harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{L}_Z(\Psi^*_\sigma\theta )= \frac{d}{dt}\big(\varphi^t_{(u,\eta)}\big)^*(\Psi^*_\sigma\theta)|_{t=0}=\frac{d}{dt}\Psi^*_\sigma\circ\big(\varphi^t_{(u,\eta)}\big)^*\theta|_{t=0},
{\cal L}= 2 \sqrt{2}\epsilon^{\mu \nu \rho}[ \eta^{4}A_{\rho}- 2 i ( \eta^{2}- \frac{1}{2}\vert \phi \vert^{2}) \phi ( D_{\rho}\phi )^{*}] F_{\mu \nu}+ 4 ( \eta^{2}- \vert \phi \vert^{2})^{2}\vert D_{\mu}\phi \vert^{2}- V
\Delta_{\tilde{n}}(fg^{\tilde{n}}) =\sum_{k_1+\dots+k_{\tilde{n}+1}=\tilde{n},\atop{}^\forall k_i\ge0}\frac{\tilde{n}!}{k_1!\dots k_{\tilde{n}+1}!}\Delta_{k_1}(f)\Delta_{k_2}(g)\dots\Delta_{k_{\tilde{n}+1}}(g) &=\tilde{n}!\,\Delta_1(g)^{\tilde{n}}=\frac{\tilde{n}!}{2^{\tilde{n}}},
\begin{aligned}C\geq\mathbb{E}\int_0^T\Big[ q_1(t)(b^{v_1}(t)-b(t))-(x^{v_1}(t)-x(t))H_{1x}(t)+k_1(t)(\sigma^{v_1}(t)-\sigma(t)) + \sum_{j=1}^2k_{j1}(t)(\sigma_j^{v_1}(t)-\sigma_j(t))\Big]dt.\end{aligned}
\mathrm{d}\mu = e^{- 8 \pi^{2}/ g^{2}}\, ( M_{\mathrm{P V}})^{n_{b}- \frac{1}{2}n_{f}}\, ( 2 \pi )^{- n_{b}/ 2}\, \prod_{i}\mathrm{d}\eta_{i}\, \left\{\mathrm{B e r}\, \left\langle \frac{\partial \Phi ( \eta )}{\partial \eta_{j}}\bigg | \frac{\partial \Phi ( \eta )}{\partial \eta_{k}}\right\rangle \right\}^{\frac{1}{2}}\, ,
\tilde{D}(r_{K-2}) \geq (1-r_{K-2}) \sum_{j=0}^{1}\frac{1}{N^j}- r_{K-2}\sum_{j=0}^{0}\frac{1-j}{N^j}& = \left( \frac{N^2+(K-2)N}{N^2+(K-2)N+(K-2)}\right) \left(1+\frac{1}{N}\right) - \frac{K-2}{N^2+(K-2)N+(K-2)}& = \frac{N^2 +(K-1)N}{N^2+(K-2)N+(K-2)}& = \bar{D}(r_{K-2})
\begin{array}{rcl}V_{12}= \displaystyle 2^2\cos[k_{1\lambda_1}\wedge k_{2\lambda_2}+(p_{\lambda}+q_{\lambda})\wedge(k_{1\lambda_1}-k_{2\lambda_2})]\cos(p_{\lambda}\wedge q_{\lambda}), V_{34}= \displaystyle 2^2\cos[k_{3\lambda_3}\wedge k_{4\lambda_4}+(p_{\lambda}+q_{\lambda})\wedge(k_{3\lambda_3}-k_{4\lambda_4})]\cos(p_{\lambda}\wedge q_{\lambda}), \end{array}
\sigma_{\rho_g}= \sqrt{\frac{1}{T}\int_{0}^{T}\left( \rho_g(t)-\rho_g \right)^2 \dt}\simeq \sqrt{\frac{1}{N_T}\sum_{i=1}^{N_T}\left( \rho_g(t_i)-\rho_g \right)^2}
d_{\kappa_n}^{(n)}= \left| \begin{array}{ccc}\Delta_{Q, \kappa_n-m}^{(s, 0)}( 1 + r_1^{(n)}) \cdots \Delta_{Q, \kappa_n-m}^{(s, m-1)}( 1 + r_1^{(n)}) \vdots \vdots \Delta_{Q, \kappa_n-m}^{(s, 0)}( 1 + r_m^{(n)}) \cdots \Delta_{Q, \kappa_n-m}^{(s, m-1)}( 1 + r_m^{(n)}) \end{array}\right|;
\frac{f^{3}}{p\dot{p}}=\lambda_{1}\frac{p^{3}}{f\dot{p}}+\lambda_{2},\text{and\}\left( \frac{g}{r}\right) \frac{d}{dg}\left( \frac{g^{2}}{r}\right)=-\lambda_{1}\frac{r\dot{r}}{g}+\lambda_{3},
m_{\bf n}\approx \iint_{\mathbb{R}^2}\hat{f}(x,y) \pi_{\bf n}(x,y) e^{-(x^2+y^2)/2}\;\mathrm{d}x\mathrm{d}y &= \iint_{\mathbb{R}^d}\sum_{\bf t}\hat{c}_{\bf t}\pi_{\bf t}(x,y) \pi_{\bf n}(x,y) e^{-(x^2+y^2)}\;\mathrm{d}x\mathrm{d}y &= \sum_{{\bf t}\in W}\hat{c}_{\bf t}\left \langle \pi_{\bf t}, \pi_{\bf n}\right\rangle_\omega,
\delta{\cal L}_{total}= \omega\left[J^{\mu,a}(D_{\mu}c^{a}) -i\overline{\eta}^{a}(-{1\over 2}f^{abc}c^{b}c^{c}) + iF^{a}\eta^{a}+H^{\mu}{\partial{\cal L}_{total}\over \partial n^{\mu}}\right]
\Delta_{\mathsf{P}}(\mu,C_F,P) = \limsup_{L\rightarrow\infty}~ \frac{(B+1)\max\left(T_F^{(B)}, T_E^{(B)}\right)}{L}= \limsup_{L\rightarrow\infty}~\frac{(B+1)}{B}\frac{\max\left(T_F,T_E\right)}{L}.
