harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V_{n m}^{<}( B ) = \frac{- 1}{2 \pi^{2}l^{3}}\int_{0}^{1}d \eta \int_{0}^{1}d u \, \frac{u^{2}\left[ f ( u ; r ) \right]^{n - 2 m}\left[ g ( u ; r ) \right]^{m}}{\left[ ( r u + \frac{\pi}{2})^{2}+ \bar{c}^{2}\right]^{n}}.
_0 \left[ \left(U^{(12)}_h\right)^2\right] = \mathbb{E}_0 \left[ \left(U^{(12)}_h\right)^2\right]= \begin{cases}\frac{k^2 \zeta_1^h}{n}+ O\left(n^{-2}\right) d = 1, \frac{2\zeta_2^h}{n^2}{k \choose 2}^2 + O(n^{-3}) d = 2. \end{cases}
{{M}^{\kappa \mu s}_{\gamma \left| \theta \right.}}(s)=\frac{{{\left( 1-\frac{\overline{{{\gamma}}}_{\theta}}{\mu \left( 1+{{\kappa}_{\theta}}\right)}s \right)}^{m-\mu}}}{{{\left( 1-\frac{\mu{{\kappa}_{\theta}}+m}{m}\frac{\overline{{{\gamma}}}_{\theta}}{\mu \left( 1+{{\kappa}_{\theta}}\right)}s \right)}^{m}}}
\Delta(F,\phi_0)(\phi) := \frac{1}{\phi-\phi_0}\int_{\phi_0}^\phi \delta(F,\phi_0)(\psi)\, d\psi= \frac{1}{\phi-\phi_0}\int_{\phi_0}^\phi \frac{F(\psi)-F(\phi_0)}{\psi-\phi_0}\, d\psi.
\Psi ( x ) = \sum_{\sigma = + j}^{- j}\int{\frac{d^{3}p}{( 2 \pi )^{3}}}{\frac{1}{2 \omega_{\vec{p}}}}\Big [ u_{\sigma}( \vec{p}) a_{\sigma}( \vec{p}) e^{- i p \cdot x}+ v_{\sigma}( \vec{p}) b_{\sigma}^{\dagger}( \vec{p}) e^{i p \cdot x}\Bigr ] \quad ,
|\phi(k)|\leq \frac{|\tilde{P}_{x_2-k}(\theta_{k+1})|}{|\tilde{P}_{2q_n-1}(\theta_{x_1})|}\prod_{j=x_1}^k |\cos(\pi \theta_j)|\, |\phi(x_1-1)|+\frac{|\tilde{P}_{k-x_1}(\theta_{x_1})|}{|\tilde{P}_{2q_n-1}(\theta_{x_1})|}\prod_{j=k}^{x_2}|\cos(\pi \theta_j)|\, |\phi(x_2+1)|
\bar{\pi}_k := (\pi|_{\hat{V}_k\otimes\mathbb{C}[[w]]})\otimes\mathrm{id}_{\mathbb{C}[w]/(w^{mr-r+1})}\colon V_k\otimes_{\mathbb{C}[z]/(z^m)}\mathbb{C}[w]/(w^{mr-r+1})\longrightarrow (w^k)/(w^{k+mr-r+1}) \bar{\nabla}_k := \nabla|_{z^m=0}\colon V_k \longrightarrow V_k\otimes\frac{dz}{z^m}.
\sum_{k=0}^{n}\lambda^{n-k}\left(\frac{\lambda^{k+1}}{z^{k+1}}A^k-H^k\right)+\frac{\lambda^{n+1}}{z^{n+2}}\left(A^{n+1}+\frac{A^{n+2}}{z}+\dots\right)=\frac{1}{\lambda}\left(H^{n+1}+\frac{H^{n+2}}{\lambda}+\dots\right),
\mathcal{B}= \left\{(u,q, p) \in (\mathcal{O}_S^\times\cap k)\times\mathbb{Z}^2 : 0<|\alpha_1 q u_1+\cdots+\alpha_d q u_d-p|<\frac{1}{H^\varepsilon(u_1)|q|^{d+\varepsilon}}\right\}
p^*(\mathcal{B}_r(\mu^*(t),z^*(t))) = \frac{1}{\tilde{\mathcal{T}}}\meas \{t': \t'\in [0,\tilde{\mathcal{T}}], \ \ (\mu^*(t'),z^*(t'))\in \mathcal{B}_r(\mu^*(t),z^*(t))\}.
\mathcal V=\left (\begin{array}{cc}\mathcal V_0(h_1, \mathbf t^1, d) \mathcal V_0(h_2, \mathbf t^1, d) \mathcal V_0(h_1, \mathbf t^2, d) \mathcal V_0(h_2, \mathbf t^2, d) \end{array}\right)
\underset{\{\mathbf{p}_i\}_{i=1}^N,\, \{\mathbf{q}_i\}_{i=1}^N}{\min}\sum_{i=1}^{M}C_i(\mathbf{p}_i,k) - \sum_{i=M+1}^N U_i(\mathbf{q}_i,k) & \underline{\mathbf{p}}_i^{[k]}\leq \mathbf{p}_i \leq \overline{\mathbf{p}}_i^{[k]}, \, i =1,\hdots,N & \sum_{i=1}^N \mathbf{q}_i = \mathbf{P}^{[k]}\mathbf{q}_i = \mathbf{p}_i,\, i =1,\hdots,N.
\sum_{l=1}^2 \partial u_l \cdot ( (J^s-1) \partial_l)^{\partial}u_2= -s \sum_{l=1}^2 \partial u_l \cdot J^{s-2}\partial \partial_l u_2, & \sum_{l=1}^2 ( (J^s-1) \partial_l)^{\partial}u_l \cdot \partial u_2= \sum_{l=1}^2 ( (J^s-1) u_l) \partial_l u_2.
