harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\delta^2\lvert\textbf{s}\rvert^2}{32C_{\mathbf{I}}C_e^2c(0)\sigma_z^2}\!\cdot\! \frac{\epsilon_S^2\sigma_z^2}{\lvert\textbf{s}\rvert^2}\! \ge\! \frac{\delta^2}{\sum\limits_{i \ge 1}\! \frac{32C_{\mathbf{I}}e^{2L (T+\frac{\epsilon_{b}}{K_{S,0}+1})}}{(i+K_{S,0})^2}}\overset{\Delta}{=}R.
\mu X_{j-1}(u) e^{-\int_{\tau_{j+1}}^u G_j(v) \: dv}\leq (1 + 3 \delta) s e^{\int_{\tau_j}^u G_{j-1}(v) \: dv}e^{-\int_{\tau_{j+1}}^u G_j(v) \: dv}&= (1 + 3 \delta) s e^{\int_{\tau_j}^{\tau_{j+1}}G_j(v) \: dv}e^{-s(u - \tau_j)}.
\Big | \sum_{j=1}^\infty (M_{a_j}P_\alpha M_{1-b_j}f)(z) \Big |e^{-\frac{\alpha}{2}|z|^2}&\leq \Big ( \frac{\alpha}{\pi}\Big )^n \sum_{j=1}^\infty |a_j(z)| \int_{\mathbb{C}^n}|1-b_j(w)| |f(w)| e^{-\frac{\alpha}{2}|w|^2}e^{-\frac{\alpha}{2}|w-z|^2}dw &\leq 2^n \| f\|_{\infty,\alpha}
G((v,0);(v,0))=2m^{-2}\left(\sum_{i\notin I_2}\left| 4^{-1}m-\lambda_i \right| a_i^2+\sum_{i\in I_2}a_i^2 \right) \geq c \sum_{i=1}^\infty a_i^2=c\|v\|^2_{L^2},
\mathcal{R}_{kl}^{\nu}(\alpha_{1,\nu})=\{\xi\in\mathcal{O}_{\nu}:|\langle k,\omega_{\nu}(\xi)\rangle+\langle l,\bar{\Omega}_{\nu}(\xi)\rangle|<\alpha_{1,\nu}\frac{\langle l\rangle_{\infty}}{\langle k\rangle^{\tau}}\},
m_{\phi_{S}^{0}}^{2}= p_{r}^{2}= a ( p_{r}) = 4 \left. \frac{( k_{r}+ h ) ( \tilde{k}_{r}+ \tilde{h}) + ( k_{i}+ s^{I}) ( \tilde{k}_{i}+ \tilde{s}^{I})}{( k_{r}+ h )^{2}+ ( k_{i}+ s^{I})^{2}}\right|_{p = p_{r}}.
X(t) = \begin{cases}x (-r \le t \le 0), \displaystyle x \, \exp \! \Bigg[ \! \int_0^t \!\! A_1 \big( X(s-r) \big) \mbox{d}W(s) \! + \! \int_0^t \! \Bigg\{\! A_0 \big( X(s-r) \big) - \frac{A_1 \big( X(s-r) \big)^2}{2}\! \Bigg\}\mbox{d}s \Bigg] \ (0 \le t \le T), \end{cases}
\hat{H}(k_x, k_y, k_z) =\left(\begin{array}{cc}\sin k_y - \cos k_x 2 + \cos k_x + \cos k_y + e^{ik_z}2 + \cos k_x + \cos k_y + e^{-ik_z}- \sin k_y + \cos k_x\end{array}\right) &=\hat{H}_{\mathrm{loc}}(a(k_x, k_y), b(k_x, k_y), k_z),
\begin{array}{l}V_1(Y_1)=\exp\left(i\left(\lambda T-\int_0^T p(t)dt\right)\right)(1+O(\lambda^{-1})), V_1(Y_2)=\exp\left(-i\left(\lambda T-\int_0^T p(t)dt\right)\right)(1+O(\lambda^{-1})).\end{array}
\det R(z) =\exp\big(\frac{1}{2n}\sum_{k=0}^{n-1}\log |z_0^2-z_1^2-z_2^2-2z_1z_2\cos\theta_k|\big) &=\big(\prod_{k=0}^{n-1}|z_0^2-z_1^2-z_2^2-2z_1z_2\cos\theta_k|\big)^{1/2n}.
