aligned}
\rho_{s,3}(\widetilde{\mathcal A}_{s}\varphi) \, :=\,{\mathbb E}_{0}\Big[ Z_{s}^{-1}\Big( \varphi^{\prime}( \widetilde{X}_{s}) b ( s, \widetilde{X}_{s},{F}_{s, 2}) + \frac{1}{\, 2\,}\varphi^{\prime\prime}( \widetilde{X}_{s}) \Big) \Big \vert \mathcal F_{T}^{X}\Big] \, , 0 \le s \le T \, .
\left\{\frac{1}{\sqrt{N}}\sum_{a}( \nabla \Phi_{a}( x ) ) \cdot ( \nabla \Phi^{a}( x ) ) \right\}= \frac{( \nabla q ( x ) ) \cdot ( \nabla q ( x ) )}{4 q ( x )}\hspace *{0 . 5 c m}.
p_{\gamma\dot{\alpha}}A_{\epsilon_{1}\epsilon_{2}\cdot\cdot\cdot\epsilon_{n}}^{\dot{\alpha}\dot{\beta}_{1}\dot{\beta}_{2}\cdot\cdot\cdot\dot{\beta}_{n}}=mcB_{\gamma\epsilon_{1}\epsilon_{2}\cdot\cdot\cdot\epsilon_{n}}^{\dot{\beta}_{1}\dot{\beta}_{2}\cdot\cdot\cdot\dot{\beta}_{n}}
\frac{d W}{dt}\le \left| \frac{d W}{dt}\right| = | \nabla w_\Omega (x_0 + t n(x_0)) \cdot n(x_0)| &\le |\nabla w_\Omega (x_0 + t n(x_0)) | \le G(w_\Omega (x_0 + t n(x_0))) &= G (W(t)).
\alpha(\varepsilon)=\frac{b\,m}{d-b}+\frac{b\, d\,m\,(N+1)\, \varepsilon}{(d-b)^2 \,N+2\,(d-b)\,m-(d-b)^2}, \beta(\varepsilon)=-\frac{2\,b\,d\,m\, \varepsilon}{(d-b)^2 \,N+2\,(d-b) \,m-(d-b)^2}.
E = - \frac 1{2 a}\sum_{l = 1}^{\infty}( 2 l + 1 ) \int_{- \infty}^{\infty}\frac{d y}{2 \pi}e^{i \delta y}x \frac d{d x}\ln \left[ 1 - \xi^{2}\lambda_{l}^{2}( x ) \right] .
\int_{\mathbb{S}^2}e^{- 2 \, p \, (r \circ \hat \varphi)}\, e^{2 \, p \, \hat r}= O ( e^{2 \, p \, \hat r}\, e^{- 2 \, \hat r}) \int_\Sigma e^{- 2 \, p \, r}= O (1).
{\rm II}^{p'}\leq c\sum_{j=0}^{n-1}\int_{t_j}^{t_{j+1}}\| \bar E(t_n-t) - B_{n-j}P_h \|_{\mathcal{L}_2^0}^{p'}\d t & \leq c\| A^{-\frac{s}{2}}\|_{\mathcal{L}_2^0}^{p'}\sum_{j=0}^{n-1}\int_{t_j}^{t_{j+1}}\|A^{\frac{s}{2}}(\bar E(t_{n}-t)-B_{n-j}P_{h}) \|^{p'}\, \d s.
\sigma_{y}^{2}(n\tau_{0},N)=AVAR(n\tau_{0},N)=\frac{1}{2n^{2}\tau_{0}^{2}(\frac{N-1}{n}-1)}\sum_{i=0}^{\frac{N-1}{n}-2}(x_{ni+2n}-2x_{ni+n}+x_{ni})^{2}
\tilde{H}_{p,m}(t)={\rm Res}_{x_{0}, \cdot, x_{2m}}\frac{(1+x_{0})^{m^2p+mp-(m+1)t}\big(2+3x_{0}+3x_{0}^{2}+x_{0}^{3}\big)}{x_{0}^{2m^{2}p+5}}(x_{1}\cdots x_{2m})^{-2mp}\phantom{\tilde{H}_{p,m}(t)=}{}\times\prod\limits_{i=1}^{2m}(1+ x_{i})^{t}\prod\limits_{1\leq i<j\leq 2m}(x_{i}-x_{j})^{2p}\prod\limits_{i=1}^{2m}\left(1-\frac{x_{i}}{x_{0}}\right)^{-2mp}.