\mu_1\mathbb{D}_{t}^{(\alpha+1)}u+\mu_2\mathbb{D}_{t}^{(\alpha)}u+uu_x=\lambda w(x,t), (x,t)\in (0,L)\times (0,T],\alpha\in (0, 1), w(x,t)=u_{xx}(x,t),(x,t)\in (0,L)\times (0,T], u(x,0)=\varphi_1 (x),v(x,0)=\varphi_2 (x), x \in [0,L], u(0,t)=u(L,t)=0, t \in [0, T].
W^{( 1 )}= \frac{1}{2 !}\int_{1 \ldots 6}V_{1 2 3}V_{4 5 6}( - 2 ) \frac{\delta W_{A}^{( 0 )}}{\delta D_{3 6}^{- 1}}\left( \frac{\delta^{2}W_{\psi}^{( 0 )}}{\delta S_{1 2}^{- 1}\delta S_{4 5}^{- 1}}+ \frac{\delta W_{\psi}^{( 0 )}}{\delta S_{1 2}^{- 1}}\frac{\delta W_{\psi}^{( 0 )}}{\delta S_{4 5}^{- 1}}\right) .
\left(\sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}\right)\langle x, x \rangle^\frac{1}{2}+\langle x, x \rangle^\frac{1}{2}\sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}=\sqrt{n}\langle x, x \rangle^\frac{1}{2}(2-c_x)\langle x, x \rangle^\frac{1}{2}.
f\star_{\vec{c}}^{n}g\ (\mathbf{n})\ =\iint_{S^{2}\times S^{2}}f(\mathbf{n}_{1})g(\mathbf{n}_{2})\mathbb{L}_{\vec{c}}^{j}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n})d\mathbf{n}_{1}d\mathbf{n}_{2}\ ,
L_0 = \sum_{i=1}^\ell \sum_{m=0}^\infty \xi_{i,m}\Big(m\partial_{\xi_{i,m}}-(1-\delta_{i,1})\partial_{\xi_{i-1,m}}\Big) +\sum_{i=1}^\ell\sum_{m=1}^\infty\xi_{\ell+i,m}\Big(m\partial_{\xi_{\ell+i,m}}+(1-2\delta_{i,1})\partial_{\xi_{\ell+i-1,m}}\Big) -\xi_{1,0}\xi_{2\ell,0}.
{\mathrm{\boldmath \alpha}}_{1}^{*}\cdot{\bf h}- \frac{{\mathrm{\boldmath \alpha}}_{1}^{*}\cdot{\mathrm{\boldmath \alpha}}_{2}^{*}}{4 \pi r_{1 2}}\sqrt{1 -{\bf v}_{2}^{2}}\sqrt{g_{2}^{2}+ q_{2}^{2}}.
w^T\mathcal{I}_{\phi\phi}w= \frac{1}{4\beta_a}(w_0+w_1)^2 + \frac{1}{4h}\sum_{j=1}^{N}\epsilon_{j}(w_{j-1}-w_{j+1})^2 + \frac{1}{4\beta_b}( w_N+w_{N+1})^2\geq 0.
\sum_n h(n)\frac{|\rho_j(n)|^2}{\cosh(\pi t_j)}=\frac{2}{\zeta(2)}N+\frac{1}{2\pi i}\int_{(1/2)}\tilde{h}(s)\frac{L(u_{j}\otimes u_{j},s)}{\cosh(\pi t_j)}ds.
\begin{dcases}\rho_t+\rho_x +u_{x}=0 (0,T)\times (0,1), u_{t}-u_{xx}+u_x +\rho_x =0 (0,T)\times (0,1), \rho(t,0)=\rho(t,1)+ p(t) t\in (0,T) , u(t,0)=0, \u(t,1)=0 t\in (0,T) , \rho(0,x)= \rho_0(x), \u(0,x)= u_0(x) x\in(0,1), \end{dcases}
(\gamma\cdot f)(x) := \langle \frac{d}{dt}\Big|_{t=0}\gamma(\exp^g_y(t\cdot f(y)\nu(y))), \nu(\gamma(y)) \rangle_g\Big|_{y=\gamma^{-1}(x)}= f(\gamma^{-1}(x))\cdot \langle \gamma_* \nu(\gamma^{-1}(x)), \nu(x)\rangle_g,
\mathcal{L}_{k}^{\epsilon, v_k}\psi(x)= \bigl \langle \bigtriangledown_x \psi(x), f_k(x, v_k^{\epsilon}(x)) \bigr\rangle + \frac{\epsilon}{2}\operatorname{tr}\bigl \{a_k(x)\bigtriangledown_x^2 \psi(x)\bigr\},
{1\over4}{\partial v(x)\over\partial x}\Bigg\vert_{x=x^{*}}={1\over G}-{\mathrm{e}}^{-x^{*}}+x^{*}E_{1}(x^{*})-{3\alpha(\Lambda)\over8(3\pi-\alpha(\Lambda)\ln x^{*})x^{*}}=0
EXPL(X) := \neg \left| \frac{S_1[M]\upharpoonright X}{=}\right| \leq |M| TOP(X) : = EXPL(X) \wedge EXPL(M-X) E_0(X,Y) := (TOP(X) \wedge TOP(Y) \wedge X \boxminus Y) \vee{}& (\neg (TOP(X) \vee TOP(Y)) \wedge X=Y)
\mathbf{W}(\mathbf{z},\mathbf{\phi}) =\mathbf{W}_0(\mathbf{z})+\sum_{l=1}^{\infty}\varepsilon^l \mathbf{W}_l(\mathbf{z},\mathbf{\phi}), \mathbf{R}(\mathbf{z},\mathbf{\phi}) =\mathbf{R}_0(\mathbf{z})+\sum_{l=1}^{\infty}\varepsilon^l \mathbf{R}_l(\mathbf{z},\mathbf{\phi}),