\lim_{n \uparrow \infty}\mathbb E\left[e^{s \underline{X}^n_{e(q)}}\right] =q\lim_{n \uparrow \infty}\int_{0}^{\infty}e^{-qt}\mathbb E\left[e^{s\underline{X}^n_{t}}\right]dt =\mathbb E\left[e^{s \underline{X}_{e(q)}}\right],
D= G_{uu}u_1^2-\frac{g'\gamma^1}{2}\frac{A}{w_1}+G_{ux_1}u_1+\frac{u_1}{w_1}\sum_{1\leq k\leq n}w_kG_{ux_k}+\frac{g'}{2}\sum_{2\leq i\leq n}w_i\gamma^i\frac{A}{w_1^2}&-\frac{u_1}{w_1}\sum_{2\leq i\leq n}G_{ux_i}w_i+\frac{1}{w_1}\sum_{1\leq k\leq n}w_kG_{x_kx_1}-\frac{1}{w_1^2}\sum_{1\leq k,i\leq n}w_iw_kG_{x_kx_i}.
\Big\langle \delta_0 \Big\rangle_{3,(1,d)}^{A,}= \frac{1}{2}\Big\langle \tau_0( \Delta ) \Big\rangle_{2, (1,d)}^{A,}= \frac{1}{2}\cdot 2d \cdot \Big\langle \mathsf{1}\Big\rangle_{2, (1,d)}^{A,}= d \cdot d^2 \sum_{m|d}m \,,
\tilde{\mu}=\,\sqrt{-\det \eta_4}\,dx^1\,\wedge dx^2\,\wedge dx^3 \,\wedge dx^4\wedge\left(\sqrt{-\det \eta_4}\,\delta y^1_k \wedge\, \delta y^2_k \wedge\, \delta y^3_k\wedge\, \delta y^4_k\right).
\frac{\partial^3}{\partial \rho^3}H(t,\rho)=3!\int\limits_{-1}^tf(s)H(s,\rho)ds + 3!\int\limits_{-1}^tf(s)\frac{\partial}{\partial \rho}H(s,\rho)ds +\int\limits_{-1}^tR_2(s,\rho)ds
\hat F^{ij}_{\mu\nu}(z)=\int_{T^4}d^4x\int_{T^4}d^4x'\,\psi^i{}^\dagger(x;z)\sigma_\mu (D_z^\dagger D_z)^{-1}(x,x';z)\sigma_\nu^\dagger\psi^j(x;z)-[\mu\leftrightarrow\nu].
X_1 := \{w \in W_{\textrm{per}}^{1,r}({\mathcal{S}}) : w = 0 \textrm{on}\{\zeta = -1\}\}X_2 := L^\infty([-1,0]) Y := \{u \in (W_{\textrm{per}}^{1,r'}(\mathcal{S}))^* : u = \mathcal{A}+ \partial_\xi \mathcal{B}_1 + \partial_\zeta \mathcal{B}_2, \textrm{for}\mathcal{A}, \mathcal{B}_1, \mathcal{B}_2 \in L^r({\mathcal{S}}) \},
0=-\big(D_{2}\big)^{-1/2}A_{2}^{T}P_{1 0}^{*}x_{0}^{o}(t)-\big(D_{2}\big)^{1/2}y_{0}^{o}(t)-\big(D_{2}\big)^{-1/2}A_{2}^{T}h_{1 0}(t).
RQ_4(a,b;p;z)=q^{(a^2-b^2)/(2p)-(a-b)/2}\frac{\vartheta_4\left(\left(\frac{p}{2}-a\right)\pi z;q^{p/2}\right)}{\vartheta_4\left(\left(\frac{p}{2}-b\right)\pi z;q^{p/2}\right)}.
E ( R ) = - i 1 6 \pi^{2}G_{\mathrm{N}}L \int_{0}^{R}d r \int_{0}^{r}d \ell \ \rho ( r ) \rho ( \ell ) r \ell \ln{\frac{\sin{\frac{r + \ell}{2 L}}}{\sin{\frac{r - \ell}{2 L}}}}, \quad 0 < R \le \pi L
\frac{d}{dt}|v(t)|_{H^{-1}}^2 + 2|v(t)|_H^2 = -2b(v(t) + z(t), \mathcal{M}(\Phi)(t), (-A)^{-1}v(t)) - 2b(\mathcal{M}(\Psi)(t), v(t) + z(t), (-A)^{-1}v(t)).
\Big| \int f_k(y) \,d(c_{Q^{(j)}}\mathcal{H}^n_{\mid L_{Q^{(j)}}}- c_{Q^{(j-1)}}\mathcal{H}^n_{\mid L_{Q^{(j-1)}}})(y)\Big | \le c_{Q^{(j)}}\Big| \int f_k(y)\,d(\mathcal{H}^n_{\mid L_{Q^{(j)}}}- \mathcal{H}^n_{\mid L_{Q^{(j-1)}}})(y)\Big| &+ |c_{Q^{(j)}}-c_{Q^{(j-1)}}| \Big| \int f_k(y)\, d\mathcal{H}^n_{\mid L_{Q^{(j-1)}}}(y)\Big | = K_1 + K_2.
