harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\overline{p}_{n,k}= \left[ \frac{c_k + \delta_k}{\theta \log 2 \beta_{n,k}}- 1 \right]^+ = \left[ \frac{c_k + m_k \theta - c_k}{\theta \log 2 \beta_{n,k}}- 1 \right]^+ = \left[ \frac{m_k}{\log 2 \beta_{n,k}}- 1 \right]^+
\begin{array}{l}{M_{\gamma_{D,2}^{M,r}}}(s) = \prod\nolimits_{k = 1}^M{{\delta_k}^{- m}{{\left({{\delta_k}^{- 1}+ s}\right)}^{- m}}}\times \prod\nolimits_{l = r + 1}^M{{\alpha_l}^{- m}{{\left({{\alpha_l}^{- 1}+ s}\right)}^{- m}}}\end{array}
\sum_{m=0}^\infty\frac{(-\rho)^{m}}{m!}\Biggr[ -4\sum_{i=m}^{2m}\left(\frac{-1}{1-s^2}\right)^{i+1}4^{2m}2^{2(i-m)}(2m)!\left(\binom{i+1}{2(i-m)+1}+\binom{i}{2(i-m)+1}\right) &-\frac{2}{(1-s^2)^{2m+1}}\int_1^\infty\frac{\d{t}}{t}t^{\frac{1}{4}}\Delta_4^{2m+1}\bigr(\Psi(\cdot)\ln^{2m}(\cdot)\bigr)(t)\cosh\left(\frac{\ln(t)s}{4}\right)\Biggr]
\sum_{p = 0}^n{\binom{m+n-p-1}{m-1}\frac{{\mu^m}}{{\nu^{p + 1}(\nu + \mu )^{m + n - p}}}}& = \frac{1}{{\nu^{n + 1}}}- \sum_{i = 1}^m{\binom{i+n-1}n\frac{{\mu^{i - 1}}}{{(\nu + \mu )^{i + n}}}}\,,
s_d(n) = [x^n] y \sim \sqrt{\frac{\phi(s)}{2 \pi \phi''(s)}}\; n^{-3/2}\, \phi'(s)^n &= \sqrt{\frac{s / M_d(s)}{2 \pi \big( -s M_d''(s) / M_d(s)^2 \big)}}\; n^{-3/2}\left( \frac{1}{M_d(s)}\right)^n &\;=\; \frac{1}{\sqrt{-2\pi M_d''(s)}}\, n^{-3/2}\, M_d(s)^{1/2-n}.
\left\lbrace\begin{array}{l}\dot{R}+ H_{px}(Y,P) R + R H_{xp}(Y,P)+R H_{pp}(Y,P) R+ H_{xx}(Y,P)=0, R(\xi_0, 0)= P_{\xi,t}(\xi_0,0)Y^{-1}_{\xi,t}(\xi_0,0) .\end{array}\right.
\sum_{j=1}^{p-1}\omega_q^{\left(a^t_{\sigma, r^j}-a^t_{\sigma, s^j}\right)-\left(a^t_{\sigma, r}-a^t_{\sigma, s}\right)}=\sum_{j=1}^{p-1}\omega_p^{j\left(s_{\pi_{\hat{\beta}}(\hat{\gamma})}-r_{\pi_{\hat{\beta}}(\hat{\gamma})}\right)}=-1.
A(j_1,k_1)=\begin{cases}2(2c)^{2t_1+k_1+1}t_1! (k_1+t_1)! \frac{\pi^{m/2+1}}{\Gamma(m/2)}j_1=2t_1, 2(2c)^{2t_1+k_1+2}t_1! (k_1+t_1+1)! \frac{\pi^{m/2+1}}{\Gamma(m/2)}j_1=2t_1+1.\end{cases}
P_{v}= \exp \left\{- \mu \tilde{n}^{c r}\int d{\bf x}\right\}, \mu = \sum_{n = 0}^{\infty}\frac{\beta ( n + 1 )}{( n + 1 )^{3 / 2}\beta ( 1 )}\exp \left\{- n \pi \frac{( a M )^{2}}\rho \right\}.
\Phi_2(\sigma_2;U_a)=(u_{a}e^{\sigma_2},v_{a}e^{\sigma_2},p_a,\rho_a,Z_a), &\Phi_3(\sigma_3;U_a)=(u_a,v_a,p_a,\rho_ae^{\sigma_3},Z_a), &\Phi_4(\alpha_4;U_a)=(u_a,v_a,p_a,\rho_a,Z_a+\alpha_4).
\lambda = \left( 4 g_{2}w ( p - k ) w ( k +{\frac{q}{2}}) \right)^{2}={\frac{1}{N_{f}^{2}}}a^{- 4}\left( 4 \tilde{g}_{2}w ( p - k ) w ( k +{\frac{q}{2}}) \right)^{2}. \nonumber
l_2 = \sum_{i=1}^3 \sigma_i b_i + \sum_{j=1}^3 \tau_j y_j \equiv \sum_{i=1}^3 \sigma_i \varphi(a_i) \equiv \varphi\left( \sum_{i=1}^3 \sigma_i a_i \right)\equiv \varphi(l_1) \pmod{\ell^t}.