T_{(\mu \nu )\beta}=-\frac{1}{2e}\epsilon_{\,\,\,\,\,\,\,\mu \nu}^{\alpha\sigma}\partial_{\alpha}h_{\sigma \beta}-\frac{1}{2eM^{2}}\epsilon_{\,\,\,\,\mu \nu \beta}^{\sigma}\partial_{\eta}\Theta_{\sigma}^{\eta}\,,
B_j^+=\mathbf{p}B_j^++\sum_{i=1}^{j-1}\sum_{s=2}^{j-i+1}\sum_{I\in \mathcal{I}_{ij}(s)}E^{e_I}\mathbf{p}B_i^+\frac{\prod_{\ell\in I^\complement}(h_i-h_\ell+1)}{\prod_{\ell=i+1}^{j}(h_i-h_\ell)}
\int d^{2}\theta \theta^{2}\Bigr \{\frac{\delta W_{m}}{\delta \bar{\Phi}_{A}( \theta )}+ \Phi^{A}( \theta ) \Bigr \}= 0 \qquad \Longleftrightarrow \qquad \frac{\delta W_{m}}{\delta \eta_{A}}= \phi^{A},
\Gamma_{\mu}= \left( \begin{array}{c c}{{0}}&{{\gamma_{\mu}}}\\{{\gamma_{\mu}}}&{{0}}\\ \end{array}\right) \hspace{. 2 c m}, \hspace{. 2 c m}\Gamma_{5}= \left( \begin{array}{c c}{{0}}&{{i I}}\\{{- i I}}&{{0}}\\ \end{array}\right) \hspace{. 2 c m}, \hspace{. 2 c m}\Gamma_{6}= \left( \begin{array}{c c}{{0}}&{{\gamma_{5}}}\\{{\gamma_{5}}}&{{0}}\\ \end{array}\right)
I_{1}= \int_{0}^{M}x^{\alpha}e^{-\frac{x}{n}}e^{I(x;r)}dx, I_{2}= \int_{M}^{\infty}\frac{x^\alpha e^{-\frac{x}{n}}}{\widehat{Z}_{n-1,\alpha+2}}Z_{n-1,\alpha}\left(\frac{x}{4(n-1)^{2}};r \right) dx,
a_m^{i,p;j,q}=0,\textrm{if}\i=0\ \textrm{or}\j=0, b^{0,q}_{m;i,p}\neq 0, \qquad\textrm{only if}\i=0\ \textrm{and}\q=p+m, b^{i,q}_{m;i,p}= \delta^q_{p+m}\left( p+\mu_i+\frac12 \right)^{[m+1]},
\tfrac{d}{d\tau}|_{\tau=0}\Pi_+^{\psi_+ + \phi_+ + \tau (\zeta_++{\mathfrak a}_+),T}(v) = \sum_i\langle \tfrac{d}{d\tau}|_{\tau=0}e_i(\psi_++\phi_+ + \tau (\zeta_++{\mathfrak a}_+),T),v\rangle e_i(\psi_++\phi_+,T) & + \sum_i \langle e_i(\psi_++\phi_+,T),v \rangle \tfrac{d}{d\tau}|_{\tau=0}e_i(\psi_++\phi_+ + \tau (\zeta_++{\mathfrak a}_+),T),
\begin{array}{l}{{\tau_{1}\equiv \pi_{0}+ \eta^{1}\approx 0}}\\{{\tau_{2}\equiv E^{\prime}+ \phi^{\prime}+ \pi + A_{1}+ \eta^{2}\approx 0}}\\{{\tau_{3}\equiv E + \eta^{3}\approx 0}}\\{{\tau_{4}\equiv - \pi - \phi^{\prime}- 2 A_{1}+ A_{0}+ \eta^{4}\approx 0}}\\ \end{array}
S_{\^{1}TT^{\ast}G}^{\prime}=\left\{\left( g,\mu ,\frac{\delta \bar{H}}{\delta \mu},-T^{\ast}R_{g}\frac{\delta \bar{H}}{\delta g}\right)\in \^{1}TT^{\ast}G:\left( g,\mu \right) \in G\circledS\mathfrak{g}^{\ast}\right\}.
\bigl ( u_{n}\cdot \epsilon_{\Omega}^{U_{n - 1}G}( u_{n - 1}^{\prime}\cdot u_{n - 1}) \cdot \ldots \cdot \epsilon_{\Omega}^{U_{1}G}( u_{1}^{\prime}\cdot u_{1}) , e , \ldots , e \bigr ) \ \ .
\{p_i,u_j\}_D=\frac{1}{2}\delta_{ij}, \{u_i,u_j\}_D=-\frac{1}{2}d_i \omega_{ij},\{p_i,p_j\}_D=\frac{1}{2d_j}b_{ij}, \{x_i,x_j\}_D=-2d_i\omega_{ij}x_ix_j, \{\hat{y}_i,\hat{y}_j\}_D=2d_i b_{ij}\hat{y}_i \hat{y}_j,\{\hat{y}_i,x_j\}_D=2 d_i \delta_{ij}\hat{y}_i x_j,
e^h\mathrm{div}\Big(e^{-h}V\,\mathcal{L}_hV \Big) =\big(\mathcal{L}_hV \big)^2+V\cdot\nabla \mathcal{L}_hV = =\big(\mathcal{L}_hV \big)^2+V^i\,\partial_i\partial_j V^j-\partial_j h\,V^i\,\partial_i V^j-\partial_{ij}h\,V^i\,V^j.
\mathrm{depth}\,(R/\big<J,y\big>) =1+\mathrm{depth}\,(K[x_{k+1},\ldots,x_{n}]/I(D_{F})) &=1+\mathrm{dim}\,(K[x_{k+1},\ldots,x_{n}]/I(D_{F})) &=1+(n-k)-(g-k) &=1+n-g.