\Theta_{\; \nu}^{\mu}( x ) = \frac{\partial{\cal L}( x )}{\partial ( \partial_{\mu}\alpha )}\partial_{\nu}\alpha + \frac{\partial{\cal L}( x )}{\partial ( \partial_{\mu}n_{k})}\partial_{\nu}n_{k}-{\cal L}( x ) \, \delta_{\nu}^{\mu}
J(u_1^n, u_2^n)= J(\varphi_1^n, \varphi_2^n) =J(\varphi_1, \varphi_2) + J(\varphi_1^n - \varphi_1, \varphi_2^n-\varphi_2) + o_n(1) &\geq m(a_1, a_2) + \frac12 \|(\varphi_1^n-\varphi_1, \varphi_2^n-\varphi_2)\|^2_{\dot H \times \dot H}+o_n(1).
\Phi_{a,b,c}(g^ih^j,g^sh^t,g^kh^l)=\zeta_m^{a[\frac{k+s}{m}]i}\zeta_n^{b[\frac{k+s}{m}]j}\zeta_n^{c[\frac{t+l}{n}]j}, \forall 0\leq i,s,k< m, \ 0\leq j,t,l< n
\mathbf{i}_{k,d}\ = \ \left(i_{k,d}^{n,n}\,,\, \ldots \,,\i^{0,0}_{k,d}\right) \ = \ \big( \underbrace{0,\ldots,0}_{k \ \mathrm{terms}}\,,\, d , \ldots , d \,,\, \underbrace{0,\ldots,0}_{k \ \mathrm{terms}}\big) \,;
( \bar{\sigma}_{\nu \mu})_{u}^{v}= \left( \begin{matrix}{0_{[ N - 2 ] \times [ N - 2 ]}}&{0_{[ N - 2 ] \times [ 2 ]}}\\{0_{[ 2 ] \times [ N - 2 ]}}&{( \bar{\sigma}_{\nu \mu})_{\dot{\alpha}}^{\dot{\beta}}}\\ \end{matrix}\right) ,
\widetilde{\epsilon}_n= \sqrt{n}\sup_{|y|\leq k_n}\left| \Phi \bigg( \frac{s_n t-y}{( s_n^2-s_{k_n}^2)^{1/2}}\bigg) -\Phi(t) - \phi(t) \bigg( \frac{s_n t-y}{( s_n^2-s_{k_n}^2)^{1/2}}- t\bigg) \right| \leq \sqrt{n}\sup_{|y| \leq k_n}\bigg| \frac{s_n t-y}{( s_n^2-s_{k_n}^2)^{1/2}}- t\bigg|^2 \xrightarrow{n\rightarrow\infty}0.
\frac{d e}{d \omega}= \frac{\gamma^{3}}{2 \pi^{3}R_{0}^{3}}\frac{\omega^{2}+ ( \gamma v \omega_{0})^{2}}{\omega^{2}}\frac{v^{3}w^{2}}{w^{2}+ ( 2 \gamma v )^{2}}h_{\gamma}( w ) ,
E[W^k_{\ell_2\Delta,t}(j)W^k_{\ell_1\Delta,t}(j)]=P(W^k_{\ell_2\Delta,t}(j)=1,W^k_{\ell_1\Delta,t}(j)=1)+O\left(e^{\lambda\Delta \ell_1}e^{2\lambda \Delta\ell_2}\Delta^3\right), & E[Z_0(\ell_2\Delta)W^k_{\ell_1\Delta,t}(j)]=E[Z_0(\ell_2\Delta);W^k_{\ell_1\Delta,t}(j)=1]+O\left(e^{\lambda\Delta \ell_1}e^{2\lambda \Delta\ell_2}\Delta^2\right).
(Vf)(x,t)=\int_{0}^{t}\int_{\Omega}\varepsilon_{n, a}(x-\xi,b(t,\tau))f(\xi,\tau)d\xi d\tau &=\int_{0}^{t}\int_{\Omega}\varepsilon_{n, a}(x-\xi,b(t,\tau))\bigg(\frac{\partial}{\partial \tau}-a(\tau)\Delta_{\xi}\bigg)(Vf)(\xi,\tau)d\xi d\tau.
P=(\frac{2(\frac{m}{n})}{(\frac{m}{n})^{2}+1},\frac{(\frac{m}{n})^{2}-1}{(\frac{m}{n})^{2}+1})=(\frac{2mn}{m^{2}+n^{2}},\frac{m^{2}-n^{2}}{m^{2}+n^{2}}).
\frac{c_i+\lambda_i}{w_iB}\left(\frac{B\left(|\mathcal{S}|-1\right)}{\sum_{j \in \mathcal{S}}\frac{c_j+\lambda_j}{w_j}}\right)^2=\frac{B\left(|\mathcal{S}|-1\right)}{\sum_{j \in \mathcal{S}}\frac{c_j+\lambda_j}{w_j}}-w_id_i^{ne},
\sum_{i \in e}\Theta^\Delta_\epsilon(H, e, i) \left ( \partial \Theta^\Delta_{\epsilon}(H, i) + \epsilon \log \Theta^\Delta_{\epsilon}(H, e, i) \right ) &= \sum_{i \in e}\Theta^\Delta_{\epsilon'}(H, e, i) \left ( \partial \Theta^\Delta_{\epsilon}(H, i) + \epsilon \log \Theta^\Delta_{\epsilon}(H, e, i) \right ),
\simeq \varepsilon_{\mu_{1}\mu_{2}\mu_{3}\mu_{4}. . . \mu_{4 p}}[ T r F_{\mu_{1}\mu_{2}}F_{\mu_{3}\mu_{4}}. . . F_{\mu_{4 p - 1}\mu_{4 p}}+ 2 \partial_{\mu_{1}}T r F_{\mu_{3}\mu_{4}}. . . F_{\mu_{4 p - 1}\mu_{4 p}}z^{\dagger}D_{\mu_{2}}z ]
g=g_R g_L=\left( \begin{array}{cc}A U -V B \end{array}\right)=\left( \begin{array}{cc}a\tilde{a}-u\tilde{v}a\tilde{u}+u\tilde{b}-v\tilde{a}-b\tilde{v}b\tilde{b}-v\tilde{u}\end{array}\right)
\epsilon_{i_{1}i_{2}\ldots i_{n}}=\delta_{i_{1}i_{2}\ldots i_{n}}^{1\,2\ldots n}\,\,\,\,\,\,\,\,\,\,\&\,\,\,\,\,\,\,\,\,\,\epsilon^{i_{1}i_{2}\ldots i_{n}}=\delta_{1\,2\ldots n}^{i_{1}i_{2}\ldots i_{n}}
M^{\mu_{n}( i )}{}_{( j )}\left( p \right) f_{\left[ k \right]}^{\mu_{n}( j )}\left( p \right) = \lambda_{\left[ k \right]}^{\mu_{n}}\left( p \right) f_{\left[ k \right]}^{\mu_{n}( i )}\left( p \right) \; \, , \; \, p \in V_{\mu_{n}}^{+}\; \, ,
V(\tilde{x}(T))-V(x^* (T)) = 2\sum_{i=1}^r (a_i-u_i^*(\tau))\lambda(\tau)A_i x^*(\tau) + o(\varepsilon) = 2\sum_{i=1}^r (a_i-u_i^*(\tau))m_i(\tau) + o(\varepsilon).