\begin{array}[c]{c}\dfrac{\partial u}{\partial t}+u.\nabla u=\nu\triangle u+\nabla p+f,\text{}t>0, u=0,\Omega\times\left( 0,\infty\right) \text{,}u=0\partial\Omega\times\left( 0,\infty\right) \text{and}u\left( x,0\right) =u_{0},\Omega\end{array}
0 \rightarrow{\cal O}_{\pi^{*}{\cal S}}( \sigma |_{\pi^{*}{\cal S}}) \rightarrow{\cal O}_{\pi^{*}{\cal S}}( 2 \sigma |_{\pi^{*}{\cal S}}) \rightarrow{\cal O}_{\pi^{*}{\cal S}}( 2 \sigma |_{\pi^{*}{\cal S}}) |_{\sigma |_{\pi^{*}{\cal S}}}\rightarrow 0
\frac{\Delta_k^-(\lambda;x)}{\Delta(\lambda)}\underset{b\to+\infty}{\longrightarrow}\sum_{\ell=1}^{N}\alpha_{k\ell}\,\mathrm{e}_{\lambda}^{\theta_{\ell}\,(x-a)}\quad\frac{\Delta_k^+(\lambda;x)}{\Delta(\lambda)}\underset{b\to+\infty}{\longrightarrow}0
\Vert W \Vert_\square = \sup_{S, T\subset [0,1]}\left\vert \int_{S\times T}W(x,y)dxdy \right \vert &=\sup_{f,g\colon [0,1]\to [0,1]}\left\vert \int W(x,y)f(x)g(y)dxdy\right \vert,
\ddot{x}^{2}\big [ \acute{x}{}^{2}x \! \! \dot{}\, \acute{}\,{}^{a}- ( \acute{x}x \! \! \dot{}\, \acute{}\, ) \acute{x}{}^{a}\big ] - \acute{x}^{2}( \ddot{x}x \! \! \dot{}\, \dot{}\, \acute{}\, ) \dot{x}{}^{a}= 0 .
2m = \sum_{v\in V'}d(v) + \sum_{v \in V''}d(v) = d(w) + \sum_{v \in V'\setminus \{w\}}d(v) + \sum_{v \in V''}d(v) &\geq 1 + 3|V' \setminus \{w\}| + 4 |V''| &= 1 + 3\big(|V'|-1\big) + 4\big(n-|V'|\big) &= 4n - |V'| - 2.
\frac{\partial \tilde H}{\partial m}(x,p,m)= \frac{\partial \tilde f}{\partial m}(x,v^*,m) + \nabla_v \tilde f (x,v^*,m) \cdot \frac{\partial v^*}{\partial m}+ p \cdot \frac{\partial v^*}{\partial m}= \frac{\partial \tilde f}{\partial m}(x,v^*,m) ,
\sigma^{\prime}\equiv \sigma +{\frac{1}{2}}C_{n}\alpha +{\frac{1}{2}}C_{p}\alpha +{\frac{1}{2}}m_{e}\ , \quad \delta^{\prime}\equiv{\frac{1}{2}}\delta +{\frac{1}{2}}C_{n}\alpha -{\frac{1}{2}}C_{p}\alpha -{\frac{1}{2}}m_{e}\ ,
S_{\chi}= \pi^{2}\chi_{0}^{2}\int_{0}^{\infty}d r r^{3}\left\{[ f^{\prime}( r ) ]^{2}+ m_{\chi}^{2}[ f ( r ) ]^{2}- \mu_{0}^{2}\theta ( r - r_{0}) [ f ( r ) ]^{2}\right\}
H [ \Psi , \xi ] \! \! : = \! \! \frac{i}{4}\int [ \bar{\Psi}( t , \vec{\bf r}) \vec{\gamma}\cdot \vec{\nabla}\Psi ( t , \vec{\bf r}) - \vec{\nabla}\bar{\Psi}( t , \vec{\bf r}) \cdot \vec{\gamma}\Psi ( t , \vec{\bf r}) ] d r^{3}
{\cal L}= \frac{1}{2}( \partial_{\mu}\Phi_{1}^{\prime})^{2}+ \frac{1}{2}( \partial_{\mu}\Phi_{2}^{\prime})^{2}+ \frac{1}{2}( \partial_{\mu}\Phi_{3}^{\prime})^{2}+ \frac{m}{\pi}\mu \cos ( 2 a \sqrt{\pi}\Phi_{1}^{\prime}) ,
b\left( x,t,\nabla u\left( x,t\right) \right) =\int_{\mathcal{Y}_{2,3}}a\left( y^{2},s^{3},\nabla u\left( x,t\right)+\nabla_{y_{1}}u_{1}\left( x,t,y_{1},s_{1}\right) +\nabla_{y_{2}}u_{2}\left( x,t,y^{2},s^{3}\right) \right) dy^{2}ds^{3}
\,D_{i_s}^{(k)}=\left(\frac{\mathsf{d}}{\mathsf{d}x_s}\right)^{(k)}\circ\dots\circ \left(\frac{\mathsf{d}}{\mathsf{d}x_s}\right)^{(k)}=\left( \left(\frac{\mathsf{d}}{\mathsf{d}x_s}\right)^{(k)}\right)^{\circ i_s},
d \overline{{s}}^{2}= \overline{{g}}_{\mu \nu}d x^{\mu}d x^{\nu}= d t^{2}- \overline{{a}}^{2}( t ) d l^{2}, d l^{2}= [ d r^{2}/ ( 1 - k r^{2}) + r^{2}d \Omega^{2}] ,
\overline{R}(\overline{X},\overline{Y})\overline{Z}=c\{\overline{g}(\overline{Y},\overline{Z})\overline{X}-\overline{g}(\overline{X},\overline{Z})\overline{Y}\},\quad\forall\, \overline{X},\overline{Y},\overline{Z}\in\Gamma(T\overline{M}).