W_2(u,z)=zK_\mu(uz)\left(\sum_{s=0}^{N-1}\frac{A_s(z)}{u^{2s}}+g_2(u,z)\right) \hphantom{W_2(u,z)=}{}-\frac{z}{u}K_{\mu+1}(uz)\left(\sum_{s=0}^{N-1}\frac{B_s(z)}{u^{2s}}+zh_2(u,z)\right),
M_{BPS}^2 ={\textstyle{1\over 2}}e^{-2\phi_\infty}{\vec \beta}^T \mu_R{\vec \beta}+{\textstyle{1\over 2}}e^{2\phi_\infty}{\vec{\tilde\alpha}^T}\mu_R{\vec{\tilde\alpha}}+ \left [({\vec \beta}^T\mu_R{\vec \beta})({\vec \alpha}^T\mu_R{\vec \alpha})-({\vec \beta}^T\mu_R{\vec \alpha})^2\right]^{1\over 2},
\psi = \begin{cases}b_2 - \frac{b_2 b_3}{1+ \eta}, \mbox{if}\eta \ne -1 \infty, \mbox{if}\eta = -1 \end{cases}\psi' = \begin{cases}b_2 - \frac{b_2 b_3}{1+ \eta'}, \mbox{if}\eta' \ne -1 \infty, \mbox{if}\eta' = -1. \end{cases}
\geq \zeta_{1}(q)\int_{\rho B_{1}}\int_{\rho B_{1}}\left[ \psi(x)w(s,x) - \psi(y)w(s,y) \right]^{2}k(x,y)dxdy & - \zeta_{2}(q)\int_{\rho B_{1}}\int_{\rho B_{1}}(\psi(x)-\psi(y))^{2}(w^{2}(s,x) + w^{2}(s,y))k(x,y)dxdy,
u_{j , \alpha}^{( \infty )}( n ) = \frac{P_{j}( - \hbar \tilde{\partial}_{t}) \tau ( n )}{\tau ( n + 1 )^{\frac{1}{2}- \alpha}\tau ( n )^{\frac{1}{2}+ \alpha}}~ , ~ ~ u_{j , \alpha}^{( 0 )}( n ) = \frac{P_{j}( - \hbar \tilde{\partial}_{\bar{t}}) \tau ( n + 1 )}{\tau ( n + 1 )^{\frac{1}{2}- \alpha}\tau ( n )^{\frac{1}{2}+ \alpha}}~ .
Q^i_{n_0;l_1,\ldots,l_{n_0}}:=\{A\in P_{n_0;l_1,l_2,\ldots,l_{n_0}}:\substack{\alpha^{(l_i)}_i,\ldots,\alpha^{(l_{n_0})}_{n_0}\alpha^{(l_{i+1})}_{i+1},\ldots,\alpha^{(l_{n_0})}_{n_0}}\}, i=1,2,\ldots, n_0
\begin{aligned}R_{\mathrm{c},k}=\log_2\det\left(\mathbf{I}+\mathbf{P}_{\mathrm{c}}^H\mathbf{H}_k\left(\mathbf{R}_{\mathrm{c},k}\right)^{-1}\mathbf{H}_k^H\mathbf{P}_{\mathrm{c}}\right), R_{k}=\log_2\det\left(\mathbf{I}+\mathbf{P}_k^H\mathbf{H}_k\left(\mathbf{R}_{k}\right)^{-1}\mathbf{H}_k^H\mathbf{P}_k\right), \end{aligned}
\left(\begin{bmatrix}R_q\otimes P^\perp U + M_zR_q\otimes P U 0 0 W_1\end{bmatrix}, \begin{bmatrix}R_{\overline q}\otimes U^*P+R_{\overline q}M_z\otimes U^*P^\perp 0 0 W_2\end{bmatrix}\right)
C_{r s}^{m n}= i [ \pi_{r t}^{m}, F_{t s}^{n}] = i [ \pi_{t s}^{n}, F_{r t}^{m}] = [ A_{r t}^{m}, ( A^{n \dagger})_{t s}] \stackrel{_{\textstyle =}}{_{_{N}}}A_{r t}^{m}( A^{n \dagger})_{t s}
L_{0}=\frac{1}{2}\alpha_{0}^{I}\alpha_{0}^{I}+\sum_{p=1}^{\infty}\alpha_{-p}^{I}\alpha_{p}^{I},=\alpha^{\prime}p^{I}p^{I}+\sum_{p=1}^{\infty}p\alpha_{p}^{I\dagger}\alpha_{p}^{I}+c
\left(U\cdot(\nabla_X\cdot\psi),Y\cdot\psi\right)+\left(\nabla_X\psi,Y\cdot U\cdot \psi\right) =-g\left(U\times T(X),Y\right)-g\left(T(X),Y\times U\right) &=-2\varphi(T(X),Y,U),
\hspace{-1cm}\frac{2}{k^+k^{'+}}\left[\left(m^2+k_{\perp}k'_{\perp}e^{+i\lambda(\varphi-\varphi')}\right)\delta^{\lambda}_{\lambda'}-m\lambda\left(k'_{\perp}e^{+i\lambda \varphi'}-k_{\perp}e^{+i\lambda\varphi}\right)\delta^{\lambda}_{-\lambda'}\right]
\begin{array}{ccc}\psi^{\dagger}\psi \Rightarrow \beta\int_{\alpha_-}^{\alpha_+}{dp \over2\pi}={\beta\over 2\pi}(\alpha_+-\alpha_-){1\over \beta^2}{\partial \psi^{\dagger}\over \partial \lambda}{\partial \psi \over \partial \lambda}\Rightarrow{\beta\over 2\pi}\int dp \, p^2={\beta \over 6\pi}(\alpha_+^3 +-\alpha_-^3) \cdots \cdots \cdots .\end{array}
\left|\Phi_n(t,h,\mathcal C)-\Phi_{n,N}(t,h,\mathcal C)\right| \leq \|h\|_{\mathfrak B_{\frac12}}^2\frac{16{\rm e}}{\sqrt{2\pi}}\sum_{m=N+1}^\infty m^{3/2}2^{-m}
\beta_i:=\alpha_i|_{\frak{t}^\sigma}\ (1\leq i\leq l), \beta_{l+1}:=(\alpha_{l+1}+\alpha_{l+2}+\cdots+\alpha_n)|_{\frak{t}^\sigma}, &\gamma_i:=\alpha_{i+l+1}|_{\frak{t}^\sigma}\ (1\leq i\leq n-l-1).
