harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Pr(Y \ge X - \ell) \le \exp\left(C \ell \sqrt{\frac{\log m}{mp}}\right) \sum_{k=\lfloor mp - 4 \sqrt{mp \log m}\rfloor}^{\lceil mp + 4 \sqrt{mp \log m}\rceil}\Pr(Y \ge k) \Pr(X = k) + 2m^{-2}\le \Pr(Y \ge X) \exp\left(C \ell \sqrt{\frac{\log m}{mp}}\right) + 2m^{-2}.
J(\Psi,\widetilde \Psi)=\Vert \bar \Phi'(\mathcal C_1\bar{\mbox{\bf a}},\mathcal C_2\bar{\mbox{\bf b}})-\Psi\Vert^2_{L^2(\widetilde \Gamma)}+\Vert \Phi'(\mathcal P_1{\mbox{\bf a}},\mathcal P_2{\mbox{\bf b}})-{\widetilde\Psi}\Vert^2_{L^2(\widetilde \Gamma)}\rightarrow \inf,
T_Q = \begin{bmatrix}Q_0 Q_{-1}Q_{-2}\cdots \cdots Q_{1}Q_0\otimes I_d Q_{-1}\otimes I_d \cdots \cdots Q_{2}Q_{1}\otimes I_d Q_0\otimes I_d\otimes I_d \cdots \cdots \vdots \vdots \vdots \ddots \end{bmatrix}
\int_{\Omega}|u(x)|^{p_s}dx =\int_{\Omega}|u(x)|^{p_s}w(x)^\frac{p_s}{p}w(x)^{-\frac{p_s}{p}}\,dx &\leq(\int_{\Omega}|u(x)|^pw(x)\,dx)^\frac{p_s}{p}(\int_{\Omega}w(x)^{-s}\,dx)^\frac{1}{s+1},
\hat{\phi}_{( I )}( x ) = \int_{( \infty )}\frac{d^{3}\vec{k}}{( 2 \pi )^{3}2 \omega ( \vec{k}\, )}\left[ a ( \vec{k}\, ) e^{- i k \cdot x}\, + \, a^{\dagger}( \vec{k}\, ) e^{i k \cdot x}\right] \ ,
\left(\sum_{t=1}^{T}\eta_t\right)^{-1}= \frac{1}{\min\left(\frac{T}{8\kappa \bar{L}d},Ta_T\right)}= \max\left(\frac{8\kappa \bar{L}d}{T},\frac{1}{Ta_T}\right) \leq \frac{8\kappa \bar{L}d}{T}+\frac{1}{Ta_T}\enspace.
a_{30}+3a_{31}+3a_{32}+a_{33}+9a_{21}+9a_{22}+3a_{23}+9a_{11}+9a_{12}+3a_{13}+3a_{01}+3a_{02}+a_{03}=1, &a_{30}+3a_{31}+3a_{32}+a_{33}+6a_{21}+6a_{22}+2a_{23}+3a_{11}+3a_{12}+a_{13}\leq\mu_{R}, &a_{31}+2a_{32}+a_{33}+3a_{21}+6a_{22}+3a_{23}+3a_{11}+6a_{12}+3a_{13}+a_{01}+2a_{02}+a_{03}\leq\mu_{T}.
1 \leq \alpha + \beta^{1/2}\biggl(\frac{1}{N}\sum_{j=1}^{N}\bigl(M_{t+h}^{j}\bigr)^{2}\biggr)^{1/2}+Ch\bigl(1+\sup_{s \in [0,t+h]}\vert Z_{s}^{i_{0}}\vert\bigr)+\sup_{s \in [t,t+h]}\bigl\vert W_{s}^{i_{0}}-W_{t}^{i_{0}}\bigr\vert.
\left|\left\lbrace(x,y,t)\in K:\left|\widetilde{P}_1'(t)\alpha(x+\widetilde{P}_1(t),y)-\widetilde{P}_2'(t)\beta(x,y+\widetilde{P}_2(t))\right|\leq \varepsilon\right\rbrace\right| \lesssim 2^{c|l|}\varepsilon^{\frac{1}{7(8(d_1+d_2)-17)}}
|f_0(w_\delta) - f_0(W_\delta)|_{\frac{(p+1)(1+\gamma)}{p}}=\begin{cases}O(\delta|\log \delta|^\frac{1}{4}) n=4,\ \gamma=0, O(\delta) n\geq 5,\ \gamma=0, O(\delta^{1-\frac{(n+2)\gamma}{2(1+\gamma)}}) n\geq 4,\ \gamma>0,\end{cases}
\delta \varphi^{i}( y ) = \sum_{k}\int d^{2}x \left( \Lambda_{k}( x ) \phi_{k}^{i ( 0 )}+ \dot{\Lambda}_{k}( x ) \phi_{k}^{i ( 1 )}( x , y ) + \ddot{\Lambda}_{k}( x ) \phi_{k}^{i ( 2 )}( x , y ) \right) ,
F(m+n) =\nu(g_1^{[s_1(m+n)]}\cdots g_k^{[s_k(m+n)]}) &\leq\nu(g_1^{[s_1m]+[s_1n]}\cdots g_k^{[s_km]+[s_kn]})+\nu((g_1^{[s_1m]+[s_1n]}\cdots g_k^{[s_km]+[s_kn]})^{-1}(g_1^{[s_1(m+n)]}\cdots g_k^{[s_k(m+n)]})) &\leq\nu(g_1^{[s_1m]+[s_1n]}\cdots g_k^{[s_km]+[s_kn]})+\sum_{i=1}^k\nu(g_i).