\nabla^Q t= ev \circ T \circ d \phi(t)=ev \circ T \left(x,{T}d\left(ev^{\ast}(t) \right) \right) = ev \circ T \left(x,{T}\left(\nabla ev^{\ast}\right)(t)\right)+ev \circ T \left(x,{T}\circ ev^{\ast}(\nabla t)\right), = ev \circ T^2\circ \left(\nabla ev^{\ast}\right)(t)+\phi^{\ast}\phi(\nabla t),
m_{1}\frac{d^{2}r_{1}}{dt^{2}}=-\frac{m_{1}m_{2}G(r_{1}-r_{2})}{|r_{1}-r_{2}|^{3}};m_{2}\frac{d^{2}r_{2}}{dt^{2}}=-\frac{m_{1}m_{2}G(r_{2}-r_{1})}{|r_{2}-r_{1}|^{3}},
-\frac{\epsilon_m^2\beta_m^2}{\alpha_m}\Delta u_m(x_m +\epsilon_m\beta_mx)+\frac{\beta_m^2}{\alpha_m}\lambda u_m(x_m +\epsilon_m\beta_mx) &+\beta_m^2\rho(x_m +\epsilon_m\beta_mx)\phi_{u_m}(x_m +\epsilon_m\beta_mx)\frac{1}{\alpha_m}u_m(x_m +\epsilon_m\beta_mx)=\frac{\beta_m^2}{\alpha_m}u_m^p(x_m +\epsilon_m\beta_mx).
det(A)=\sum_{\begin{matrix}k_{1},k_{2},...,k_{n}\ge0\\k_{1}+2k_{2}+\cdot\cdot\cdot+nk_{n}=n\end{matrix}}\prod_{l=1}^{n}\frac{(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}tr(A^{l})^{k_{l}},
\binom{k+(i+j+1)p}{m+ip}=\frac{(k+p+(i+j)p)\cdots(p+(i+j)p)}{(m+ip)\cdots(1+ip)}\binom{k+p-m+(i+j)p}{ip}&\equiv_{p^2}(i+j+1)\binom{k+p}{m}\binom{k-m+p+(i+j)p}{ip}&\equiv_{p^2}(i+j+1)\frac{p(-1)^{m-(k+1)}}{(k+1)\binom{m}{k+1}}\binom{i+j}{i}.
\hat{V}_{\rm cyl}=\frac{v_{n+1}r^{n+1}L}{\frac12 v_{n+2}L^{n+2}}=\frac{2 v_{n+1}}{v_{n+2}}\left( \frac{r}{L}\right)^{n+1},\;\;\;(0 \leq r < +\infty), \hat{A}_{\rm cyl}=\frac{a_n r^n L}{\frac12 a_{n+1}R^{n+1}}=\frac{2a_n}{a_{n+1}}\left( \frac{r}{L}\right)^n \left( \frac{L}{R}\right)^{n+1},
\Gamma_{\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}\left( k_{1}, k_{2}, k_{3}\right) = \frac{g^{3}}{\left( 2 \pi \right)^{3}}\int \frac{\mathrm{d}^{3}{\bf Q}}{2 |{\bf Q}|}N \left( |{\bf Q}| \right) \left[{\cal S}_{\mu_{1}\mu_{2}\mu_{3}}^{a_{1}a_{2}a_{3}}\left( k_{1}, k_{2}, k_{3}, Q \right) + Q \rightarrow - Q \right] ,
[\begin{matrix}a&v\\w&b\end{matrix}][\begin{matrix}a^{\prime}&v^{\prime}\\w^{\prime}&b^{\prime}\end{matrix}]=[\begin{matrix}aa^{\prime}+v\cdot w^{\prime}&av^{\prime}+b^{\prime}v+w\times w^{\prime}\\a^{\prime}w+bw^{\prime}-v\times v^{\prime}&bb^{\prime}+v^{\prime}\cdot w\end{matrix}]
\tilde{\ell_{1}}(x) \leq \bar{\ell_{1}}(x)+\frac{2\gamma\varepsilon}{\lambda_{1}^2}(\lambda_{1}+ \|x_{1}\|), \ -\tilde{a_{1}}\|x\|^2=-\bar{a}_{1}\|x\|^2+\frac{\varepsilon}{\lambda_{1}^2}\|x\|^2\leq-\bar{a}_{1}\|x\|^2+\frac{\varepsilon}{\lambda_{1}^2}\gamma^2 ,
1=\sum_{n,n_1}\prod_{i=1}^n \sum_{\underline{m}_i}\prod_{j=1}^{n_1}\sum_{\underline{l}_j}\left|\tilde{\phi}^{(n,n_1)}(\underline{m}_i,\underline{l}_j, \underline{K}-\sum_i\underline{m}_i-\sum_j\underline{l}_j) \right|^2\,.
\mathbb E [ X_{s}^{\dagger}X_{t}^{\dagger}] \, =\, \mathbb E [ \widetilde{X}_{s}^{\dagger}\widetilde{X}_{t}^{\dagger}]\, =\, e^{-(t-s)}\int^{s}_{0}e^{-2 v}I_{0}( 2 \sqrt{(t -s + v) v}){\mathrm d}v \, ; 0 \le s \le t \, .
\mathfrak{P}_k(z_0\times z_0)^{-1}\overline{\mathbf{f}(g_{\mathbf{h}}\times h_{\mathbf{h}})}=\sum_{j=1}^e\mathfrak{P}_k(z_0)^{-1}\overline{\mathbf{g}_j(g_{\mathbf{h}})}\mathfrak{P}_k(z_0)^{-1}h_j(\mathbf{h}).
\xi_{\nu}^{+}(\nu^{+})= 2^{[\frac{l}{2}]}(\sqrt{-1})^{\frac{\nu_1+\cdots+\nu_m-m}2}\cdot\sqrt{\nu_1\cdots\nu_m}&\cdot(\sqrt{-1})^{\frac{\nu_{m+1}+\cdots+\nu_l}2}\cdot\sqrt{\nu_{m+1}\cdots\nu_l}= 2^{\frac{l}{2}}(\sqrt{-1})^{\frac{n-m}{2}}\sqrt{\frac{\nu_1\nu_2\cdots\nu_l}{2}}.