\frac{2\cdot 4\,n^3\,\pi^3\,e^{4n\pi y}}{(e^{2n\pi y}-1)^3}-\frac{4\,n^3\,\pi^3\,e^{2n\pi y}}{(e^{2n\pi y}-1)^2}=\frac{4\,n^3\,\pi^3\,e^{2n\pi y}}{(e^{2n\pi y}-1)^3}\left(e^{2n\pi y}+1\right)
\hat{V}_{\mu \nu \rho}^{( 1 \, x , y ) \, A B}= - 2 \cdot 2 \, \mathrm{T r}\, T^{A}T^{B}\, ( Y_{R}^{( d )}- Y_{L}^{( d )}) \, V_{\mu \nu \rho}^{( 1 )}= 4 \, \mathrm{T r}\, T^{A}T^{B}\, V_{\mu \nu \rho}^{( 1 )}\, .
A_{0}^{3}= 4 g_{d}\left( 2 \alpha^{\prime}\right)^{- \frac{1}{2}}( z_{1}- z_{2})^{1 + 2 \alpha^{\prime}p_{1 \cdot}p_{2}}( z_{1}- z_{3})^{1 + 2 \alpha^{\prime}p_{1 \cdot}p_{3}}( z_{2}- z_{3})^{1 + 2 \alpha^{\prime}p_{2 \cdot}p_{3}}e^{\sum_{i = 1}^{3}\alpha^{\prime}p_{i}^{2}\ln V_{i}^{\prime}( 0 )}
{\cal{L}}={i\over 2}\ [\{\bar{\psi}}\gamma^\mu(\overrightarrow{\partial}{}_\mu +ieA_\mu )\psi-{\bar{\psi}}\gamma^\mu(\overleftarrow{\partial}{}_\mu -ieA_\mu )\psi \ ]-m{\bar{\psi}}\psi-{1\over 4}F^{\mu\nu}F_{\mu\nu}\ ,
\begin{aligned}-|\ll|-m\cdot n + l^2 + s\cdot(m-n) + 2\cdot\left(\sum_{i=1}^{n-l}t_i\right) = -s^2-(t_1-s)\cdot(t_1-z_n) + l^2 + \left(\sum_{i=1}^{n-l}t_i\right) + \left(\sum_{i=l+1}^{n}z_i\right) - \left(\sum_{i=1}^{n-l}(t_{i+1}-t_i)\cdot(t_{i+1}-z_{n-i})\right) \end{aligned}
I_{G}(p)=\sum_{i=1}^{J}(p_{i}\sum_{k\ne i}p_{k})=\sum_{i=1}^{J}p_{i}(1-p_{i})=\sum_{i=1}^{J}(p_{i}-p_{i}^{2})=\sum_{i=1}^{J}p_{i}-\sum_{i=1}^{J}p_{i}^{2}=1-\sum_{i=1}^{J}p_{i}^{2}.
(x_1,\cdots,x_n,y_1,\cdots, y_n,z)\circ(\widetilde{x_1},\cdots,\widetilde{x_n},\widetilde{y_1},\cdots,\widetilde{y_n},\widetilde{z}) = (x_1+\widetilde{x_1},\cdots,x_n+\widetilde{x_n},y_1+\widetilde{y_1},\cdots,{y_n}+\widetilde{y_n},z+\widetilde{z}+\sum_{j=1}^{n}(y_j\widetilde{x_j}-x_j\widetilde{y_j}))
\varphi(s_i)\varphi(s_j)=s_1^{a_{1i}}...s_r^{a_{ri}}s_1^{a_{1j}}...s_r^{a_{rj}}=s_1^{a_{1j}}...s_r^{a_{rj}}s_1^{a_{1i}}...s_r^{a_{ri}}=\varphi(s_j)\varphi(s_i).
C_{\beta(\alpha)}x^{\beta(\alpha)}+ f_\alpha x^\alpha = C_{\beta(\alpha)}\left(x^{\frac{\beta(\alpha)}{2}}+ \frac{f_\alpha}{2C_{\beta(\alpha)}}x^{\frac{\beta'(\alpha)}{2}}\right)^2 - \frac{f_\alpha^2}{4C_{\beta(\alpha)}}x^{\beta'(\alpha)}.
{\mit{\Gamma}}_{\alpha , \beta}^{\theta}~ = ~{\textstyle{\frac{1}{4}}}\, \Bigl ( \mathrm{\s l ~ g}_{\alpha + \beta}\, N_{\alpha , \beta}\, + \, \mathrm{\s l ~ g}_{\theta \alpha + \beta}\, N_{\theta \alpha , \beta}\, + \, \mathrm{\s l ~ g}_{\alpha + \theta \beta}\, N_{\alpha , \theta \beta}\, + \, \mathrm{\s l ~ g}_{\theta \alpha + \theta \beta}\, N_{\theta \alpha , \theta \beta}\Bigr ) ~ .
( - 1 )^{F}= \Gamma_{1 1}( - 1 )^{\sum_{m = 1}^{\infty}\psi_{- m}\cdot \psi_{m}}~ ~ , ~ ~{G}= - \gamma_{0}\beta_{0}- \sum_{m = 1}^{\infty}\left[ \gamma_{- m}\beta_{m}+ \beta_{- m}\gamma_{m}\right] ~ ~ .