( \omega_{1}, \{\omega_{1}, \omega_{1}\}) + \{\omega_{1}, ( \omega_{1}, \omega_{1}) \}= ( - 1 )^{\varepsilon_{A}( 1 + \varepsilon_{B})}\left( \omega_{1}, \left.{\frac{\partial_{r}^{2}\Gamma}{\partial \Phi^{A}\partial \Phi^{B}}}\right|_{K}\right) \left.{\frac{\partial \omega_{1}}{\partial J_{B}}}\right|_{K}\left.{\frac{\partial \omega_{1}}{\partial J_{A}}}\right|_{K}.
\lim_{N\in\mathcal{N}}\frac{E_{s}(\alpha_{N})-I_{s}(\sigma) N^{2}}{N^{1+s}}=\left(\sum_{k=1}^{p-1}(2\theta_{k})^{s}\sum_{j=k+1}^{p}2\theta_{j}+\sum_{k=1}^{p}\theta_{k}^{s}\,\Large(\theta_{k}-\sum_{j=k+1}^{p}2\theta_{j}\Large)\right)\frac{2\zeta(s)}{(2\pi)^{s}}.
{\!\hat{s}}_{k}\!\!=\!\!\left(\boldsymbol{v}_{k}\right)\!^{\mathrm{H}}\!\boldsymbol{g}_{k}\sqrt{p_{k}}s_{k}\!+\!\left(\boldsymbol{v}_{k}\right)\!^{\mathrm{H}}\!\sum_{l \neq k}\!\boldsymbol{g}_{l}\sqrt{p_{l}}s_l\!+\!\left(\boldsymbol{v}_{k}\right)^{\mathrm{H}}\!\boldsymbol{z}, \ \ \forall k \! \in \! \mathcal{K}.
\lim_{\epsilon \to + 0}\left[ K_{\mu \nu}\right]_{x^{\perp}= - \epsilon}^{x^{\perp}= \epsilon}= \left. \frac{1}{m_{4}^{3}}\left( T_{\mu \nu}- \frac{1}{d - 1}g_{\mu \nu}g^{\alpha \beta}T_{\alpha \beta}\right) \right|_{x^{\perp}= 0}
\int_{C}{d^{4}}{\eta}= a^{4}\frac{2 \pi^{2}}{\Gamma ( 2 )}[ - \frac{1}{3}s i n^{2}\theta_{4}^{0}c o s \theta_{4}^{0}- \frac{2}{3}c o s \theta_{4}^{0}+ \frac{2}{3}]
\sup_{(t,x)\in R^-}u(t,x) = \sup_{(t,x)\in R^-}U(t,x,0)\leq\sup_{(t,x,y)\in R^-\times(-1,1)}U(t,x,y) &\leq c\inf_{(t,x,y)\in R^+\times(-1,1)}U(t,x,y)\leq c\inf_{(t,x)\in R^+}U(t,x,0)=c\inf_{(t,x)\in R^+}u(t,x),
\beta (a^r c^s) = (Q \otimes id ) \big( \Delta_\mathcal{C}(a^r c^s) \big) = (Q \otimes id ) \big( \Delta_\mathcal{C}(a)^r \Delta_\mathcal{C}(c)^s \big) = (Q \otimes id ) \Big( (a \otimes a - q \, c^* \otimes c)^r (c \otimes a + a^* \otimes c)^s \Big)
C_j < p_1+3\sum_{k=2}^{j_2}p_k+2\sum_{k=j_2+1}^{j_s}p_k + \sum_{i=1}^{s-1}p_{j_{i+1}-1}+ \sum_{k=j_s+1}^{j-1}p_k+ 2p_j &\leq p_1+3\sum_{k=2}^{j_2}p_k+2\sum_{k=j_2+1}^{j_s}p_k + \sum_{i=2}^{s-1}p_{j_{i+1}-1}+ \sum_{k=j_s+1}^{j-1}p_k+ 3p_j &\leq 3\sum_{k=1}^j p_k.
{{\bar{R}}_{\mu \nu \lambda}}^{\; \; \; \; \; \; \kappa}= \partial_{\mu}{\bar{\Gamma}}_{\nu \lambda}^{\; \; \; \; \kappa}- \partial_{\nu}{\bar{\Gamma}}_{\mu \lambda}^{\; \; \; \; \kappa}- [ \bar{\Gamma}_{\mu}, \bar{\Gamma}_{\nu}{]_{\lambda}}^{\kappa}.
(\omega^2 -{\omega^\prime}^2) \int_\theta^{2\pi}\!\phi^L_\omega(\sigma)\:\phi^L_{\omega^\prime}(\sigma)\,d\sigma + \int_\theta^{2\pi}\!\partial_{\sigma}(\phi^L_{\omega^\prime}(\sigma)\:\partial_{\sigma}\phi^L_\omega(\sigma) - \phi^L_\omega(\sigma)\:\partial_{\sigma}\phi^L_{\omega^\prime}(\sigma))\,d\sigma = 0
O^w(G(n)+H) = \left(\frac{3}{2}\right)^w (G(n)+H+1)-1 &= \left(\frac{3}{2}\right)^w G(n)+ \left(\frac{3}{2}\right)^w H + \left(\frac{3}{2}\right)^w -1 = (O^w \circ G)(n) + \left( \frac{3}{2}\right)^w H
e_{C,2}=-\frac{5t}{2}-\frac{5|J|}{4}-\frac{|\alpha_J|}{2}+\frac{3}{4}-\frac{5(g_0-t)}{2}-\frac{5(|I_0\setminus J|-1)}{4}-\frac{|\alpha_{I_0\setminus J}|-\alpha_i}{2}+\frac{3}{4}&\ge e_1+\frac14
R = \left( \begin{array}{c c c c c}{N - 1}&{- 1}&{.}&{.}&{- 1}\\{- 1}&{N - 1}&{- 1}&{}&{.}\\{.}&{- 1}&{.}&{.}&{.}\\{.}&{}&{.}&{.}&{- 1}\\{- 1}&{.}&{.}&{- 1}&{N - 1}\\ \end{array}\right) .