{}I_3 \le \frac{c}{T_n^2}\int_0^{T_n}\int_{\frac{1}{2}(\prod_{l \ge 3}h_l^*)^{\frac{2}{d-2}}}^D (\frac{c}{s'^{\frac{d}{2}}}+ c) ds' dt = \frac{c}{T_n}((\prod_{l \ge 3}h_l^*)^{-1}+ D),
\phi(\mathbf{v}(a\oplus b)) =\boldsymbol{e}_{23}(a+b), \phi(\mathbf{w}(a\oplus b)) =\boldsymbol{e}_{32}(a+b), \phi(\mathbf{f}(a\oplus b)) =\boldsymbol{e}_{12}(a)+\boldsymbol{e}_{31}(\rho(b)), \phi(\mathbf{g}(a\oplus b)) =\boldsymbol{e}_{21}(a)-\boldsymbol{e}_{13}(\rho(b)),
\frac{2C_1C_2}{\pi}{\cal B}\left(\frac{1}{2},-\frac{\lambda+m+1}{2}\right){\cal B}\left(\frac{1}{2},-\frac{\lambda+n+1}{2}\right){\sigma}_1^{\frac{\lambda+m+1}{2}}{\sigma}_2^{\frac{\lambda+n+1}{2}}\leq
\mathrm{[}\hat{\Lambda}( k^{1}) , \hat{\Lambda}( k^{\prime \, 1}) ] = [ \hat{\Lambda}^{\dagger}( k^{1}) , \hat{\Lambda}^{\dagger}( k^{\prime \, 1}) ] = 0 ; \, \, \, [ \hat{\Lambda}( k^{1}) , \hat{\Lambda}^{\dagger}( k^{\prime \, 1}) ] = \delta ( k^{1}- k^{\prime \, 1}) ,
\limsup_{\varepsilon\to0}\int_{y_\varepsilon}^{x_\varepsilon}\frac{p_\varepsilon}{z^{\alpha+1}}dz\le \limsup_{\varepsilon\to0}\frac{p_\varepsilon}{(x_\varepsilon\wedge y_\varepsilon)^{\alpha+1}}\int_{y_\varepsilon}^{x_\varepsilon}dz=\limsup_{\varepsilon\to0}\frac{(x_\varepsilon-y_\varepsilon)^2}{\varepsilon(x_\varepsilon\wedge y_\varepsilon)^{\alpha+1}}=0.
\begin{array}{l}{\dot{o}^{A}= \frac{1}{2}\dot{m}o^{A}- \i \phi \iota^{A},}{\dot{\iota}^{A}= \i \mu o^{A}- \frac{1}{2}\dot{\overline{{m}}}\iota^{A}+ \acute{o}^{A}.}\end{array}
C3D_{L}=\sum_{m=l}^{m}\sum_{n=l}^{n}tcrx_{m,n,l}+\sum_{m=l+1}^{m}\sum_{q=l+1}^{n}tcrx_{m,l,q}+\sum_{n=l+1}^{m}\sum_{q=l+1}^{m}tcrx_{l,n,q}
\frac{n!}{\prod_{k=1}^{n}(k!)^{j_{k}}}\prod_{k=1}^{n}(\frac{k!}{k})^{j_{k}}\prod_{k=1}^{n}\frac{1}{j_{k}!}=\frac{n!}{\prod_{k=1}^{n}k^{j_{k}}j_{k}!}.
g(u_1,u_2)= \alpha_1 |u_1^2+u_2^2|^2 + \alpha_2 |u_1^2-u_2^2|^2 -4\alpha_2 |u_1|^2|u_2|^2 +2\alpha_3 \cos \eta (|u_1|^4-|u_2|^4) &+4 \alpha_3 \sin \eta (|u_1|^2+|u_2|^2) \Re (\overline{u_1}u_2) + (2\alpha_1 + r) (|u_1|^2+|u_2|^2)^2,
{\rm co}_{-}(M)\ = \ \Bigl\{\, x - y \ \Bigl|\x \in{\rm co}\, (M) , \y \ge 0\Bigr\}\ ;{\rm co}_{+}(M)\ = \ \Bigl\{\, x + y \ \Bigl|\x \in{\rm co}\, (M) , \y \ge 0\Bigr\}\, .
\sum\limits_{n=1}^{N}c_1\| \mu_1 \delta_t e^{n-\frac{1}{2}}+\mu_2 e^{n-\frac{1}{2}}\|^2-\sum\limits_{n=1}^{N}\Lambda_4^{n-\frac{1}{2}}\leq \sum\limits_{n=1}^{N}\Lambda_2^{n-\frac{1}{2}}+\sum\limits_{n=1}^{N}\Lambda_3^{n-\frac{1}{2}}+\sum\limits_{n=1}^{N}\Lambda_5^{n-\frac{1}{2}}.
{\bf{\hat \Lambda}}= diag\left[{1,1,1,1,{{\hat s}_e},{{\hat s}_\upsilon},{{\hat s}_\upsilon},{{\hat s}_\upsilon},{{\hat s}_\upsilon},{{\hat s}_\upsilon},{{\hat s}_q},{{\hat s}_q},{{\hat s}_q},{{\hat s}_q},{{\hat s}_q},{{\hat s}_q},{{\hat s}_\pi},{{\hat s}_\pi},{{\hat s}_\pi}}\right],
f_1 = 1728^{1/6}\eta^{4}K^{-1/6}\_{3}F_2(-1/6, 1/6, 1/2; 3/4, 1/4; K), f_2 =1728^{-1/12}\eta^{4}K^{1/12}\_{3}F_2(1/12, 5/12, 3/4; 5/4, 1/2; K), f_3 =1728^{-7/12}\eta^{4}K^{7/12}\_{3}F_2(7/12, 11/12, 5/4; 7/4, 3/2; K).