\int_{0}^{T_{*}}\frac{\|\nabla \Pi(\cdot,t)\|_{\dot{B}^{0}_{\infty,\infty}}^{\frac{2}{3}}}{\big(1+\ln(e+\|\nabla \Pi(\cdot,t)\|_{\dot{B}^{0}_{\infty,\infty}})\big)^{\frac{2}{3}}}dt=\infty.
\nabla_{\partial_t}\nabla_{\partial_t}Y =\nabla_{\partial_t}\nabla_Y\partial_t=\nabla_{\nabla_Y\partial_t}\partial_t=\nabla_{\nabla_{Y^T}\partial_t}\partial_t-\frac{Y^{\perp}}{u}\nabla_{\nabla_{\partial_t}\partial_t}\partial_t &=\nabla_{\nabla_{Y^T}\partial_t}\partial_t-\frac{Y^{\perp}}{u}\nabla_{u\nabla u}\partial_t.
f=\sum_{m,l}b_{ml}\,L({\bf z}_{ml})-5\,\phi\,\partial_{(\zeta^\prime)_n}V^4+\sum_{i,j}c_{ij}\,\partial_{(\zeta^\prime)_n}\bigl[w_{\mu_i^{\prime},\zeta_i^{\prime}}^4\,z_{ij}\bigr].
\hat{\rho}(s) =e^{\gamma+I(-s)}= \begin{cases}O\left(\exp\left(I(\xi)-\frac{t^2u}{2\pi^2}\right)\right) \mbox{if}|t| \le \pi, O\left(\exp\left(I(\xi)-\frac{u}{\pi^2+\xi^2}\right)\right) \mbox{if}|t| \ge \pi, \frac{1}{s}+ O\left( \frac{1+u \xi}{|s|^2}\right) \mbox{if}1+u\xi=O(|t|).\end{cases}
D^{1}\check \varOmega_3(u)={}\frac{3\check\varOmega_3(u)}{2}D^1\log\frac{1}{u^2 (4-u) ( 16-u)}{}+\frac{3}{2}\frac{\log u}{u^2 (4-u) ( 16-u)}\det\begin{pmatrix}D^0\mu^1_{2,2}(u) D^0\mu^1_{2,3}(u) D^1\mu^1_{2,2}(u) D^1\mu^1_{2,3}(u) \end{pmatrix},
\lim_{C\to +\infty}C \int_{\{u\leq V_{\theta}-C\}}\theta_{u^C}^k \wedge \theta_{v^C}^{n-k}=0,\ \ \lim_{C\to +\infty}C \int_{\{v\leq V_{\theta}-C\}}\theta_{u^C}^k \wedge \theta_{v^C}^{n-k}=0, \ \forall k.
U_{j,m}^{n-1}=u-l\frac{\partial u}{\partial t}+\frac{l^{2}}{2}\frac{\partial^{2}u}{\partial t^{2}}-\frac{l^{3}}{6}\frac{\partial^{3}u}{\partial t^{3}}+\frac{l^{4}}{24}\frac{\partial^{4}u}{\partial t^{4}}+...
\sum_{s_1\geq\dots\geq s_{r}}\frac{a^{s_1+\dots+s_r}q^{s_1^2+\dots+s_{r}^2-s_1-\dots-s_{i}}}{(q)_{s_1-s_2}\dots(q)_{s_{r-1}-s_r}}\beta_{s_r}=\frac{1}{(aq)_\infty}\sum_{j\in\mathbb{Z}}a^{rj}q^{rj^2-ij}\frac{1-a^{i+1}q^{2j(i+1)}}{1-aq^{2j}}\alpha_j,
v_{3,2}(t,r) = \frac{-1}{r}\int_{t}^{\infty}ds \int_{0}^{s-t}\rho d\rho \left(\frac{1}{\sqrt{(s-t)^{2}-\rho^{2}}}-\frac{1}{(s-t)}\right)\lambda''(s) &\left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(-1-\rho^{2}+r^{2})^{2}+4r^{2}}}+F_{3}(r,\rho,\lambda(s))\right)
\hat{L}\Psi^{( 1 )}= - \sqrt{2}\left( \begin{array}{c}{D_{i}^{( 0 )}\phi^{( 0 )}}\\{( D_{i}^{( 0 )}\phi^{( 0 )})^{\ast}}\\{\lambda F_{i k}^{( 0 )}}\\ \end{array}\right) K_{a}^{i a},
\begin{pmatrix}\xi^1_1(x,u) \dots \xi^N_1(x,u) \vdots \vdots \xi^1_r(x,u) \dots \xi^N_r(x,u)\end{pmatrix}=\begin{pmatrix}\xi^1_1(x,u) \dots \xi^N_1(x,u) \phi^1(x,u) \vdots \vdots \vdots \xi^1_r(x,u) \dots \xi^N_r(x,u) \phi^1(x,u)\end{pmatrix}=s,
E=F=\langle d_1^{2^{n-2}}\rangle \times \prod_{2l+1\in A_0\setminus\{1\}}\langle d_{2l+1}^{-1}d_{2^{n-1}-(2l+1)}\rangle\times &\times\prod_{k=1}^{n-3}\prod_{2l+1\in A_k}\langle d_{2l+1}^{-2^k}d_{2^{n-1}-(2l+1)}^{2^k}\rangle.