\theta = \left( \begin{array}{c}{{- \frac{\sqrt{2 \omega_{o}}}{\mu + \lambda + \mu_{o}}d_{4}U ( a , x )}}\\{{d_{2}U ( a + 1 , x )}}\\{{- \frac{\sqrt{2 \omega_{o}}}{\mu + \lambda + \mu_{o}}d_{2}U ( a , x )}}\\{{d_{4}U ( a + 1 , x )}}\\ \end{array}\right) .
G_{M N}= \left( \begin{array}{c | c}{\begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}}&{0}\\ \hline{0}&{\mathrm{{\bf 1}}}\\ \end{array}\right) , \B_{M N}= 0 , \D = X^{1}\quad ( M , N = 0 , 1 , \dots ) .
\stackrel{\wedge}{\psi}^{\dagger}= \sigma_{-}= \left( \begin{array}{c c}{{0}}&{{0}}\\{{1}}&{{0}}\\ \end{array}\right) , \qquad \stackrel{\wedge}{\psi}= \sigma_{+}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{0}}&{{0}}\\ \end{array}\right)
(s_1,\dots,s_r) \cdot [p_1,\dots,p_r] = [s_1p_1, s_1^{-1}s_2p_2,\dots,s_{r-1}^{-1}s_rp_r] &= [s_1p_1s_1^{-1},\dots,s_rp_rs_r^{-1}].
(\kappa^{*}_{2}-\kappa^{*}_{1})^{2}g = -\frac{4R^{4}}{3\sqrt{c}}\cdot\frac{\|\nabla K^{*}\|^{2}}{(1+\rho_{*}\kappa^{*}_{1})^{3}(1+\rho_{*}\kappa^{*}_{2})^{3}}
\dim_H E_\varphi(\Phi) = \liminf_{\ell\to\infty}\frac{\sum_{j=1}^\ell \beta^{j/b}}{d\sum_{j=1}^{\ell+1}\beta^{j/b}- \beta^{(\ell+1)/b}}= \frac 1{d\beta^{1/b}-\beta^{1/b}+1}.
\begin{cases}\Delta^2 u=\lambda u,{\rm in}\ \Omega, (1-\sigma)\frac{\partial^2 u}{\partial\nu^2}+\sigma\Delta u=0,{\rm on}\ \partial\Omega, \frac{\partial\Delta u}{\partial\nu}+(1-\sigma){\rm div}_{\partial\Omega}\left(D^2 u\cdot\nu\right)_{\partial\Omega}=0,{\rm on}\ \partial\Omega,\end{cases}
(\iota,\phi)^{(1)}\big(\psi\big(as_v^{\Lambda^i}b\big)\big)=\sum_{\mu\in v\Lambda^{e_i}}\phi(a)s_\mu^\Lambda{s_\mu^\Lambda}^*\phi(b)=\phi(a)s_v^\Lambda \phi(b)=\phi(as_v^{\Lambda^i}b).
\mathcal{E}_{1}(\tau):= \frac{1}{8}\sum_{\chi(-1) = -1}E_{\chi,k}(\tau),E_{\chi,k}(\tau) = 1 + \frac{2}{L(1 - k, \chi)}\sum_{n=1}^{\infty}\chi(n) \frac{n^{k-1}q^{n}}{1 - q^{n}},
\sigma_{Dd_{n+1}^-(x)}- \sigma_{Dd_{n}^+d_{n+1}^-(x)}\:= \: D(x) - Dd_{n}^+(x) \sigma_{Dd_{n+1}^-(x)}- \sigma_{Dd_{n}^-d_{n+1}^-(x)}\:= \: D(x) - Dd_{n}^-(x) - Dd_{n+1}^+(x) + Dd_{n}^-d_{n+1}^+(x)
\langle s_1, p(v_j)-p(v_1)\rangle =0 (2\leq j\leq t) \langle s_2, p(v_j)-p(v_1)\rangle =0 (2\leq j\leq t-1) \vdots \langle s_{t-1}, p(v_j)-p(v_1)\rangle =0 (j=2)
p_\psi(z) =\mathcal{CN}(z;0,\sigma_\mathrm{N}^2)*\left[\sum\limits_{k=1}^{Q-1}\tilde{\omega}_k\mathcal{CN}(z;0,\tilde{\sigma}_k^2)+\tilde{\omega}_Q\delta(z)\right] &=\sum\limits_{k=1}^{2^{N_B-1}}{\omega}_k\mathcal{CN}(z;0,\sigma_k^2),