P_{j , n , s}^{S P F j^{\prime}, n^{\prime}, s^{\prime}}= P_{j}^{j^{\prime}}P_{n}^{n^{\prime}}P_{s}^{s^{\prime}}+ ( - 1 )^{( 2 j - n + s ) / 2}P_{j}^{k / 2 - j^{\prime}}P_{n}^{n^{\prime}+ k + 2}P_{s}^{s^{\prime}+ 2},
\lim_{x_{n+1}\rightarrow0}x_{n+1}^{1-2\overline{s}}\partial_{n+1}\overline{v}\bigg|_{C_{\overline{s},1/2}'}=\lim_{x_{n+1}\rightarrow0}x_{n+1}^{1-2\overline{s}}\partial_{n+1}\tilde{v}\bigg|_{C_{\overline{s},1/2}'},
[A, F_{1,0}]_{q}=[[\cdots [[F_{2,1},F_{1,0}]_q,F_{3,0}]_{q},\cdots,F_{n-2,0}]_q,F_{n-1,0}]_{q}\, &=(-d)^{-1}[[\cdots [[F_{1,1},F_{2,0}]_{q},F_{3,0}]_{q},\cdots,F_{n-2,0}]_q,F_{n-1,0}]_{q}\,.
\delta \psi_{A\mu}= \mathcal{D}_\mu \, \epsilon_A - \frac{1}{3}\, \mathcal{T}_{AB}^{\rho \sigma}\left( g_{\mu \rho}\, \gamma_\sigma - \frac{1}{8}\, \epsilon_{\mu \rho \sigma \lambda \nu}\, \gamma^{\lambda \nu}\right) \, \epsilon^B
S_2(t) = i\eta t^{-1}(|e^{3i\lambda_1 (\Phi_1(t)-\theta_1)}v_1(t)|^2 - |W_1|^2) w (t) & -i\lambda_6 t^{-1}\{(e^{3i\lambda_1 (\Phi_1(t)-\theta_1)}v_1(t))^2 - W_1^2\}\overline{w (t)}.
\left( M \epsilon_{I ( 0 )}+{\frac{1}{\sqrt 2}}( Q + i P )_{I J}\gamma^{0}\epsilon_{( 0 )}^{J}\right)^{\prime}= e^{i \arg S_{0}}\left( M \epsilon_{I ( 0 )}+{\frac{1}{\sqrt 2}}( Q + i P )_{I J}\gamma^{0}\epsilon_{( 0 )}^{J}\right) = 0 \ ,
\hbar \sum_{r \geq 0}X_{1,r}^{\pm}u^{-r-1}= x_1^{\pm}(u - \dfrac{1}{2}(\varepsilon_1 - \varepsilon_2)), 1 + \hbar \sum_{r \geq 0}H_{1,r}u^{-r-1}= h_1(u - \dfrac{1}{2}(\varepsilon_1 - \varepsilon_2)),
\begin{matrix}{\zeta^{+}= \eta_{z}^{+}\mathrm{e}^{\varphi}d z ,}&{\zeta^{-}= \eta_{z}^{-}\mathrm{e}^{\varphi}d z ,}&{\tilde{\zeta}_{+}= \eta_{+ \bar{z}}\mathrm{e}^{\varphi}d \bar{z},}&{\tilde{\zeta}_{-}= \eta_{- \bar{z}}\mathrm{e}^{\varphi}d \bar{z},}\\ \end{matrix}
|n_{1}(x_{1})-n_{1}(x_{2})| \leq |x_{1}-x_{2}|_{\theta \in [0,1]}|n_{1}'(\theta x_{1}+(1-\theta)x_{2})| &\leq |x_{1}-x_{2}|_{\theta \in [0,1]}|4 \sin^{2}(\theta x_{1}+(1-\theta)x_{2})| &\leq C |x_{1}-x_{2}| \left(|x_{1}|^{2}+|x_{2}|^{2}\right)
P(l) =\frac{1}{\zeta(2)\sqrt{l_1}}\sum_{\substack{m=1 m \textrm{odd}}}^\infty\frac{A(l_1m^2,1)}{m}(\prod_{p\mid 2lm}\frac{p}{p+1})\check{\Phi}_{l_1m^2}(0) &\hskip 30pt +O(\frac{1}{Y\sqrt{l_1}}\sum_{\substack{m=1 m \textrm{odd}}}^\infty\frac{|A(l_1m^2,1)|}{m}|\check{\Phi}_{l_1m^2}(0)|).
\sup_{M\times[0,T)}|\tilde{F}(x,t)| \leq \|F\|_{L^{\infty}(M)}+\sup_{M\times[0,T)}\left|\frac{\partial\varphi}{\partial t}(x,t)\right| & \leq 2\|F\|_{L^{\infty}(M)}+\left\|\log\frac{(\omega+\sqrt{-1}\partial\overline{\partial}\varphi_{0})^{n}}{\omega^{n}}\right\|_{L^{\infty}(M)}.
\widetilde{\mathcal{M}}_{\gamma_{i_0-1}}:=\sup_{0<|h|<h_0}\left[\frac{\delta_h (u_R\,\widetilde\eta)}{|h|^\frac{\gamma_{i_0-1}+t}{p}}\right]^p_{W^{s,p}(B_{r_{i_0-1}})}.