\begin{aligned}\hat{\mathcal{S}}_{\alpha\beta}(r,s) =\mathcal{\hat{S}}_{\beta\alpha}(s,r), \mathcal{S}_{\alpha\beta}(r,s) =\hat{\mathcal{S}}_{\alpha\beta}(r,s)-A\theta^{(\alpha)}(r)\rho\hat{\eta}^{(\beta)}(s),\end{aligned}
N^*\psi(h_1)^\beta N \equiv N^* \prod_{i=-\infty}^{-1}\psi(R(a))^{(ab)^{-i}}\cdot \left[\psi(R(a)),\psi(L(b))\right] \cdot \prod_{i=1}^\infty \psi(L(b))^{-(ab)^i}N \pmod{2^{t}}.
\tilde{\Delta}((\tilde{x}^i)^p (\tilde{x}^j)^q)= p(p-1) (\tilde{x}^i)^{p-2}(\tilde{x}^j)^q+q(q-1) (\tilde{x}^i)^p (\tilde{x}^j)^{q-2}& -(p+q)(p+q+1) (\tilde{x}^i)^p (\tilde{x}^j)^q.
\begin{aligned}\int_{B_1 (s,x_0)}(\partial_t \phi) (\partial_t \psi) + (\partial^i \phi) (\partial_i \psi) d x = \int_{\Sigma_0}(\partial_t \phi) (\partial_t \psi) + (\partial^i \phi) (\partial_i \psi) d x - \int_0^s \int_{\Sigma_t}(\Box \phi) (\partial_t \psi) d x d t, \end{aligned}
\left( \frac{C^{2}\, e^{2\, \alpha}\, n^{4}}{\left( \left( x_{0}+y_{0}+i\, \epsilon \right)^{2}-\left( x_{s}+y_{s}\right)^{2}\right)^{2}}+\frac{C^{2}\, e^{2\, \alpha^{*}}\, n^{4}}{\left( \left( x_{0}+y_{0}-i\, \epsilon \right)^{2}-\left( x_{s}+y_{s}\right)^{2}\right)^{2}}\right.
\left. \tilde{\Phi}[ \tilde{x}, \phi , e_{0}] \right|_{\phi = n x_{c l}}= \Phi [ x_{c o m b}, e_{0}] , \qquad \left. \tilde{I}[ \tilde{x}, \phi , e_{0}] \right|_{\phi = n x_{c l}}= I [ x_{c o m b}, e_{0}] .
\frac{1}{2}ix\zeta^2-\frac{1}{2}ix\alpha^2\ln\zeta=-\frac{x}{2}\lambda+\frac{i}{4}x+\frac{x}{4}+\pi a b-\frac{1}{4}\ln x+\frac{1}{2}\ln(b{e}^{-2is})+o(1),
\bar{b}_{c}=\frac{\lambda}{\lambda+\rho}\nabla U_{c}+\frac{1}{\lambda+\rho}\sum_{k=1}^{m}\rho_{k}\nabla U_{k}\bar{\gamma}_{c_{1},c_{2}}=\frac{\lambda}{\lambda+\rho}(\nabla U_{c_{1}}-\nabla U_{c_{2}}).
(\Delta_h \phi)_{jk}=-\sum_{p=-M/2}^{M/2 -1}\sum_{q=-M / 2}^{M/2-1}\left((\varrho^{x}_{p})^{2}+(\varrho^{y}_{q})^{2}\right) \widehat{\phi}_{pq}\, e^{i \frac{2j p \pi}{M}}e^{i \frac{2k q \pi}{M}}, &(L_{z}^h\phi)_{jk}=-i\Big(x_{j}(\partial_{y}^{h}\phi)_{jk}-y_{k}(\partial_{x}^{h}\phi)_{jk}\Big), j,k=0,1,\cdots, M-1,