\pi_1(\varphi) = \alpha f(\alpha) - c_{\bar{\alpha}}\bar{\alpha}A'_{\bar{\alpha}}= \alpha f(\alpha) - c_{\bar{\alpha}}A_{\bar{\alpha}}= 0, \pi_1(\psi) = - c_{{\alpha}}{\alpha}A'_{{\alpha}}+ \bar{\alpha}f(\bar{\alpha}) = - c_{{\alpha}}A_{{\alpha}}+ \bar{\alpha}f(\bar{\alpha}) = 0,
\begin{aligned}\frac{2}{r}-\alpha < \frac{2}{r_0}\leq 2, \qquad\frac{n}{q}-(2-\alpha) < \frac{n}{q_0}\leq \frac{n}{q}, \frac{2}{r}-\beta < \frac{2}{r_1}\leq 2, \qquad\frac{n}{q}-(1-\beta) < \frac{n}{q_1}\leq \frac{n}{q},\end{aligned}
+ \oint_{\partial{\Omega}}\Biggl [{\frac{\delta^{\wedge}F}{\delta q ( x )}}\Bigg \vert_{\partial{\Omega}}{\frac{\delta^{\vee}G}{\delta p ( x )}}+{\frac{\delta^{\vee}F}{\delta q ( x )}}{\frac{\delta^{\wedge}G}{\delta p ( x )}}\Bigg \vert_{\partial{\Omega}}\Biggr ] d S
\phi_{w,\xi,\eta}(w',z')=2^{-3/2}\pi^{-2}\cdot \langle\eta\rangle^{1/2}\cdot \exp\left(i\eta \cdot z'+i\xi \cdot (w'-(w/2))-\langle \eta\rangle\cdot \|w'-w\|^2/2 \right).
{\cal L}= \frac{1}{2}\left\{\partial_{m}\phi_{i}^{\prime}\partial^{m}\phi_{i}^{\prime}+ \imath \bar{\psi}_{i}\gamma^{m}\partial_{m}\psi_{i}-{\cal W}_{i}{\cal W}_{i k l}\phi_{k}^{\prime}\phi_{l}^{\prime}-{\cal W}_{i k}{\cal W}_{i l}\phi_{k}^{\prime}\phi_{l}^{\prime}-{\cal W}_{i j}\bar{\psi}_{i}\psi_{j}\right\},
\int_{r\leq r(x)\leq r_j}|\nabla v|^2 e^{-2\rho}dx+ \int_{S_{r}}\frac{\partial v}{\partial r}v e^{-2\rho}dx-\int_{S_{r_j}}\frac{\partial v}{\partial r}v e^{-2\rho}dx=\int_{r\leq r(x)\leq r_j}[\lambda-V_0-V_1-V_2]v^2e^{-2\rho}dx.
\sum_{i\in N}\varphi_{i}(v)=\frac{1}{|N|!}\sum_{R}\sum_{i\in N}v(P_{i}^{R}\cup\{i\})-v(P_{i}^{R})=\frac{1}{|N|!}\sum_{R}v(N)=\frac{1}{|N|!}|N|!\cdot v(N)=v(N)
\max_{\substack{p_i\in\mathbb{R},R_i\ge0, i=1,\dots,K}}~ \sum_{i=1}^K\alpha_iR_i ~ \rho_i\exp\left(\frac{(2^{R_i}-1)\sigma_i^2}{Q_{ii}p_i}\right) &~\times\prod_{k{\ne}i}\left(1+\frac{(2^{R_i}-1)Q_{ki}p_k}{Q_{ii}p_i}\right)\le1, &0 ~{\le}~ p_i~{\le}~ P_i,~i=1,\dots,K.
A_0 \ = \ \{2n+1, 3n+2, 4n+3 \}, A_j \ = \ \{2n+1-j, 3n+2-j, 4n+3+2j \}, A_n \ = \ \{n+1, 2n+2, 6n+3 \}, A_{n+1}\ = \ \{n, 4n+2, 4n+4 \}, A_{n+j}\ = \ \{n+1-j, 4n+3-j, 4n+2+2j \}, A_{2n}\ = \ \{1, 3n+3, 6n+2 \},
{\hat{H}}= \frac{H}{M}= \frac{2 \pi}{l}\left( H_{\mathrm{C F T}}+ \kappa ( h ) \frac{l^{2 - h}}{\left( 2 \pi \right)^{1 - h}}B \right) \, , \qquad H_{\mathrm{C F T}}= L_{0}+ \bar{L}_{0}- \frac{c}{1 2}{\mathrm{I d}}\: .