x_i : = \frac{\langle u, x^{**}\rangle - \langle u, \tilde{x}_i \rangle}{\langle u, x^{**}\rangle - \langle u, x_0 \rangle}x_0 +\frac{\langle u, \tilde{x}_i \rangle - \langle u, x_0\rangle}{\langle u, x^{**}\rangle - \langle u, x_0\rangle}x^{**}
\mathcal{Li}_{K,\mathfrak{s}}({\bf z})^{(-1)}=\frac{z_r}{(\theta-t)^{s_r}}\mathcal{Li}_{K,(s_1, \ldots, s_{r-1})}(z_1, \ldots, z_{r-1})+\mathcal{Li}_{K,\mathfrak{s}}({\bf z})
F_{\kappa}(x)=(1+\kappa\nu)(2\kappa)^{\nu}\frac{\Gamma(\frac{1}{2\kappa}+\frac{\nu}{2})}{\Gamma(\frac{1}{2\kappa}-\frac{\nu}{2})}\frac{\alpha\beta^{\nu}}{\Gamma(\nu)}\int_{0}^{x}z^{\alpha\nu-1}exp_{\kappa}(-\beta z^{\alpha})dz
\inf_{\overline{M}\times \{t=0\}}u_t - t\sup_{\overline{M_T}}|\psi_t| \leq u_t \leq \sup_{\overline{M}\times \{t=0\}}u_t + t\sup_{\overline{M_T}}|\psi_t|, \forall\,(x, t)\in \overline{M_T}.
v = v_{1}xv_{2}\, \, \, \, \, v_{1}\in Y_{k-q-1}\, \, \, \, \, x \in X_{i}^{0}\, \, \, \, \, v_{2}\in Y_{q}&w = w_{2}yw_{1}\, \, \, \, w_{1}\in Y_{l - q - 1}\, \, \, \, \, y \in X_{j}^{0}\, \, \, \, \, w_{2}\in Y_{q}
\hat{\mathbb{E}}_{\tau+}[\eta X +Y] =\mathbb{L}^1\lim_{n\rightarrow\infty}(\eta^+\hat{\mathbb{E}}_{\tau_n+}[ X ]+\eta^- \hat{\mathbb{E}}_{\tau_n+}[ -X])+Y &=\eta^+\hat{\mathbb{E}}_{\tau+}[ X ]+\eta^- \hat{\mathbb{E}}_{\tau+}[ -X]+Y.
E_{2k}^{(p,q)}(z,s) = \sum_{\gamma \in (\Gamma_{p,q})_\infty \backslash \Gamma_{p,q}}{\rm Im}(z)^{s-k}\bigg|_{2k}(\sigma^{-1}\gamma) &= \frac{\,1\,}{\lambda^s}\sum_{\gamma \in (\Gamma_{p,q})_\infty \backslash \Gamma_{p,q}}\frac{{\rm Im}(\gamma z)^{s-k}}{j(\gamma, z)^{2k}}.
2\sum_{1\leq i< j\leq n}f_k(i)f_k(j)=\big(\sum_{1\leq i\leq n}f_k(i)\big)^2-\sum_{1\leq i\leq n}f_k^2(i)=\big(\sum_{1\leq i\leq n}f_k(i)\big)^2+O_{\varepsilon}(n^{1+\varepsilon}),
y^rz^l\mathbb{H}_r^k(y, tz-y)= z^l\sum_{i=0}^{k}{\dbinom{k+r}{i+r}y^{i+r}(tz-y)^{k-i}}= z^l\sum_{i=r}^{k+r}{\dbinom{k+r}{i}y^i(tz-y)^{k+r-i}}.
\int_{-\infty}^{0}V_q(b+z)\frac{(\vartheta_k)^j(-z)^{j-1}}{(j-1)!}e^{\vartheta_k z}dz=\sum_{i=1}^{M}\frac{U_i(\vartheta_k)^j}{(\vartheta_k+\beta_{i,\xi})^j}.
\int_{\mathbb R_+^d}R_{\alpha,i}f(x) g(x)x^{2\alpha}dx=-4\int_0^\infty \int_{\mathbb{R}^d_+}\partial_{x_i}P^\alpha_t (f)(x)\partial_t \mathbb{P}^{\alpha,i}_t (g)(x)x^{2\alpha}dx t dt.
\cong\frac{\{f\in L^2_{\mathrm{loc}(p,n)}(X,E,h,\omega);\overline{\partial}f=0\}}{\{g\in L^2_{\mathrm{loc}(p,n)}(X,E,h,\omega);\mathrm{there ~is~ an~}\gamma\in L^2_{\mathrm{loc}(p,n-1)}(X,E,h,\omega) \,\mathrm{satisfying}\, \overline{\partial}\gamma=g\}}
i \Pi_{S \mu \nu}^{\underline{{{a}}}\underline{{{b}}}}( q ) = N g^{2}\delta_{\underline{{{a}}}\underline{{{b}}}}\mu^{2 \epsilon}\int \frac{d^{n}\ell}{( 2 \pi )^{n}}\left\{\frac{\ell_{\mu}\ell_{\nu}- m_{G}^{2}g_{\mu \nu}}{( \ell + q )^{2}( \ell^{2}- m_{G}^{2})}- \frac{1}{8}\frac{( 2 \ell + q )_{\mu}( 2 \ell + q )_{\nu}}{\ell^{2}( \ell + q )^{2}}\right\}
I_{m,n}=\frac{\partial\mu}{\partial\theta_{m}}\Sigma^{-1}\frac{\partial\mu}{\partial\theta_{n}}+\frac{1}{2}tr(\Sigma^{-1}\frac{\partial\Sigma}{\partial\theta_{m}}\Sigma^{-1}\frac{\partial\Sigma}{\partial\theta_{n}})
\int_\Sigma \left( 4 - 2 \, |\nabla_\Sigma r|^2\right) \, e^{- 2 \, r}+ \frac{4 \, \pi}{|\Sigma|}\, \int_\Sigma \left( 1 - \langle \nu, \nabla r \rangle \right) + \int_\Sigma \left( 1 - \langle \nu, \nabla r \rangle \right)^2 = 4 \, \pi + O (|\Sigma|^{-1/2}).