S^{-1}\left(\begin{bmatrix}\frac{1}{\sqrt{2 |U_n| \sqrt{U_1}}}0 0 \frac{1}{\sqrt{2 |U_n| \sqrt{U_1}}}\end{bmatrix}\begin{bmatrix}U_1^2+U_n^2 U_n U_{\tau}\end{bmatrix}-\begin{bmatrix}R_1 R_2\end{bmatrix}\frac{1}{\sqrt{2 |U_n| \sqrt{U_1}}}U_1^2 \right) = G.
T_{\phi \phi}= \frac{1}{( k - 2 )}g_{a b}{\large :}J_{\phi}^{a}( z ) J_{\phi}^{b}( z ){\large :}= \frac{1}{k - 2}\Sigma{\large :}J_{n - m}^{a}J_{m}^{a}{\large :}z^{- n - 2}= \Sigma L_{n}z^{- n - 2},
Z_L^{(n)}(\{q_{N_s}\})=\tilde Z^{(n)}(\{q_{N_s}\},L)H(\{q_{N_s}\}) \prod_{s=1}^n\frac{(-1)^{2l_{1s}+2l_{2s}-1}16^{2l_{1s}}Z^{(1)}(k_s;l_{1s},l_{2s})}{k_s^{(3-2l_{1s})/2}}
\mathrm{Z}_{\mathrm{R}}^{I}( r ) = \frac{1}{2}\, \mathrm{T r}_{\mathrm{R}}\left( r^{2 L_{0}}\right) = \mathrm{T r}_{\mathrm{R}}\left( r^{2 L_{0}}{\frac{1 + \Omega}{2}}\right) ~ ~ ,
\mathcal{R}_{(2),P}(R_1')= \left\{\begin{aligned}(R_2,R_3): R_2,R_3\geq 0,\R_2\leq I(U_1;Y_2|U_2)\triangleq r_e,\R_3\leq I(X_2;Y_3) \triangleq r_f, R_2+R_3\leq I(U_1,U_2;Y_2)+I(X_2;Y_3|U_2) \triangleq r_g \end{aligned}\right\}.
I_2 = \int_{E_2}\big\lvert H\big(u+\mathrm{i}a(u) +\mathrm{i}\tau\big)\big\rvert^q \cdot \lvert 1+\mathrm{i}a'(u)\rvert \,\mathrm{d}u \leqslant M_2^q\cdot \sqrt{1+M^2}\cdot 2\delta_0.
\lim_{n\to\infty}\,\frac{1}{\mu_n}\int_{\mu_n}^{2\mu_n}\int_{|x|<4t}\,\left(\partial_tu+\frac{x}{t}\cdot\nabla u+\left(\frac{d}{2}-1\right)\frac{u}{t}\right)^2(x,t)\,dx\,dt=0,
\bar{a}^1 = \left(a_1^1,\dots, a_{n_1}^1\right) , \ \bar{b}^1 = \left(b_1^1,\dots, b_{n_1}^1\right) & \bar{a}^2 = \left(a_1^2,\dots, a_{n_2}^2\right) , \ \bar{b}^2 = \left(b_1^2,\dots, b_{n_2}^2\right)
( \mathcal{O}_{\pm}^{i n}( \Omega ) )^{\dag}= \mathcal{O}_{\pm}^{o u t}( P \Omega_{A}) \ , ( \mathcal{O}_{\pm}^{o u t}( \Omega ) )^{\dag}= \mathcal{O}_{\pm}^{i n}( P \Omega_{A})
b^{+}_2 (\tilde{X^Y}) = b_1 (\tilde{X^Y}) - 1 + \frac{nlr}{2}( \chi ( X^Y) + \sigma(X^Y)) = (l-1)(r-1) - 1 + \frac{nlr}{2}(\chi(X^Y) + \sigma(X^Y)) &> (l-1)(r-1)-1 + \frac{nlr}{2}( 2b(2g-2) - \frac{(b-1)(g-1)}{2}) > 1
\leq f(Z_T) + G_0 G^2 \eta_0^2 &\leq f(G_0^{-2}\sigma^2 \log(1 / \delta)) + G_0 G^2 \eta_0^2 &= G_0^{-1}\sigma^2 \log(1 / \delta) + G_0 G^2 \eta_0^2
\left( \begin{array}{c}{\vec{x}}\\{\vec{p}}\\ \end{array}\right) = \left( \begin{array}{c c}{{\mathbf a}}&{{\mathbf b}}\\{{\mathbf d}}&{{\mathbf c}}\\ \end{array}\right) \left( \begin{array}{c}{\vec{\alpha}}\\{\vec{\beta}}\\ \end{array}\right)
\lim_{c\to +\infty}e^{-c\langle \xi ,\rho_M\rangle}\Delta_\emph{U}(\xi) \phi_\xi^{\emph{bc}}(x+c\rho_M; g^{(c)})= \Delta_\emph{U}(\xi) \phi^{\emph{t}}_\xi (x;g) .
\Gamma_{\vec{v}}(\varphi, \sigma) = b_{\vec{v}}(\varphi) \mathcal{T}(\sigma) + f_{\vec{v}}(\varphi), \Gamma_{\varphi}(\varphi, \sigma) = b_{\varphi}(\varphi) \mathcal{T}(\sigma) + f_{\varphi}(\varphi).
n_{\ast}\left( \left( \frac{r}{K_{1}}\right)^{\frac{1}{2}},p_{0}U_{\perp}p_{0}\right) := n_{+}\left( \frac{r}{K_{1}}, \big( p_{0}U_{\perp}p_{0}\big)^{\ast}p_{0}U_{\perp}p_{0}\right).