\theta_1 = \frac{\sqrt{-1}}{2}tr(\pi_1\wedge d\pi_1^{\dagger}+\pi_2\wedge d\pi_2^{\dagger}+\pi_3\wedge d\pi_3^{\dagger}+\pi_4\wedge d\pi_4^{\dagger}& +\pi_5'\wedge d\pi_5^{\prime\dagger}+\pi_6'\wedge d\pi_6^{\prime\dagger}+\pi_7\wedge d\pi_7^{\dagger}+\pi_8\wedge d\pi_8^{\dagger}).
\sqrt{- g}= \exp{\frac{1}{2}}\mathrm{t r}\log g = 1 +{\frac{1}{2}}( h^{( 1 )}+ h^{( 2 )}) -{\frac{1}{4}}h_{\alpha \beta}^{( 1 )}h^{( 1 ) \alpha \beta}+{\frac{1}{8}}h^{( 1 ) 2}+ \ldots
{\cal A}^\mu(\hat p, \hat q)=\int^{+\infty}_{-\infty}dt \int_0^1 dy\,\varepsilon^{dil}_{\eta\nu}\,\varepsilon^{dil}_{\lambda\sigma}\,\langle\,V_{-1}^\eta(iy,-\hat p-\hat q)\,V_{-1}^\nu(-iy,-\hat p-\hat q)\,V_0^\mu(t, 2\hat q)\,V_0^\lambda(i, \hat p)\,V_0^\sigma(-i, \hat p)\,\rangle\quad.
\int_{\mathbb{T}^2}(\chi_{\rho_x}(\omega_1,\omega_2))^m (\chi_{\rho_y}(\omega_1,\omega_2))^n \mathrm{d}\varepsilon_{x,y}(\omega_1,\omega_2) & = \frac{1}{8}\frac{1}{64 \kappa^2 \pi^4}\sum_{(\omega_1,\omega_2) \in \mathcal{C}_k^{W,Sp(2)}}(\chi_{\rho_x}(\omega_1,\omega_2))^m (\chi_{\rho_y}(\omega_1,\omega_2))^n J_{x,y}(\omega_1,\omega_2)^2
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^{f}, X ]}.
G^{(p)}_r(t,u,1,z) = \frac{t^{r}z^ru(1-\delta_1^p)}{\gamma_1\delta_1^p}+ \sum_{k=2}^\infty \frac{t^{r}z^ru^{k}\left(1-\frac{\delta_{k}^p}{\delta_{k-1}^p}\right)}{\gamma_1\cdots \gamma_{k}\delta_k^p}.
-\frac{q c_k}{(q-1)^2}+ \sum_{\ell=k}^{-1}\left(\ell-\frac{1}{q-1}\right) c_\ell + \sum_{\ell = n}^{k-1}\left(k-\frac{1}{q-1}\right) c_\ell=-\frac{1}{q-1}.
{\frac{M_{P l}}{m}}= \left({\frac{3 n^{3}P_{k}^{S}}{1 2 8 \pi}}\right)^{{\frac{1}{n - 4}}}\left({\frac{\phi}{M_{P l}}}\right)^{-{\frac{n + 2}{n - 4}}}= \left({\frac{3 n^{3}P_{k}^{S}}{1 2 8 \pi}}\right)^{{\frac{1}{n - 4}}}\left({\frac{4 \pi}{n L}}\right)^{{\frac{n + 2}{2 ( n - 4 )}}},
\theta^\infty_1=-\varepsilon^{-1},\theta^\infty_2=\tilde{\theta}^\infty_1+\varepsilon,t=\varepsilon\tilde{t},H=\varepsilon^{-1}\tilde{H}, q_1=\tilde{q}_1,p_1=\tilde{p}_1+\frac{\tilde{\theta}^\infty_1+\varepsilon^{-1}}{\tilde{q}_1},Y=t^{-\varepsilon^{-1}}\begin{pmatrix}t 0 0 0 1 0 0 0 1\end{pmatrix}\tilde{Y}.
\alpha_2(y) := (\lambda_0a+\lambda_1a^2)y^2+((a-b)\lambda_0+(2ab-1)\lambda_1)y+\lambda_0b+\lambda_1b^2 \alpha_1(y) :=-a^2y^3+(a^2-2ab+1)y^2+(2ab-b^2-1)y+b^2 \alpha_0(y) := -(\lambda_0a+\lambda_1a^2)y^3+((a+b)\lambda_0+(1-2ab)\lambda_1)y^2-(\lambda_0b+\lambda_1b^2)y \;.