\left| \frac{{\partial \varphi}_{i}}{{\partial y}_{j}}\right|\le \frac{l_{i}^{x}}{l_{1}^{x}}\frac{\left( M_{1}^{x}M_{i}^{x}-\mu_{1}^{x}\mu_{i}^{x}\right)}{{(\mu_{i}^{x})}^{2}\sum\limits_{j=1}^n y_{j}}.
F_{N}(a,b;t)= \frac{1-tq^{N+1}}{1-t}-\frac{(b-a)(1-tq^{N+1})t}{(1-t)(b-at)}+\frac{(b-a)(1-bq^{N+2})t}{(1-bq)(b-at)}F_{N}(a,bq;t).
[ 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}
g = - \kappa^{\frac{4}{3}}\left( \begin{matrix}{- 2 ( \psi^{2}-{\frac{f}{\kappa^{2}}})}&{0}&{-{\frac{1}{\sqrt{2}}}\psi}\\{0}&{{\frac{2}{f \kappa^{2}}}}&{0}\\{-{\frac{1}{\sqrt{2}}}\psi}&{0}&{-{\frac{1}{4}}}\\ \end{matrix}\right) .
O(\l,D)=e^{\int \left(\frac 1{\xi-\l}-\frac 1{\xi-\l_*}\right)\chi_D(\xi)d\xi}, \chi_D(\xi)=\begin{cases}1/2, \xi\in(a_j,\l_j) -1/2, \xi\in(a_j,\l_j) 0, \end{cases}
H_{\sigma , \alpha}\ : = \ \left\{\Psi \in{\mathcal H}_{\sigma}^{( 0 )}\bigm | Q_{\alpha}\ \Psi = 0 \right\}\ \big / \ \left\{\Psi \mid \Psi = Q_{\alpha}\Phi , \Phi \in{\mathcal H}_{- \sigma}^{( 0 )}\right\},
\begin{array}{l l l l}{{\mathrm{f o r ~}\phi = + 1}}&{{~ : ~}}&{{( i^{k}, 1 , \pm 1 , \pm 1 , \pm 1 , 1 )}}&{{, ( k = 0 , . . . , 3 ) ~ ,}}\\{{\mathrm{f o r ~}\phi = - 1}}&{{~ : ~}}&{{( i^{k}e^{i \pi / 4}, 1 , \pm 1 , \pm 1 , \pm 1 , 1 )}}&{{, ( k = 0 , . . . , 3 ) \; ,}}\\ \end{array}
G_{\delta}=e^{\delta A}G\bigl(e^{\delta A}\cdot\bigr)=e^{\delta A}F_1\bigr(e^{\delta A}\cdot\bigr)+Be^{\delta A}F_2\bigl(e^{\delta A}\cdot\bigr) \quad,\sigma_{\delta}=e^{\delta A}\sigma\bigl(e^{\delta A}\cdot\bigr)e^{\delta A}.
\begin{aligned}|\tilde Y_{s,t}(x,v)| \lesssim \frac{\delta (t-s)}{1+\frac{\langle x^\perp\rangle^2}{\langle v^\perp\rangle^2}}, |\nabla_v \tilde Y_{s,t}(x,v)| \lesssim \log(2 + t) \frac{\delta (t-s)}{1+\frac{\langle x^\perp\rangle}{\langle v^\perp\rangle}}. \end{aligned}
dX^{\epsilon,\hat{v}}(t) = f_{\zeta_t^{\epsilon}}\bigl(X^{\epsilon, \hat{v}}(t), \hat{v}_{{\zeta_t^{\epsilon}}}^{\epsilon}(X^{\epsilon, \hat{v}}(t))\bigr) dt + \sqrt{\epsilon}\sigma_{{\zeta_t^{\epsilon}}}\bigl(X^{\epsilon,\hat{v}}(t)\bigr) d W(t),
\Psi = \int d^{3}x \{\overline{{C}}\chi + \overline{{P}}N + \overline{{C}}^{i}\chi_{i}+ \overline{{P}}^{i}N_{i}+ \overline{{C}}_{( 1 )}\chi_{( 1 )}+ \overline{{P}}_{( 1 )}N_{( 1 )}+ \overline{{C}}_{1}^{1}\chi_{1}^{1}\},
\begin{cases}U(\xi) \sim \left(\dfrac{c}{k}\xi\right)^{-1/p}, U'(\xi) \sim -\dfrac{1}{p}\left(\dfrac{k}{c}\right)^{-\frac{1}{p}}\xi^{-\frac{p+1}{p}},\end{cases}{\rm{as}}\xi\to +\infty.