{\cal L}( 1 , \alpha - \frac{1}{2}, g ) = - i \epsilon_{\mu \nu \lambda}B_{\mu}^{*}( \partial_{\nu}+ i a_{\nu}+ i C_{\nu}+ M ) B_{\lambda}+ \frac{g}{4}( B_{\mu}^{*}B_{\mu})^{2}- \frac{i}{8 \pi ( \alpha - 1 / 2 )}\epsilon_{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda}\; .
\vert A \vert_{ij}=\begin{vmatrix}a_{11}\cdots a_{1j}\cdots a_{1n}\vdots \vdots \vdots \vdots \vdots a_{i1}\cdots \boxed{a_{ij}}\cdots a_{in}\vdots \vdots \vdots \vdots \vdots a_{in}\cdots a_{ni}\cdots a_{nn}\end{vmatrix}.
\Vert w(x,t,\cdot) \Vert_{L^p(\Gamma\backslash\Gamma_{\epsilon})}=O(\epsilon^{\frac{1}{p}}\tau^{-1}), p=1,2, \Vert w(x,t,\cdot) \Vert_{L^\infty(\Gamma\backslash\Gamma_{\epsilon})}=O(\tau^{-1}).
\Delta^{I}n_{2}=\Delta^{I}\Big(-\frac{bv}{\sqrt{\Phi}}\Big)=\lambda_{21}\Big(-\frac{au}{\sqrt{\Phi}}\Big)+\lambda_{22}\Big(-\frac{bv}{\sqrt{\Phi}}\Big)+\lambda_{23}\Big(\frac{\sqrt{\omega}}{\sqrt{\Phi}}\Big),
\int_{\Omega\cap B_{\eta}}n c^{2}\psi^{2}+\int_{\Omega\cap B_{\eta}}|\nabla c|^{2}\psi^{2}=\int_{\partial\Omega}\nabla c\cdot\nu c\psi^{2}-2\int_{\Omega\cap B_{\eta}}\nabla c\cdot \nabla \psi c\psi.
\langle s_1 \cdot \cdot \cdot s_N|\Psi \rangle =\left({B \over 4}\right)^{N/4}e^{-B \sum_i s_i^2/2}e^{-{1 \over 4B}\sum_i \left({\partial \over \partial s_i}\right)^2}\Psi(\sqrt{2B}s_1, \cdot \cdot \cdot ,\sqrt{2B}s_N).
\tilde{\psi}_{\mu}( x ) = a^{1 / 2}( t ) e^{- K / 1 2}\hat{\psi}_{\mu}( x ) , \, t i l d e \psi_{5}( x ) = \sqrt{\frac{3}{2}}a^{- 1 / 2}( t ) e^{K / 1 2}\hat{\psi}_{5}( x ) .
\frac{r(|W_k|+q_k+q_{k-1}-2,{\mathbf x})}{|W_k|+q_k+q_{k-1}-2}\le \frac{|W_k| + 2q_k + q_{k-1}-2}{|W_k|+q_k+q_{k-1}-2}&= 1 + \frac{1}{t_k+1 +\eta_k-\varepsilon_k}< 1 + \frac{1}{t_k+1 +\eta_k}+ \varepsilon_k.
{\cal E}= \int_{0}^{\infty}d u \int_{- \pi}^{\pi}d \phi \, u \left\{\left[ b - b_{\mathrm{e x t}}\right]^{2}+ \left( \partial_{u}f \right)^{2}+ \frac{1}{u^{2}}\left( \partial_{\phi}f \right)^{2}+ f^{2}\left( j^{2}+ \frac{1}{u^{2}}g^{2}\right) + \frac{1}{2}\kappa^{2}\left( 1 - f^{2}\right)^{2}\right\}.
< O_{k_{0}}O_{k_{1}}. . . O_{k_{p - 1}}P >_{0}= - \mid \ln \mu \mid \left( \begin{array}{c}{{2 \Sigma}}\\{{\Sigma}}\\ \end{array}\right) \frac{1}{\pi \sqrt{2 | t_{2}|}}\frac{1}{( - 4 t_{2})^{\Sigma}}\frac{\partial^{p - 1}}{\partial t_{0}^{p - 1}}t_{0}^{\Sigma},
\tilde{g}= \frac{( R_{s}M )^{\frac{3 - d}{4}}}{\prod r_{i}}\ , \quad \tilde{\alpha}^{\prime}= \frac{R_{s}^{1 / 2}}{M^{3 / 2}}\ , \quad \tilde{R}_{i}= \frac{1}{r_{i}M}\ , \quad R_{s}\rightarrow 0 \ ; \M , r_{i}\mathrm{~ f i x{\ ' e}s}\ .
II=((\omega\,k+\nabla \varphi)\cdot \nabla ){\bf E}=(\omega\,k_1+\varphi_{x_1}){\bf E}_{x_1}+(\omega\,k_2+\varphi_{x_2}){\bf E}_{x_2}+(\omega\,k_3+\varphi_{x_3}){\bf E}_{x_3}.
\mathcal{B}_\eta\left(\hat{D}_A(\mathbf{x})\,\hat{D}_B(\mathbf{y})\,\hat{\rho}_{AB}\,{\hat{D}_A(\mathbf{x})}^\dag\,{\hat{D}_B(\mathbf{y})}^\dag\right) &= \hat{D}_C\left(\sqrt{\eta}\,\mathbf{x}+ \sqrt{\eta-1}\,T\mathbf{y}\right)\,\mathcal{B}_\eta(\hat{\rho}_{AB})\,{\hat{D}_C\left(\sqrt{\eta}\,\mathbf{x}+ \sqrt{\eta-1}\,T\mathbf{y}\right)}^\dag
T_\alpha=\left(\begin{smallmatrix}0 0 t_{02}t_{10}0 0 0 t_{21}0 \end{smallmatrix}\right),\,\,\,\,T^2_\alpha=\left(\begin{smallmatrix}0 t_{01}0 0 0 t_{12}t_{20}0 0\end{smallmatrix}\right),\,\,\,\,t_{ij}=\sqrt{\frac{\alpha_{1n}(j)}{\alpha_{1n}(i)}},\,\,i,j\in{\mathbb F}_p.