(\lambda_{1}-\lambda_{2})+\frac{1}{2}(\lambda_{5}-\lambda_{6})-\frac{1}{2}(\lambda_{7}-\lambda_{8})+ & +\mu \lbrack (\lambda_{1}+\lambda_{2})-(\lambda_{3}+\lambda_{4})+(\lambda_{5}+\lambda_{6})-(\lambda_{7}+\lambda_{8})]=0.
f(x_1,x_2) =\frac{3}{4}\exp\left[-\frac{(9x_1-2)^2+(9x_2-2)^2}{4}\right]+\frac{3}{4}\exp\left[-\frac{(9x_1+1)^2}{49}-\frac{9x_2+1}{10}\right] &+\frac{1}{2}\exp\left[-\frac{(9x_1-7)^2+(9x_2-3)^2}{4}\right]-\frac{1}{5}\exp\left[-(9x_1-4)^2-(9x_2-7)^2\right].
\frac{F(t_{n+1})}{F(t_n)}= \Big(\frac{t_{n+1}}{t_n}\Big)^{\beta'}\cdot\frac{A(t_{n+1})}{A(t_n)}\cdot\exp\Big(\int_{t_n}^{t_{n+1}}\frac{\mathcal E(y)}ydy\Big).
T_{1}\equiv \frac{U_{3p}(\{1\}^{n-1})}{(n-1)!}-3p \sum\limits_{1\le i <j \le n-3}\Big(H_{p}(\{1\}^{i-1},3,\{1\}^{j-i-1},2,\{1\}^{n-3-j}) &+H_{p}(\{1\}^{i-1},2,\{1\}^{j-i-1},3,\{1\}^{n-3-j}) \Big) &\equiv -3p \frac{2{n-3 \choose 2}}{(n-3)!}U_{p}(2,3,\{1\}^{n-5}) &\equiv 0 \pmod{p^2}.
\mathcal{L}_{t}\Phi(v,t) =\frac{\partial}{\partial t}\Phi(v,t)+\frac{1}{2}Tr[\Phi''(v,t)\Sigma_{t}(v)Q\Sigma^{\ast}_{t}(v)]+(Av, \Phi'(v,t)) &\quad+(F_{t}(v), \Phi'(v,t))+\int_{Z}[\Phi(v+\Gamma_{t}(v,z),t)-\Phi(v,t)-\big(\Gamma_{t}(v,z), \Phi'(v,t)\big)]\nu(dz),
\begin{array}{c l}{{L_{\mu \nu}= - i g \frac{\tau_{i}}{2}W_{\mu \nu}^{i}+ i \frac{g^{\prime}}{2}B_{\mu \nu}; R_{\mu \nu}= i g^{\prime}B_{\mu \nu}}}\\{{D_{\mu}\phi = ({\partial}_{\mu}- i g \frac{{\tau}_{i}}{2}W_{\mu}^{i}- i \frac{g^{\prime}}{2}B_{\mu}) \phi .}}\\ \end{array}
[L^{\mu},L^{\nu}] = \{\partial_{\sigma}(\partial^{\lambda}A^{\sigma}\partial^{\rho}A^{m}\partial_{\rho}A^{\nu}) \partial_{\lambda}A^{\mu}- \partial_{\sigma}( \partial^{\lambda}A^{\sigma}\partial^{\rho}A^{m}\partial_{\rho}A^{\mu}) \partial_{\lambda}A^{\nu}\}\partial_{m}
p^{+}= \frac{1}{2}( 1 + K ) , \quad p^{-}= \frac{1}{2}( 1 - K ) , \quad{K_{\alpha}}^{\beta}= \frac{1}{\sqrt{- \pi^{2}}}\pi_{m}( X \tilde{\Gamma}^{m})_{\alpha}{}^{\beta}, \quad K^{2}= 1 .
\frac{\ln\zeta(s)}{s}= \int_2^\infty \frac{\pi(x)}{x(x^s-1)}dx,s>1 \zeta(s) = \prod_{p\in}\frac{1}{1-p^{-s}},s>1 \ln\zeta(s) = \ln\prod_{p\in}\frac{1}{1-p^{-s}}= \sum_{p\in}\ln\frac{1}{1-p^{-s}}
D^2_B(j) =\sum_{k=0}^{N-1}d^2_B(j,k)=\sum_{k=0}^{N-1}\sum_{i=1}^{N-1}\frac{1}{\lambda_i^2}(u_{ij}-u_{ik})^2 &=\sum_{i=1}^{N-1}\sum_{k=0}^{N-1}\frac{u^2_{ij}-2u_{ij}u_{ik}+u_{ik}^2}{\lambda_i^2}&=N\sum_{i=1}^{N-1}\frac{u^2_{ij}}{\lambda_i^2}+\sum_{i=1}^{N-1}\frac{1}{\lambda_i^2}\,.
e^{i \Gamma_{C}[ \vec{\varphi}]}= \langle T_{C}\mathrm{e x p}\left[ \frac{i}{\delta}\int_{C}\left\{{\cal L}^{\prime}( \vec{\phi}( x ) + \vec{\varphi}( x ) ) - \frac{\delta \Gamma_{C}[ \vec{\varphi}]}{\delta \vec{\varphi}( x )}\cdot \vec{\phi}( x ) \right\}\right] \rangle .
\begin{array}{c}{{R_{i j}= R_{i}{}^{k}{}_{j k}, R_{i a}= P_{i}{}^{k}{}_{a k}= - P_{i a},}}\\{{R_{a i}= P_{a}{}^{b}{}_{i b}= P_{a i}, R_{a b}= S_{a}{}^{c}{}_{b c}= S_{a b}}}\\ \end{array}
\sum_{i, k=1}^n (g_{i k}-\partial_i f \partial_k f) T^{i k}= -R+ 2R|\nabla f|^2-R|\nabla f|^4-(n-1)\operatorname{Ric}(X,X) = -R^3-(n-1)\operatorname{Ric}(X,X)=O(r^{-2}).