\textstyle W_{ab}{}^c{}_e\sigma^{de}+W_{ab}{}^d{}_e\sigma^{ce}+\frac{2}{3}\delta_{[a}{}^cW_{b]}{}_e{}^d{}_f\sigma^{ef}+\frac{2}{3}\delta_{[a}{}^dW_{b]}{}_e{}^c{}_f\sigma^{ef}=0.
I(s) \equiv \int_1^\infty{{(t^2-1)^{-s}dt}\over{{\rm e}^{2\pi a t}-1}}= \int_1^{t_0}{{(t^2-1)^{-s}dt}\over{{\rm e}^{2\pi a t}-1}}+\int_{t_0}^\infty{{(t^2-1)^{-s}dt}\over{{\rm e}^{2\pi a t}-1}}\, ,
\sum_A g_S(K_X t_A, K_Y t_A)= \sum_{r}g_S(K_X t_r, K_Y t_r)=\sum_{r,\alpha,\beta}g_S(\xi_{\alpha}^rw^{\alpha}, \eta_{\beta}^rw^{\beta}) = \sum_{r,\alpha}\xi_{\alpha}^r \eta_{\alpha}^r.
\left\langle \Psi_1,\Psi_1^*\right\rangle=1,\left\langle \Psi_2, \Psi_2^*\right\rangle=-1,\left\langle\Psi_1,\Psi_2^*\right\rangle=\left\langle\Psi_2, \Psi_1^*\right\rangle=0.
+iq_{0}^{-1}\cos\psi\cdot\left(\frac{F[\frac{1}{q_{0}}](i\mu_{0})w_{1}[\frac{1}{q_{0}}](i\mu_{0})w_{3}[\frac{1}{q_{0}}](i\mu_{0})}{w_{2}[\frac{1}{q_{0}}](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{2}\partial_{\eta}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\begin{cases}a_{i,k}+b_{i,k}+c_{i,k}= 1 d_{i,k}-x_i = r_{i,k}+s_{i,k}+t_{i,k}-Ca_{i,k}\leq r_{i,k}\leq 0 0 \leq s_{i,k}\leq \bar{y}b_{i,k}\bar{y}c_{i,k}\leq t_{i,k}\leq Cc_{i,k}\beta_{i,k}= -c\P(\omega_k)\left(\frac{s_{i,k}}{\bar{y}}+ c_{i,k}\right)\end{cases}
L = \left(\begin{matrix}0 -1 0 0 0 0 0 1 1 0 1 0 1 1 0 1 0 -1 0 -1 -1 -1 0 0 0 0 1 0 0 0 0 1 0 -1 1 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 0 -1 0 0 0 0 \end{matrix}\right).
f^{( \kappa )}( z ) = \frac{1}{\kappa}\biggl ( 1 - \exp \Bigl ( - \kappa \tan^{- 1}\frac{z}{a}\Bigr ) \biggr ) = \sum_{n = 1}^{\infty}\frac{f_{n}}{\sqrt{n}}\biggl ( \frac{z}{a}\biggr )^{n},
\delta(n,m)=\frac{1}{Q}\sum_{1 \leq q \leq Q}\frac{1}{q}\, \ \sideset{}{^\star}\sum_{a\bmod q}e\left(\frac{(n-m) a}{q}\right)\int_{\mathbb{R}}\psi(q,x)e\left(\frac{(n-m) x}{qQ}\right) d x,
\alpha_{1}= \left( \begin{array}{c}{e^{i n \theta}}\\{0}\\{0}\\{d_{+}e^{i ( n + 1 ) \theta}}\\ \end{array}\right) , \alpha_{3}= \left( \begin{array}{c}{0}\\{d_{+}e^{i ( n + 1 ) \theta}}\\{e^{i n \theta}}\\{0}\\ \end{array}\right) ,
\mathbb E_x \int_0^t \bigl|b\cdot\nabla g \bigr|(\omega_s)ds & \leq \liminf_n \mathbb E_x \int_0^t \bigl|b_n\cdot\nabla g \bigr|(\omega_s)ds = \liminf_n \int_0^t e^{-s\Lambda_{C_\infty}(b)}\bigl|b_n\cdot\nabla g \bigr|(x)ds & \leq e^{\mu T}\liminf_n (\mu+\Lambda_{C_\infty}(b))^{-1}|b_n| |\nabla g|(x)
\begin{aligned}\sum_{i=1}^{n}\sum_{j=0}^{k_i-2}Z_{I}(j+1,k_{i+1},\ldots,k_n,k_1,\ldots,k_{i-1},k_i-j;(\alpha,\beta)) = \sum_{i=1}^{n}Z_{II}(k_{i+1},\ldots,k_n,k_1,\ldots,k_{i-1},k_i+1;(\alpha,\beta)),\end{aligned}
(-1)^{k+1}G^{k+1}_m(x) = (-1)^k\int_{0}^{x}dy\int_{y}^{1}G^k_m(z)dm(z) = (-1)^{k+1}\int_{0}^{x}dy\left( G^k_m(y)\tilde{m}(y) + \int_{y}^{1}(G^k_m)^+(z)\tilde{m}(z)dz \right) \leq (-1)^{k+1}G^k_m(1)G^1_m(x) + (-1)^{k+1}\int_{0}^{x}\tilde{m}(y)dy\int_{y}^{1}(G^k_m)^+(z)dz \leq 2(-1)^{k+1}G^k_m(1)G^1_m(x) < \infty.