b_i=b_i(\mu_+,\mu_-) := -i+\sum_{k=jp-i}^{jp-1}m_k(\mu_-)+\sum_{k=jp+1}^{jp+i}m_k(\mu_+)\quad\textrm{for $0\leq i\leq h$}, j_0=j_0(\mu_+,\mu_-) := \min \{0\leq i\leq h\mid b_i=\max_{0\leq i'\leq h}b_{i'}\}
\psi_{\mathfrak{U}}^{[a]}: \; C^{k}(\mathfrak{U}, S_{X,p}) \rightarrow C^{k-m}\bigl(\mathfrak{V}', S_{C_{m}^{[a]}(X),p-m}\bigr) & \psi_{\mathfrak{U}}^{[a]}(\alpha) = \sum_{i=1}^{a}n_{i}\cdot \pi_{i}^{*}\psi_{\mathfrak{U}}(\alpha).
u_{i}(x,t) = \frac{x_i (-1)^i}{4\pi}\int_{Q(x)}\frac{y_1y_2}{|y|^4}\omega(y,t)\,dy + x_i B_i(x,t), \,\,\,\|B_i\|_\infty \leq C \|\omega_0\|_\infty,
|(A+1)^{(2^k)}/(A+1)^{(2^k-1)}| =|2^kf(B)-(2^{k}-1)f(B)|\geq \frac{|A|^{k+1}}{(C|A|^{3\delta})^{2^{k+1}-k-2}(\log |A|)^{2^{k+2}-k-4}}.
{\cal N}( e n ) \mathrm{I m}\tau \overline{{{\eta}}}^{2}\int_{C_{1}}[ \theta_{1}( z_{1}- w ) / \theta_{1}^{\prime}( 0 ) ]^{- e_{0}/ k}[ \theta_{1}( z_{2}- w ) / \theta_{1}^{\prime}( 0 ) ]^{e_{0}/ k}[ \theta_{1}( z_{1}- z_{2}) / \theta_{1}^{\prime}( 0 ) ]^{- 2 e_{0}( \widetilde{n_{0}}- 1 / 2 ) / k}
\hat{x}^{+}= - \frac{\kappa}{4 \lambda^{2}}\frac{1}{\hat{x}^{-}+ \frac{m}{\lambda^{3}x_{0}^{+}}}\: \: \: \: \: \mathrm{f o r}\: \: \: \: x^{+}> x_{0}^{+}
W_{T}\left(\Omega\right):=C\left(\left[0,T\right];H_{0}^{1}\left(\Omega\right)\cap W^{2,\infty}\left(\Omega\right)\right)\cap L^{\infty}\left(0,T;H^{2}\left(\Omega\right)\right)\cap C^{1}\left(0,T;L^{2}\left(\Omega\right)\right),
\sum_{k=1}^n{\bf E}\sup_{0\leqslant t\leqslant 1}\left|z_1(u_k,t)-z_2(u_k,t)\right|\leqslant \dfrac{2n^3}{3}\cdot \sqrt{\varepsilon}, \sum_{k=1}^n{\bf E}\sup_{0\leqslant t\leqslant 1}\left|z_i(u_k,t)-z_{i+1}(u_k,t)\right|\leqslant \dfrac{2n^4}{3}\cdot \sqrt{\varepsilon},2\leqslant i\leqslant n-1.
y\sup_{\left(y,\frac{(n+1)^2}{n^2}y\right)}\frac{{\rm d}}{{\rm d}x}\left( \left( \begin{matrix}\Psi_{12}(x) 0 \Psi_{22}(x) 0\end{matrix}\right)e^{-2x^{1/2}}\right) = \begin{cases}O(1) , O\big(y^{1/4}e^{-4y^{1/2}}\big) ,\end{cases}
\int_{0}^{1}r^{2}v(r)\;dr =\int_{-1}^{1}\left( \frac{t+1}{2}\right)^{2}v\left( \frac{t+1}{2}\right) \frac{dt}{2}\medskip & =\frac{1}{8}\int_{-1}^{1}(t+1)^{2}v\left( \frac{t+1}{2}\right)\;dt\medskip & \approx\sum_{k=1}^{q}\underset{_{=:\nu_{k}}}{\underbrace{\frac{1}{8}\nu_{k}^{\prime}}}v\left( \frac{\zeta_{k}+1}{2}\right) ,
\widetilde{\mathcal{C}+\mathcal{D}}= \widetilde{\mathcal{C}}+ \widetilde{\mathcal{D}}, \widetilde{\widetilde{\mathcal{C}}}= \mathcal{C}, \widetilde{\mathcal{C}\mathcal{D}}= \widetilde{\mathcal{D}}\widetilde{\mathcal{C}}.
\|T_{\lambda}\|_{L^1\cap L^\infty}\leq c\lambda\big(\eta_t + \|E\|_{L^1\cap L^\infty})\Big[\frac{\|E\|_{L^1\cap L^\infty}}{m^2_t}+ \frac{1}{m^4_t}+\frac{1}{m^2_t}{\bf 1}_{d=2}+ \frac{1}{m_t}{\bf 1}_{d=3}\Big].
\begin{aligned}xv_{-}= xyv_{+}=[x,y]v_{+}+yxv_{+}=\ell^s hv_{+}+O(n-2(s+3)) =\ell^s \mu_{+}v_{+}+ O(n-2(s+3)) =\ell^s \left( \ell^s+O(n-\alpha-2s-4v) \right) v_{+}+ O(n-2(s+3)) =\ell^{2s}v_{+}+ O(n-\alpha-2(s+3));\end{aligned}
S_{s i}= \oint_{0}{\frac{d z}{2 \pi i}}{\frac{1}{z^{s - 1}}}\oint_{0}{\frac{d w}{2 \pi i}}{\frac{1}{w^{i + 2}}}{\frac{( f^{\prime}( z ) )^{2}( f^{\prime}( w ) )^{- 1}}{f ( z ) - f ( w )}}\biggl ({\frac{f ( w )}{f ( z )}}\biggr )^{3}.
C_{(0)}(q)=E_2(q)-\frac{\zeta(-1)}{2}, C_{(2)}(q) = \frac{E_4(q)}{12}+\frac{E_2(q)^2}{2}, &C_{(1,1)}(q)=\frac{7E_6(q)}{180}+\frac23 E_2(q)E_4(q)-\frac83 E_2(q)^3.