\frac{\sqrt{v - \epsilon}J_{( D - 2 ) / 2}^{\prime}( \sqrt{v - \epsilon})}{J_{( D - 2 ) / 2}( \sqrt{v - \epsilon})}= \frac{\sqrt{\epsilon}K_{( D - 2 ) / 2}^{\prime}( \sqrt{\epsilon})}{K_{( D - 2 ) / 2}( \sqrt{\epsilon})}
- \frac 1 2 \Omega^{+ 2 i}e^{- 2}- \frac 1 2 \Omega^{- 2 i}e^{+ 2}- 2 i e_{q}^{1 +}\gamma_{q \dot{q}}^{i}\pi_{\dot{q}}^{1 -}+ 2 i e_{q}^{2 -}\gamma_{q \dot{q}}^{i}\pi_{\dot{q}}^{2 +}= 0 .
\ell \cdot \binom{\ell \cdot (\lfloor N /\log N\rfloor+1)}{k-1}\leq k\binom{k\cdot (\lfloor N /\log N\rfloor+1)}{k-1}\leq \frac{k^k 2^{k-1}N^{k-1}}{(k-1)^{k-1}(\log N)^{k-1}}
g(x) =\prod_{j=1}^n(1+r_j(t_0)\cos 2\pi n_jx) &=1+\sum_{A\subset \{1,2,\ldots,n\}}\left(\prod_{j\in A}r_j(t_0)\cos 2\pi n_j x\right) &=1+\sum_{A\subset \{1,2,\ldots,n\}}\left(w_A(t_0)\prod_{j\in A}\cos 2\pi n_j x\right),
0\leqslant |u_i(z_i)|^2|\Psi_i|^2- |\Psi|^2 = \Big( 1- \prod_{j\ne i}f_{ij}^2(z_i-z_j) \Big) |u_i(z_i)|^2 |\Psi_i|^2 &\leqslant \sum_{j\ne i}\frac{C\|u_i\|_{L^\infty}\mathbf{1}(|z_i-z_j|\leqslant \ell)}{N|z_i-z_j|}|\Psi_i|^2.
a^{+}(\hat{n}+1-\lambda)_{k,\lambda}a =a^{+}\sum_{l=0}^{k}\binom{k}{l}(\hat{n})_{l,\lambda}(1-\lambda)_{k-l,\lambda}a &=\sum_{l=0}^{k}\binom{k}{l}(1-\lambda)_{k-l,\lambda}\sum_{m=0}^{l}S_{2,\lambda}(l,m)(a^{+})^{m+1}a^{m+1}&=\sum_{m=0}^{k}\bigg(\sum_{l=m}^{k}\binom{k}{l}(1-\lambda)_{k-l,\lambda}S_{2,\lambda}(l,m)\bigg)(a^{+})^{m+1}a^{m+1}.
\tfrac{1}{D_{n+1}}Tr_{d_i}F(d_j) \stackrel{(Fon)}{=}\tfrac{1}{D_{n+1}}Tr_{pr_1}Tr_{ba}F(ba) F(pr_2) \stackrel{(Deg)}{=}\tfrac{1}{D_{n}}Tr_{pr_1}F(pr_2) &\stackrel{(CdB)}{=}\tfrac{1}{D_{n}}F(d_{j-1}) Tr_{d_i}
\iint_{M \times M}\frac{\left| u(x)-u(y) \right|^{p-2}(u(x)-u(y))(v(x)-v(y))}{(d_g(x,y))^{N+ps}}d\mu_g(x)d\mu_g(y) +\int_{M}\left| u(x) \right|^{p-2}u(x)v(x)d\mu_g(x) =\int_{\Omega}f(x,u(x))v(x)d\mu_g(x),
\frac{n!}{\pi^n}\frac{\prod_{i<j}|z_i-z_j|^2}{\left(\displaystyle \int \prod_{k=1}^{n}|z-z_k|^2 d\nu_{S^1}(z)\right)^{n+1}}= \frac{n!}{\pi^n}\frac{\prod_{i<j}|z_i-z_j|^2}{\|\tilde{a}\|_2^{2n+2}}
\frac{\left(\log{\frac{n}{k_j}}\right)^{1/s}}{\left(\log{\frac{\theta_j n}{k_j}}\right)^{1/s}}\le \frac{\left(\log{2^{2^j}}\right)^{1/s}}{\left(\log{2^{-2^{j-1}}\frac{1}{2}2^{2^j}}\right)^{1/s}}=\left( \frac{2^j}{2^{j}- 2^{j-1}- 1}\right)^{1/s}\le 4^{1/s},
\nu ( E_N ) = \nu ( [0_{M_N}]) + \nu ( [1_{M_N}] ) \leqslant C_\phi \, \left( e^{((N-1) \alpha - 1) (\phi (0)- P(\phi) )}+ e^{((N-1) \alpha - 1 )(\phi (1)- P(\phi) )}\right)
p^{1}=b_{1}\cdot\frac{e_{1}}{|e_{1}|}=|b_{1}|\frac{|e_{1}|}{|e_{1}|}cos\alpha=|b_{1}|\frac{dx}{dq^{1}}\Rightarrow\frac{p^{1}}{|b_{1}|}=\frac{dx}{dq^{1}}
\theta_{\infty}> \theta_0 + \sum_{j=1}^{m}\theta_j \; , \theta_{0}> \theta_\infty + \sum_{j=1}^{m}\theta_j \; , \theta_i > \sum_{i \ne j=1}^{m}\theta_j + \theta_{\infty}+ \theta_{0}, \forall \, i=0,...,m
A_{\mu}= \mp{\frac{2 i}{g}}\, ( H_{1}\,{\frac{v_{\mu}^{1}}{\rho_{1}}}+ g \, \kappa \, E_{1 3}^{\pm}\, R_{\mu}^{1}) \mp \,{\frac{2 i}{g}}\, ( H_{2}\,{\frac{v_{\mu}^{2}}{\rho_{2}}}+ g \, \kappa \, E_{2 3}^{\pm}\, R_{\mu}^{2}) .