\frac{1}{N^{h-m}}\sum\limits_{\mathbf{n}\in \mathbb{Z}^d}\Big(\prod\limits_{j=1}^d f_j(n_j)\Big) \frac{1}{C_{\Xi,\chi}\eta^h}\int\limits_{\mathbf{y}\in \mathbb{R}^h}F(\mathbf{y}) G(L\mathbf{y}) \boldsymbol{\chi}(\Xi(\mathbf{y}) + \widetilde{\mathbf{r}}- \mathbf{n}) \, d\mathbf{y}.
\times\left[1+\sum\limits_{N=1}^{\infty}\frac{\zeta^N}{N!}\left(\prod\limits_{a=1}^{N}\int d^3z_a\sum\limits_{\alpha_a=\pm 1,\pm 2,\pm 3}^{}\right)\exp\left(ig_m\int d^3x\vec\chi\vec\rho_{\rm gas}\right)\right].
(\lambda+\mu)\frac{\partial}{\partial x}(\frac{\partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y}+\frac{\partial u_{z}}{\partial z})+\mu(\frac{\partial^{2}u_{x}}{\partial x^{2}}+\frac{\partial^{2}u_{x}}{\partial y^{2}}+\frac{\partial^{2}u_{x}}{\partial z^{2}})+F_{x}=0
\Pi(i\omega_{n})=\frac{1}{\beta}\sum_{i\omega_{m}}\frac{1}{i\omega_{m}+i\omega_{n}-\epsilon}\frac{1}{i\omega_{m}-\epsilon^{\prime}}=\frac{n_{B}(\epsilon)+n_{F}(\epsilon^{\prime})}{i\omega_{n}-\epsilon+\epsilon^{\prime}}.
\|C \hat{V}\|_F^2={\rm Tr}\left( \hat{V}^{T}S^{T}S \hat{V}\right) +{\rm Tr}\left[ \hat{V}^{T}(C^{T}C - S^{T}S)\hat{V}\right]\geq \|S\hat{V}\|^2_F - \theta(k + \sqrt{k}\xi) \|C\|^2_F.
\sum_{m=-\infty}^{\ell - 1}F_{k,m}(z)e^{2\pi i m \tau}=\frac{(S_4^{\ell}\psi F_{k'})(z)}{(S_4^{\ell}\psi F_{k'})(\tau)}\frac{\psi(\tau)F_2(\tau)}{\psi(\tau)-\psi(z)}.
\,_pF_q^\alpha (a,b; X, Y) = \frac{\Gamma(m+1)}{x^m \rho_m'}\frac{1}{2 \pi i}\int_\Gamma \,_pF_q (a-m,b-m; xs) \prod_{j=1}^r \frac{1}{(s-y_j)^{1/\alpha}}{\rm d}s,
\mathcal{A}_{k}^a = \{\boldsymbol{x}\in \Omega: u_k(\boldsymbol{x}) + \mu_k(\boldsymbol{x}) < u_a(\boldsymbol{x}) \}, \mathcal{A}_{k}^b = \{\boldsymbol{x}\in \Omega: u_k(\boldsymbol{x}) + \mu_k(\boldsymbol{x}) > u_b(\boldsymbol{x}) \}, \mathcal{I}_{k}= \Omega \backslash (\mathcal{A}_k^a \cup \mathcal{A}_k^b).
{\cal L}_{Y M}^{( 1 )}( D \rightarrow 4 , 2 ) = \frac{\hbar g^{2}}{3 2 \pi^{2}( 4 - D )}\left( \frac{1 1}{3}C + \frac{1}{6}T_{s}- \frac{4}{3}T_{f}\right) F_{\mu \nu}^{a}F_{\mu \nu}^{a}.
\langle k_1, b_1, f_1 \rangle \in^* \langle k_2, b_2, f_2 \rangle \iff b_1 b_2 \in h\big(\big\{a_1 a_2 : f_1(a_1) \in f_2(a_2)\big\}\big), \langle k_1, b_1, f_1 \rangle =^* \langle k_2, b_2, f_2 \rangle \iff b_1 b_2 \in h\big(\big\{a_1 a_2 : f_1(a_1) = f_2(a_2)\big\}\big),
D^{j}_{q_1 q_2}(Z)\: =\:\left[\frac{(j+q_1 )! (j-q_1 )!}{(j+q_2)! (j-q_2 )!}\right]^{1/2}\quad\sum_{s=\max (0,q_1 +q_2 )}^{s=\min (j-q_2 , j+q_2 )}{j+q_2 \choose s}\times
\Gamma_{+}^{- 1}C_{3}^{- 2}B = \frac{1}{2}\frac{\alpha}{1 + \mu \alpha k}\left[ ( k_{2}\mu - k_{1}) + ( k - \mu^{2}k_{2}) C_{3}^{- 1}+ ( \frac{2}{\alpha}+ k \mu + k_{1}\mu^{2}) C_{3}^{- 2}\right] Y .
2 i A_{i}= \epsilon_{i a k}\tau_{a}\hat{x}_{k}( \frac{\phi_{1}( r ) - 1}{r}) - ( \tau_{i}- \hat{x}_{i}\hat{x}\cdot \vec{\tau}) \frac{\phi_{2}( r )}{r}+ \hat{x}_{i}\hat{x}\cdot \vec{\tau}H_{1}( r ) ,
{\cal V}[\phi_0] = V[\phi_0,\lambda(\mu)] +{{\hbar}\over{64 \pi^2}}\left\{{\lambda(\mu)^2 \phi_0^4(\mu) \over 4}\right\}\ln{\phi_0^2 \over \mu^2}+\hbar \alpha_3(\lambda(\mu)) \; \phi_0^4 +O(\hbar^2).