\widetilde{\kappa}=KP,\widetilde{\kappa}^{-1}=K^{-1}P,\widetilde{J}_{+}=\frac{1}{i}\left(\frac{1-q^{-1}}{q^{1/2}+q^{-1/2}}\right)A_{+},\widetilde{J}_{-}=\left(\frac{q^{1/2}+q^{-1/2}}{1+q^{-1}}\right)A_{-}P,
\mathcal{F}_{k-1}=\sigma\left( \left(\xi^{(0)}_0,{\xi^{(0+)}_{0}},\xi^{(1)}_0, \xi^{(2)}_0\right), \dots, \left(\xi^{(0)}_{k-1},{\xi^{(0+)}_{k-1}},\xi^{(1)}_{k-1}, \xi^{(2)}_{k-1}\right)\right).
\begin{cases}\tilde{F}_{3}(x,\delta)=2x^{3}\delta^{2}+(x^{4}-1)\delta-2x, \tilde{F}_{5}(x,\delta)=8x^{5}\delta^{6}-4x^{6}(x^{4}-1)\delta^{5}-2x^{3}(x^{8}+6x^{4}+5)\delta^{4}+(x^{12}+5x^{8}-5x^{4}-1)\delta^{3}+2x(5x^{8}+6x^{4}+1)\delta^{2}-4x^{2}(x^{4}-1)\delta-8x^{7}.\end{cases}
d(i,j) := \int_{[0,1]}|f_i(x) - f_j(x)|dx = \int_{[0,1]}f_i(x)dx + \int_{[0,1]}f_j(x) dx - 2\int_{[0,1]}f_i(x)\wedge f_j(x)dx = \lambda_X(i,i) + \lambda_X(j,j) - 2\lambda_X(i,j).
(\begin{matrix}1&0&0&0\\ 1&1&1&1\\ 1&-1&1&-1\\ 0&0&0&1\end{matrix})^{-1}=(\begin{matrix}1&0&0&0\\ 0&\frac{1}{2}&-\frac{1}{2}&-1\\ -1&\frac{1}{2}&\frac{1}{2}&0\\ 0&0&0&1\end{matrix}).
E_{\pi}\max_{1\leq k\leq2n}\left( \sum\nolimits_{j=1}^{k}(Q^{j}f+Q^{j+1}f)\right)^{2}\ll\left\vert \sum\nolimits_{j=1}^{2n}jE_{\pi}\left(fQ^{j}f\right) \right\vert +<f,Q^{2}f>.
\sum_{i \in M_P - i^*}\sum_{j \in N_i}a_{ij}x_{ij}+ \sum_{\substack{ij \in P, i \in M_P - M_0 - i^*}}(b-s) x_{ij}+ a_{i^* j^*}x_{i^* j^*}+ \sum_{j \in N_{i^*}- j^*}a_{i^* j^*}\max\left\{1, \frac{a_{i^* j}}{a_{i^* j^*}+ b - s}\right\}x_{i^* j}\leq b + (|M_P - M_0| - 2) (b-s)
\Gamma_{\mathrm{e f f}}^{\mathrm{P V}, \, ( \vec{E})}= \Gamma_{1 , \, \mathrm{e f f}}^{\mathrm{P V}, \, ( \vec{E})}+ \Gamma_{2 , \, \mathrm{e f f}}^{\mathrm{P V}, \, ( \vec{E})}+ \cdots
c_{2}^{T}= ( 2 \pi )^{- 1 / 6}3^{1 / 2}\exp \Bigg [ \frac{1}{3 \varepsilon}( 1 - \delta )^{2}+ \frac{1}{4}- \frac{1}{2}\, \gamma + \frac{\zeta^{'}( 2 )}{\pi^{2}}- \frac{1}{9}\delta + \frac{3 7}{8 1}{\delta}^{2}+{\cal O}\left({\delta}^{3}\right) \Bigg ] \, .
(a_{-1}-a_1)^k\rho^{(n,k)}([a_{-1},+\infty)^k) \leq \sum_{i_1,\cdots,i_{k}\}\sum_{\varepsilon_1,\cdots,\varepsilon_{k}\in\{\pm1\}}\prod_{j=1}^k\varepsilon_{j}(\tau^{(n)}_{i_j}-a_{\varepsilon_{j}})_+ \leq (a_{-1}-a_1)^k\rho^{(n,k)}((a_{1},+\infty)^k).
-2 \sum_{i=1}^8 z^2_{2,i}= (F-x_2L)^2 &= x_2^2(2g-2)-2x_2(L \cdot F) &= x_2(x_2((2g-2)-2 \left \lfloor \frac{g+1}{2}\right \rfloor)-2 (L-E \cdot F))
(X^{m})^n-1 = \prod_{d_1|m}\prod_{d_2|n}\Phi_{d_1\cdot d_2}(X) = \left[ \prod_{d_1|m}\prod_{d_2|n,d_2<n}\Phi_{d_1\cdot d_2}(X) \right]\cdot \left[ \prod_{d_1|m}\Phi_{d_1\cdot n}(X) \right]
a_{\boldsymbol{\alpha}}= \frac{1}{\gamma_{\boldsymbol{\alpha}}^2}\langle Y, \psi_{\boldsymbol{\alpha}}(\mathbf{X})\rangle = \frac{1}{\gamma_{\boldsymbol{\alpha}}^2}\int_{\mathcal{X}}Y \, \psi_{\boldsymbol{\alpha}}(\mathbf{X}) \, f_{\mathbf{X}}(\mathbf{x}) \, \mathrm{d}\mathbf{x}\, .