\Pi (\tau )= \frac{2n_{c}\,\rho_{c}^{2}\,\zeta}{m^{*2}\,\tau^{4}\sqrt{\pi}}\int_{0}^{\infty}d\alpha\int_{0}^{\infty}d\beta \,{\rm e}^{-\zeta^{2}{t^2}}\left( \zeta^{2}t^{3}-\frac{3}{2}t \right)\; \cosh\alpha\;\cosh\beta \; ,
\overline{\Pi}_{2,0}^{\sharp}-\overline{\Pi}_{2,0}^{\sharp}\circ B^{\sharp}\circ\Pi^{\sharp}=(\operatorname*{id}-\bar{p}_{V})\circ\Pi_{2,0}^{\sharp}, \overline{\Pi}_{0,2}^{\sharp}-\overline{\Pi}_{0,2}^{\sharp}\circ B^{\sharp}\circ\Pi^{\sharp}=\bar{p}_{V}\circ\Pi_{2,0}^{\sharp}+\Pi_{02}^{\sharp}.
p(y_t,x_{t},x_{t-1}|y^{t-1}) \stackrel{(a)}=p(x_{t-1}|y^{t-1})p(x_{t}|x_{t-1},y^{t-1})p(y_{t}|x_{t}) &=\beta_{t-1}(x_{t-1})u_t(x_t,x_{t-1})p(y_{t}|x_{t}),
|K_1(t;x)+\phi_1(t;x)|\leq 28(C_1q^{-1}(t)q^{-\frac{\sigma}{2}}(t)+C_2g^2q^{\frac{\sigma}{2}}(t)q^{-1-\sigma}(t)) = 28(C_1+C_2g^2)q^{-1-\frac{\sigma}{2}}(t)
\begin{array}{cccc}\partial_z\partial_{\bar z}\log |1|_h^2=\frac{|[s_1,f^{*_h}]|_h^2-|[s_1,f]|_h^2}{|s_1|_h^2}=\frac{2|1|_h^4-|1|_h^4-|1|_h^2\cdot|\gamma^{\dagger}|_h^2}{|1|_h^2}=|1|_h^2-|\gamma|_h^2.\end{array}
\overline{m}_i(B(\tilde{x}, r)) = \sum_I \frac{(I \cap U)}{\epsilon_{V_i}}\left[m_i(\tilde{z}_I) + m_i(\tilde{y}_I)\right] & \leq \sum_I \left[m_i(\tilde{z}_I) + m_i(\tilde{y}_I)\right] & \leq 2C^4\m_i(B(\tilde{x}, r + \epsilon_{V_i})) & \leq 2C^5\m_i(B(\tilde{x}, r)).
\begin{aligned}\#(\{c_{i,j}\}) \leq 2^{\frac{n(n-1)}{2}}\times (\ell+(n-1)m+1)^{\frac{(n-2)(n-1)}{2}}& \times p^{2a_1+a_2+\cdots+a_{k-1}+a_{k+1}+\cdots+a_{n-2}+(k-1)a_{n-1}+(k-1)(n-1)m+\frac{(n-2)(n-1)}{2}m},\end{aligned}
d_j^{\epsilon}d_i^{\epsilon'}=d_i^{\epsilon'}d_{j+1}^{\epsilon}~~~~ i\leq j d_i^{\epsilon}\eta_j = \begin{cases}\eta_{j-1}d_i^{\epsilon}~~i<j 1 ~~i=j \eta_jd_{i-1}^{\epsilon}~~i>j \end{cases}\eta_i\eta_j =\eta_{j+1}\eta_i ~~~~i\leq j.
\hat{h}_{m1}(z)=\frac{1}{n_1p}\sum\limits_{j=1}^{n_1}\sum\limits_{i=1}^p B_{1ij}(z)B_{2ij}(z),\hat{h}_{v1}(z_1, z_2)=\frac{1}{n_1p}\sum\limits_{j=1}^{n_1}\sum\limits_{i=1}^p B_{1ij}(z_1)B_{1ij}(z_2)
g^3 = (v_1v_2v_3v_4)\cdot(v_1v_2v_3v_4)\cdot(v_1v_2v_3v_4) &= (v_1v_2v_3\cdot v_4)\cdot (v_1v_2\cdot v_3v_4)\cdot (v_1\cdot v_2v_3v_4) &= (v_1v_2v_3\cdot v_1v_2\cdot v_1)(v_4\cdot v_3v_4\cdot v_2v_3v_4).
\begin{aligned}\frac{\lambda^2}{\rho_\lambda}\int_\Omega e^{\,u}\,dx =2c_\Omega\int_\gamma |\partial_n K^\pm_\gamma|+c_\Omega\int_\alpha \partial_n K_\alpha &= c_\Omega |\Omega^+|+c_\Omega|\Omega^-| &=1,\end{aligned}
\psi( s)+e^{-\lambda_{l}s}\phi_{l}=\sum_{n=0}^{\infty}\phi_{n}\left(e^{-\lambda_{n}( s- s_{0})}\langle\psi_{0}+e^{-\lambda_{l}s_{0}}\phi_{l},\phi_{n}\rangle+\int_{s_{0}}^{s-1}e^{-\lambda_{n}(s-\tau)}\langle f(\psi(\tau)),\phi_{n}\rangle d\tau \right)+\mathcal I.
\begin{array}{ll}(\nabla_XJ_1)Y=\frac{1}{2(2n-1)}\left[g(X,Y)p^\top_1-\overline \theta_1(Y)X+g(X,J_1Y)J_1(p^\top_1)\right. \qquad\qquad\quad\left.-\overline \theta_1(J_1Y)J_1X\right] ,\end{array}