\quad\mbox{The top expression in (b)}&=\sum_{t_1,\ldots, t_{n-1}}\sum_{\substack{g, \bar{g}\in G g\bar{g}=g_{t_i}}}\big\langle \theta(g_{t_1},\ldots, g_{t_{i-1}}, g, \bar{g}, g_{t_{i+1}},\ldots, g_{t_{n-1}}), \pi(\xi)|_{[t_1,\ldots, t_{n-1}]}\big\rangle &=\mbox{The bottom expression in (b)}.
\left\{K ( \Lambda ) , K ( \bar{\Lambda}) \right\}~ = ~ \sum_{I = 1}^{n_{1}}k_{I}^{1}\int_{\partial D_{1}}\Lambda_{I}^{1}d \bar{\Lambda}_{I}^{1}~ + \sum_{J = 1}^{n_{2}}k_{J}^{2}\int_{\partial D_{2}}\Lambda_{J}^{2}d \bar{\Lambda}_{J}^{2}~ ,
(\sqrt{2\pi})^n|S_{\gamma_n, K_i}| = \int_{\partial K_i}e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}(x)\leq \sum_{j=1}^n \int_{\omega_{i,j}}e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}(x),
\mu_{y_n}(B_{d_x}(y_n,r)) = \frac{\mu_{y_n}^U(B_{d_x}(y_n,r))}{\mu_{y_n}^U(B_{d_x}(y_n,\mathfrak r))}= \frac{\mu_y^U(B_{d_x}(y_n,r))}{\mu_{y}^U(B_{d_x}(y_n,\mathfrak r))}.
\frac{\omega_{\varphi}^n}{\omega_0^n}=e^{-\varphi -\delta|s|_h^{2\beta}+ \Psi-(1-\b)\psi}= \frac{e^{-\varphi -\delta|s|_h^{2\beta}+ \Psi+(1-\b)\log h}}{|s|_h^{2-2\beta}}.
\sigma_{i}e_{k_{1}k_{2}\ldots k_{n}}= \sum_{l_{i}, l_{i + 1}\in{\cal I}}{\hat{\cal R}}\left( k_{i}, k_{i + 1}; l_{i}, l_{i + 1}\right) e_{k_{1}k_{2}\ldots k_{i - 1}l_{i}l_{i + 1}k_{i + 2}\ldots k_{n}}.
c_{a,b}\left( k,t\right) =\frac{D^{\frac{k}{2}\left( \frac{\log |b|}{\log a}-1\right) +\frac{l_{Q,k}\pi i}{2\log a}}}{k!\log a}\sum\limits_{n\in A_{a,b}(k)}D^{\frac{n\pi i}{\log a}}\Gamma \left( \frac{\log |b|}{\log a}k+\frac{\left(2n+l_{Q,k}\right)\pi i}{\log a}\right) t^{-\frac{2n\pi i}{\log a}},
\Psi_+ \left( \Xi_{\mathfrak{l}}\prod_{i}\mathcal{I}_{\mathbf{m}_i}( \tau_i ) \right) = \Psi_+ \left( \prod_{i}\mathcal{I}_{\mathbf{m}_i}( \tau_i ) \curvearrowright \Xi_{\mathfrak{l}}\right) = \rho\left(\Psi \left( \prod_{i}\mathcal{I}_{\mathbf{m}_i}( \tau_i ) \right) \right) \Psi \left( \Xi_{\mathfrak{l}}. \right)
_3 \phi_2\left[\begin{matrix}q^{-N}, \alpha, \beta \gamma, \frac{q^{1-N}}{\tau}\end{matrix}; q, q\right]= \frac{(\frac{\gamma}{\beta}, \beta\tau;q)_N}{(\gamma,\tau;q)_N}{}_3 \phi_2\left[\begin{matrix}q^{-N}, \frac{\alpha\beta\tau}{\gamma}, \beta \beta\tau, \frac{\beta q^{1-N}}{\gamma}\end{matrix}; q, q\right]
\left\lbrace \begin{array}{cc}\partial_{\tau}r_1 +c_1\cdot\nabla r_1 = r_2r_3, \partial_{\tau}r_2 +c_2\cdot\nabla r_2 = r_1r_3, \partial_{\tau}r_3 +c_3\cdot\nabla r_3 = r_1r_2,\end{array}\right. \Omega.
0=\langle a{\partial}b,{\partial}c\rangle_{\mathcal A}-\langle{\partial}b,a^*{\partial}c\rangle_{\mathcal A}\phantom{0}=\langle{\partial}(ab)-({\partial}a)b,{\partial}c\rangle_{\mathcal A}-\langle{\partial}b,{\partial}(a^*c)-({\partial}a^*)c\rangle_{\mathcal A}\phantom{0}=\Gamma(ab, c)-b^*\Gamma(a, c)-\Gamma(b, a^*c)+ \Gamma(b, a^*) c.
\int_{M}X(R_g)dM=-\frac{n}{n-2}\int_{M}\left\langle{\stackrel{\circ}{{\rm Ric}_g}},\mathcal{L}_{X}g\right\rangle dM+\frac{2n}{n-2}\int_{\Sigma}{\stackrel{\circ}{{\rm Ric}_g}}(X,\nu)d\Sigma,
\mathrm{e}_{\pi_{\lambda,\beta}}(z;x):=\frac{\mathrm{e}^{zx}}{l_{\pi_{\lambda,\beta}}(z)}=\frac{\mathrm{e}^{zx}}{E_{\beta}(\lambda(e^{z}-1))},z\in\mathbb{C},\;x\in\mathbb{R},
0 =3[9A]-[3A]-[3C]-[3B] &=[9B]-[9A]+9[\vartheta_9] &=3[18A]-[6A]-[6B]-[6C] &=[27BC]-[27A]-3/2[\vartheta_9]-9/2[\vartheta_{81}] &=[36BC]-[36A]+3[\vartheta_4]+3[\vartheta_{36}]