\bold{pr}X=\xi^i\partial_{x^i}+\sum_{i,I}(S_I\xi^i)\left(D_{i;I}-\partial_{x^i}\right)+\sum_{\alpha,J_1,J_2}(D_{J_1}S_{J_2}Q^{\alpha})\partial_{u^{\alpha}_{J_1;J_2}}.
I_M(P_2, V) = \sum_{x \in P_2}i(x) \leq |P_2|^{1/2}\left(\sum_{x \in P_2}i(x)^2 \right)^{1/2}\leq m^{1/2}\left(\sum_{x \in P_2}i(x)^2 \right)^{1/2}.
[\begin{matrix}L_{R}\\L_{G}\\L_{B}\end{matrix}]=P^{-1}[\begin{matrix}X_{w}/X_{w}^{\prime}&0&0\\ 0&Y_{w}/Y_{w}^{\prime}&0\\ 0&0&Z_{w}/Z_{w}^{\prime}\end{matrix}]P[\begin{matrix}L_{R^{\prime}}\\L_{G^{\prime}}\\L_{B^{\prime}}\end{matrix}]
\tilde{\Theta}_\pm A\varphi =\tilde{\Theta}_\pm \gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=-\gamma^{(2n+1)}\tilde{\Theta}_\pm(P_{\rm F}-D_aP_{\rm F}D_a)\varphi=-A\tilde{\Theta}_\pm \varphi,
Z= \int \exp\{-\frac{1}{2 \pi}\sum_{\alpha}\int_{- \frac{\pi}{b}}^{\frac{\pi}{b}}dk| \tilde{\phi}_{\alpha}(k)|^2 \sin(kb)b^{-1}\}d\tilde{\phi}^*_{\alpha}(k) d \tilde{\phi}_{\alpha}(k)
\frac{t^{-\frac{N+2A_k}{2}}U_k(|x|)}{t^{-\frac{N+2A}{2}}U(|x|)}\le Ct^{A-A_k}(1+|x|)^{A_k-A}\le Ct^{-\frac{A_k-A}{2}}=o(1)\quad\mbox{as}t\to\infty
\psi_{i j}( x^{-}, 0 ) = \frac{1}{2 \sqrt{\pi}}\int_{0}^{\infty}d k^{+}\left( b_{i j}( k^{+}) \mathrm{e}^{- i k^{+}x^{-}}+ b_{j i}^{\dag}( k^{+}) \mathrm{e}^{i k^{+}x^{-}}\right) .
M(f) = \int_{G/\Gamma}f(x)\,{\mathrm{d}}\nu (x) = \int_{G/\Gamma}\left[\int_\Lambda h \circ g (x)\,{\mathrm{d}}\alpha (g)\right] \,{\mathrm{d}}\nu (x) = \int_\Lambda \left[\int_{G/\Gamma}h (g x)\,{\mathrm{d}}\nu (x)\right]\,{\mathrm{d}}\alpha (g) = M (h )\, .
X_j (\pi^* \omega|_e(X_1, \ldots, \widehat{X_j}, \ldots, X_{\ell+1})) = X_j (\sum_i \omega_i(X_1, \ldots, \widehat{X_j}, \ldots, X_{\ell+1}) \otimes \tilde{s}_{v_i}) &= \sum_i X_j (\omega_i(X_1, \ldots, \widehat{X_j}, \ldots, X_{\ell+1})) \otimes \tilde{s}_{v_i}&+ \sum_i \omega_i(X_1, \ldots, \widehat{X_j}, \ldots, X_{\ell+1}) \otimes X_j(\tilde{s}_{v_i}).
\int_{B_{\epsilon}^{+}}y^a|\nabla U_{\mu}|_{\bar{g}}^2\cdot O(|(x,y)|^5)dxdy \le C\mu^5\int_{B_{\epsilon/\mu}^{+}}y^{a}|(x,y)|^5|\nabla U_1|^2dxdy &=\mu^5[\tilde{\cal{E}}_5+o(1)],
X_1^j+ \sum_{k=1}^{m-1}(-1)^k(X_k^j+X_{k+1}^j)+(-1)^{m-1}(X_{m-1}^j+Y_m^j) +\sum_{k=m}^{p-1}(-1)^k(Y_k^j+Y_{k+1}^j)+(-1)^pY_p^j.
T P_F = \ (P_F + P_{\partial^+_R F}+ P_{X \setminus N_R^+ F}) T P_F = \P_F T P_F + P_{\partial^+_R F}T (P_{N_{R}^-F}+ P_{\partial^-_R F}) + 0 = \P_F T P_F + 0 + P_{\partial^+_R F}T P_{\partial^-_R F}\;,
\cosh{\frac{\ell(\gamma_p')}2}=\cosh{d(P_1',\widetilde{x'})}=\frac{\sinh{d(P_1',Q_1')}}{\sinh{d(O,\widetilde{x'})}}>\frac{\sinh{\log\frac4{\ell(c_i)}}}{\sinh{\log\frac2{\ell(c_i)}}}=\frac{16-\ell(c_i)^2}{8-2\ell(c_i)^2}>2
- \bar u_{x_n x_n}= - \frac{1}{\bar h_t^{2}}\sum\limits_{i=1}^{n-1}\bar h_{tX_i}\frac{\partial X_i}{\partial x_n}+ \frac{1}{\bar h_{t}^{3}}\bar h_{tt}&=- \frac{1}{\bar h_t^{3}}\left[\langle \, \nabla_{X}\bar h_t \, ( \bar h_{X_i X_j})^{-1}, \nabla_{X}\bar h_t \rangle- \bar h_{tt}\right]
\delta_{\mu}\epsilon^{\mu \nu \rho \sigma}H_{\nu \rho \sigma}[ \xi | s ] = - 4 \sqrt{6}\pi g \int d \tau I ( \tau ) \frac{d Y^{\mu}( \tau )}{d \tau}\dot{\xi}_{\mu}( s ) \delta ( \xi ( s ) - Y ( \tau ) ) .