{\bf J}={1\over 2}\left( \begin{array}{cc}p^\dagger q^\dagger \end{array}\right){\bf \sigma}\left(\begin{array}{c}p q \end{array}\right)+{1\over 2}\left( \begin{array}{cc}r^\dagger s^\dagger \end{array}\right){\bf \sigma}\left(\begin{array}{c}r s \end{array}\right)
\widehat y_i^*X\widehat x_i = y_i^*\left(\lim_{s,t\rightarrow \lambda_i}\frac{v(s)P(t)-P(s)v(t)}{s-t}\right)x_i = y_i^*\left(v'(\lambda_i)P(\lambda_i)-P'(\lambda_i)v(\lambda_i)\right)x_i = -y_i^*P'(\lambda_i)v(\lambda_i)x_i.
w' = 2 e^{- \varphi}w \tilde{F}\Theta^{-1}\frac{\sinh \Theta}{\cosh \Theta}v- e^{-\varphi}\tilde{F}\Theta^{-1}\frac{\sinh \Theta}{\cosh \Theta}v^{-1}\sinh^{-2}u \, u^2 w_k \varphi^k + R_1 + R_2,
\lim_{s\downarrow 0}\frac{\widehat{\gamma}_{n-1}(xs)-\widehat{\gamma}_{n-1}(s)}{L(1/s)/(n-1)!}=\frac{\beta(\lambda_1)}{1-\beta(\lambda_1)}\lim_{s\downarrow 0}\frac{\widehat{\alpha}_{n-1}(xs)-\widehat{\alpha}_{n-1}(s)}{L(1/s)/(n-1)!}= -(\log x)\frac{\beta(\lambda_1)}{1-\beta(\lambda_1)}c_{\alpha}.
\varphi_v(\gamma) =\varphi_v(m_1^{-1}g^{i_1}m_1)\varphi_v(m_2^{-1}g^{i_2}m_2)\cdots\varphi_v(m_k^{-1}g^{i_k}m_k)\varphi_v(m) &\in \langle M_v, \varphi_v(g)\rangle= \langle (ab)^l, B, (ab)^{4q}\rangle=H(q)
(-\Delta_p)^s U(x)= U^{p^*-1}(x)\leq \frac{C}{|x|^{N+\frac{s\,p}{p-1}}}\leq \frac{C\,\theta^{s\,p\,\frac{2-p}{p-1}}}{R^{s\,p\,\frac{2-p}{p-1}}|x|^{N+s\,p}}.
\dim \mathcal{C}_{A_2}(v) = 1 \cdot 5 + 2 \cdot 3 + 1 \cdot 6 = 20 \dim \mathcal{C}_{A_3}(v) = 1 \cdot 23 + 4 \cdot 20 + 3 \cdot 6 + 6 \cdot 12 + 1 \cdot 24 = 217 \dim \mathcal{C}_{A_4}(v) = 1 \cdot 119 + 5 \cdot 115 + 10 \cdot 50 + 10 \cdot 100 + 15 \cdot 30 + 10 \cdot 60 + 1 \cdot 120 = 3364
{\cal E}_{a s}^{s c}= - \frac 1 2 C_{s}\sum_{m = - \infty}^{\infty}\int_{m_{e}}^{\infty}d k [ k^{2}- m_{e}^{2}]^{1 - s}\frac{\partial}{\partial k}\ln f_{m}^{a s}( i k ) -{\cal E}_{d i v}^{s c},
-\Delta u_{\rho}(z)+\hat{\xi}_{\rho_0}u_{\rho}(z)=\frac{1}{1-\vartheta}[\lambda u^+_{\rho}(z)^{q-1}+f(z,u_{\rho}(z))]+\hat{\xi}_{\rho_0}u_{\rho}(z)\ \mbox{in}\ \Omega,
{\mathcal L}_{\langle X \rangle^{G}}J_{a}= \big\langle{\mathcal L}_{\langle X \rangle^{G}}J_{a}\big\rangle^{G}=\langle{\mathcal L}_{X}J_{a}\rangle^{G}-\langle{\mathcal L}_{a_{M}}Q(X)\rangle^{G}=\langle{\mathcal L}_{X}J_{a}\rangle^{G}.
(\zeta) = \frac{4l}{k^2}\left(\underset{i=1}{\overset{k}{\sum}}(p_i - \frac{1}{2}) \right)^2 \underset{i=1}{\overset{k}{\sum}}p_i(1-p_i) \leq \frac{l}{k^2}\left(\underset{i=1}{\overset{k}{\sum}}(p_i - \frac{1}{2}) \right)^2 .
ds^2=-4dx^+dx^- -\bigg[\sum\limits_{a=1}^4x_a^2 +(1+{\tilde m\over{\tilde r}^4})\sum\limits_{b=5}^7 x_b^2 + (1-3{\tilde m\over{\tilde r}^4})x_8^2\bigg](d{{\mu x^+}\over R^2})^2+ \sum\limits_{i=1}^8dx_i^2.