d^{1}(\phi, \psi)=d^{0}(\phi, \psi) +\max_{1\leq i\leq l_{1}}\sup_{u\in U_{x_{i}}}\Vert D_{u}(\tilde{\phi}_{x_{i}}) - D_{u}(\tilde{\psi}_{x_{i}})\Vert+\max_{1\leq i\leq l_{2}}\sup_{v\in V_{y_{i}}}\Vert D_{v}(\hat{\phi}_{y_{i}}) - D_{v}(\hat{\psi}_{y_{i}})\Vert,
c \ge \bar{c}= \min_{W(1,1),P(1,1),P(2,2)>0}\frac{\max \left( W(1,1),\tilde{\gamma}P(2,2) \right)}{\lambda_2\left( L_{\mathcal{N}}\right) \min_{j=1,2}\left( P(j,j)\Gamma(j,j) \right)}
\begin{aligned}AG_{1}\geq ((-\frac{4}{n}(\lambda -1)h+(\lambda -2)h'+\lambda h'')G_{1}+\frac{2}{n}(\lambda-1)^{2}h^{2}-(\lambda-1)\lambda hh'-\lambda^{2}h h'')\phi &+2K\lambda h\phi+\frac{2}{n}G_{1}^{2}\phi-2KG_{1}\phi-C_{5}\phi G_{1}(G_{1}-\lambda h)-\frac{|\nabla\phi|^{2}}{\phi}\frac{G_{1}}{C_{5}}. \end{aligned}
\Omega\cap B(x_0,r) =\left\{x=(x',x_d)\in B(x_0,r)\mid x_d>\gamma(x')\right\}, \|\gamma\|_{C^k(\mathbb{R}^{d-1})}=\displaystyle\sup_{|\alpha|\le k,~x'\in\mathbb{R}^{d-1}}|\partial_{x'}^\alpha \gamma(x')|\le K.
G_{D}({\bf r^{\prime \prime}},{\bf r^{\prime}}; E ) = \left( r^{\prime \prime}r^{\prime}\right)^{- ( D - 1 ) / 2}\, \sum_{l = 0}^{\infty}\sum_{m = 1}^{d_{l}}Y_{l m}({\bf \Omega^{\prime \prime}}) Y_{l m}^{*}({\bf \Omega^{\prime}}) G_{l + \nu}( r^{\prime \prime}, r^{\prime}; E ) \;
f = \big(x_1 + P(x_2, x_3), x_2 + Q(x_3), x_3+d\big)\in M_\alpha, g = \big(x_1 + P'(x_2, x_3), x_2 + Q'(x_3), x_3+d'\big)\in N_\alpha, f^{-1}= \Big(x_1 - P\big(x_2-Q(x_3-d), x_3-d\big), x_2 - Q(x_3-d), x_3-d\Big).
\lambda = b^{- 1}\left( \begin{array}{c c}{{( i / 2 ) \partial \varepsilon}}&{{\varepsilon}}\\{{( 1 / k ) \varepsilon L + ( 1 / 2 ) \partial^{2}\varepsilon}}&{{- ( i / 2 ) \partial \varepsilon}}\\ \end{array}\right) b
\mathcal{B}_{m+l}^{\left( l\right)}\left( \lambda\right) =l\binom{m+l}{l}\sum_{k=0}^{m}\genfrac{\{}{\}}{0pt}{}{m}{k}\frac{\left( -\lambda\right)^{k}k!}{\left( l+k\right)}\mathcal{B}_{l+k}^{\left( l+k\right)}\left( \lambda\right) .
D =\langle d_1\rangle\times\langle d_1^{-1}d_3\rangle\times\prod_{k=0}^{n-4}\prod_{2^{n-1-k}-(2l+1)\in B_k}\langle q(k,2l+1)\rangle= &=\langle d_1\rangle\times\prod_{2l+1\in A_0\setminus\{1\}}\langle q(0,2l+1)\rangle\times\prod_{k=1}^{n-3}\prod_{2l+1\in A_k}\langle q(k,2l+1)\rangle.
x \cap (u \otimes_A \nu) = \tau^A(x) \otimes_{A \otimes A}(m \otimes_{A}(\tau^A(u) \otimes_{A \otimes A}\nu)) &= (\tau^A(x) \otimes_{A \otimes A}(m \otimes_{A}(\tau^A(u))) \otimes_{A \otimes A}\nu)) &= (x \cup u) \otimes_A \nu.
Op_h^w\left(\tilde{\theta}_h(x)\chi(h^{\beta}\xi)(\xi-d\varphi(x))\right)v^0 = Op_h^w\left(\tilde{\theta}_h(x)\chi(h^{\beta}\xi)\gamma(\frac{x+p'(\xi)}{\sqrt{h}})(\xi-d\varphi(x))\right)v^0 & + Op_h^w\left(\tilde{\theta}_h(x)\chi(h^{\beta}\xi)(1-\gamma)(\frac{x+p'(\xi)}{\sqrt{h}})(\xi-d\varphi(x))\right)v^0 \, .
-{\frac{\partial^{4}\ln [ P_{\mathrm{v a c}}( x^{+}) ]}{\partial x^{+}\partial x^{-}\partial^{2}x_{\bot}}}={\frac{1}{4 \pi}}\sum_{n = 1}^{\infty}\left({\frac{e E ( x^{+})}{n \pi}}\right)^{2}\exp \left[ -{\frac{n \pi m^{2}}{\vert e E ( x^{+}) \vert}}\right] \; .