E Y_{m,n}\le \sum_{i=1}^n \sum_{\ell=1}^{m}E [ G(S_i,\tilde{S}_\ell) ] \le C_3 \sum_{i=1}^{n}\sum_{\ell=1}^{m}i^{-1/2}\ell^{-1/2}\le C \sqrt{n m}\,,
q\left( x,s\right) =\sum\limits_{n=0}^{\infty}q_{n}\left( x\right)f_{n}\left( s\right) \approx \sum\limits_{n=0}^{N-1}q_{n}\left( x\right)f_{n}\left( s\right) ,s\geq \underline{s}\left( d\right) ,
\tilde{j}_{m}(\rho) =\frac{q}{(2\pi)^3}\int_{|p|<(6\pi^2\rho/q)^{\frac{1}{3}}}\frac{1}{\sqrt{|p|^2+m^2}}{\rm d}p & = \frac{q}{4\pi^2}\left[\eta\sqrt{\eta^2+m^2}-m^2\ln{\left(\frac{\eta+\sqrt{\eta^2+m^2}}{m}\right)}\right], \eta=\left(\frac{6\pi^2\rho}{q}\right)^{\frac{1}{3}}.
d s^{2}= - 4 d x^{+}\, d x^{-}+ \sum_{i = 1}^{8}( d x^{i})^{2}- \left( \sum_{i = 1}^{4}( x^{i})^{2}+ 2 \sum_{i = 5}^{8}( x^{i})^{2}\right) ( d x^{+})^{2}
a_{23}^{2314}=a_{23}^{2341}=a_{23}^{2134}=a_{23}^{2143}=a_{23}^{2431}=a_{23}^{2413}= a_{23}^{1234}=a_{23}^{1243}=a_{23}^{1423}=a_{23}^{4231}=a_{23}^{4213}=a_{23}^{4123}
\|g(w, p)\|_{L^{q'}_{t>T}W^{2,r'}_x}\lesssim\big\Vert \sum^N_{\mu,\nu=1}\lambda_{\mu\nu}\|u_\mu\|_{W^{2,r}_x}\|u_\nu|^{p+1}|u_\mu|^{p-1}\|_{L_x^{\frac{r}{2p}}}\big\Vert_{L^{q'}_{t>T}}\lesssim\sum^N_{\mu,\nu=1}\big\Vert \|u_\mu\|_{W^{2,r}_x}\|u_\nu|^{p+1}|u_\mu|^{p-1}\|_{L_x^{\frac{r}{2p}}}\big\Vert_{L^{q'}_{t>T}},
\mathcal A\left(\mathbb{H}_{\mathbf{f}}\left([x_{\textup{f}}],[y_{\textup{f}}]\right)\right)(\kappa,\kappa-k-s) =\mathfrak{h}_{\mathbf{f}}^{\textup{c-wt}}\left([x_{\textup{f}}],[y_{\textup{f}}]\right) \cdot (\kappa-k) \,\qquad\qquad+ \frak h_p([x_{\textup{f}}],[y_{\textup{f}}])\cdot \left (s-\frac{\kappa-k}{2}\right )\mod \frak{I}^2.
\hat{\boldsymbol{\theta}}=\arg\underset{\boldsymbol{\theta}}{\min}\{-\log(p(\boldsymbol{\Upsilon}_{\mathrm{u}}\vert\mathbf{U},\boldsymbol{\theta})\}=\arg\underset{\boldsymbol{\theta}}{\min}\Bigl\{\log|\mathbf{K}_{\mathrm{u}}|+\boldsymbol{\Upsilon}_{\mathrm{u}}^{\mathrm{T}}\,\mathbf{K}_{\mathrm{u}}{}^{-1}\,\boldsymbol{\Upsilon}_{\mathrm{u}}\Bigr\}.
< \xi ( x , t ) \xi ( x^{\prime}, t^{\prime}) > = N \int D \xi \xi ( x , t ) \xi ( x^{\prime}, t^{\prime}) e^{- \frac{1}{2}\int d x^{\prime \prime}d t^{\prime \prime}\frac{\xi ( x^{\prime \prime}, t^{\prime \prime})^{2}}{\theta^{2}}}.
- \int_{1 - \delta}^1 H(u) du = \int_0^\delta F^{-1}(u) du &= \int_0^{\sqrt{F^{-1}(\delta)}}2 x^3 F'(x) dx &= \int_0^{G^{-1}(\delta)}x^2 G'(x) dx &= - \int_0^\delta \log(1-u) du
= \sum_{n=1}^N\left(\psi(n)-\psi(n+1) \right) \sum_{j=1}^n \frac{\varphi(j)}{j|{j}|_{\mathcal{D}_1}|{j}|_{\mathcal{D}_2}\cdots |j|_{\mathcal{D}_m}}+ \psi(N+1)\sum_{j=1}^N \frac{\varphi(j)}{j|{j}|_{\mathcal{D}_1}|{j}|_{\mathcal{D}_2}\cdots |j|_{\mathcal{D}_m}}.
{\frac{d}{d t}}\ln R = \mathrm{R e}( P ) + b_{\scriptscriptstyle R}\left( \frac{1}{R}\nabla^{2}R - ( \nabla S )^{2}\right) -{\frac{b_{\scriptscriptstyle I}}{R^{2}}}\nabla \cdot ( R^{2}\nabla S )
P_{N}(\alpha_{1},\alpha_{2};\beta_{1}, \beta_2;q;z)=\sum_{m=0}^{N}\frac{(q^{-N};q)_{m}(\alpha_{1};q)_{m}(\alpha_{2};q)_{m}}{(q;q)_{m}(\beta_{1};q)_{m}(\beta_{2};q)_{m}}z^{m}.
\epsilon(\alpha_0,\beta_0)=\dfrac{2L_{1M}^{\frac{5-\nu}{2}}L_{1m}^{\nu-2}\Tilde{\Phi}_1(\alpha_0,\beta_0)}{N_{1m}^{\frac{1-\nu}{2}}\bigg[\gamma\bigg(\tfrac{1-\nu}{2},\beta_0^2\tfrac{N_{1M}}{L_{1m}}\bigg)-\gamma\bigg(\tfrac{1-\nu}{2},\alpha_0^2\tfrac{N_{1M}}{L_{1m}}\bigg)\bigg]}<1,