S = \frac{k}{4 \pi}\int d^{2}z \left( E_{\mu \nu}( r ) \partial r^{\mu}\bar{\partial}r^{\nu}+ F ( r ) \partial x_{+}\bar{\partial}x_{-}\right) + \frac{1}{4 \pi}\int d^{2}z \sqrt{g}R \Phi ( r ) .
|B| \geq (1-\beta)^{k-2}\frac N2 - 2k(1-\beta)^{k-2}\frac{\varepsilon}{8^k}N &= (1-\beta)^{k-2}n \left( \left(2^{k-1}+\frac \varepsilon2\right)-2k\varepsilon \frac{2^k+\varepsilon}{8^k}\right) &\geq (1-\beta)^{k-2}2^{k-1}n.
D_{g_0^n}H^n\left(x^0, \vec{a}, t\right) &= im_n\left[ c_r k_{12}^n\left(\{x^0\}_{\tau=1}^{r+s}, \vec{a}\right) \cos(t c_r m_n) \mp c_s k_{21}^n\left(\{x^0\}_{\tau=1}^{r+s}, \vec{a}\right) \cos(t c_s m_n)\right].
\left| \sum_{n=1}^{N}n \cdot \pi(n) R_{X}^{n}- \sum_{n=1}^{N}\sum_{\lambda \in (W)}(\lambda R_{X})^{n}\right| \leq K+ \left| \sum_{n=N_{0}}^{N}R_{X}^{n}\left( n \cdot \pi(n)- \sum_{\lambda \in (W)}\lambda^{n}\right) \right| & \leq K+ 2 \epsilon \sum_{n=N_{0}}^{N}R_{X}^{\alpha n}& < K+ \frac{2 \epsilon}{1- R_{X}^{\alpha}},
\begin{cases}x \frac{\partial Y}{\partial x}= A (x,y) Y= x^{-p}(A_0(y) + A_1(y) x+ A_2(y) x^2 + \dots) Y y \frac{\partial Y}{\partial y}= B (x,y) Y= y^{-q}({B}_0(x) +{B}_1(x) y +{B}_2(x) y^2 + \dots) Y\end{cases}
\frac{\partial x}{\partial u}=\frac{|\begin{matrix}-\frac{\partial F}{\partial u}&\frac{\partial F}{\partial y}\\ -\frac{\partial G}{\partial u}&\frac{\partial G}{\partial y}\end{matrix}|}{|\begin{matrix}\frac{\partial F}{\partial x}&\frac{\partial F}{\partial y}\\ \frac{\partial G}{\partial x}&\frac{\partial G}{\partial y}\end{matrix}|}.
L_{r}(k) = (-1)^{r'}2^{r}\left(2\pi\right)^{r(r-1)/2 - kr}\times D_{\boldsymbol{K}}^{(k+1/2)r' - r'(r'+1)}\prod_{i=0}^{r-1}\Gamma(k-i)\prod_{i=1}^{r}L(i-k, \chi_{\boldsymbol{K}}^{i-1})^{-1}.
\mathbf B_{\lambda-\frac{n}{2}+1}F (\mathbf x) =x_n \vert x_n\vert^{-\lambda+\frac{n}{2}+1}\square \big(\vert x_n\vert^{\lambda-\frac{n}{2}+1}F\big)(\mathbf x)+\left(\lambda-\frac{n}{2}+1\right)\left(\lambda-\frac{n}{2}\right)\frac{1}{x_n}F(\mathbf x).
y'(0) =\frac{d}{dx}(b_0E_{\alpha, 1}(\lambda x^{\alpha}))|_{x=0}+\frac{d}{dx}(b_1xE_{\alpha, 2}(\lambda x^{\alpha}))|_{x=0}+ \frac{d}{dx}\left(\int_0^x (x-t)^{\alpha-1}E_{\alpha, \alpha}[\lambda(x-t)^{\alpha}]f(t)dt\right)\Big|_{x=0}.
Q_{d}=p^{d-s+1}\left( \sum_{i=d}^{s-1}p^{s-1-i}T_{i}^{p^{i-d}}\left( (1+S_{i})^{p^{i-d}}-1\right)-\sum_{i=d+1}^{s-1}p^{s-1-i}Q_{i}^{p^{i-d}}\right).
\mathbb{E}\left[ \sum_{j=1}^{d_1}\langle\phi_j-\phi_{j,\Pi_2},(1-\hat{\Pi}_{V_2})f_2\rangle_n^2\right] \leq \frac{\varphi^2d_1}{n}\mathbb{E}\left[ \Vert (1-\hat{\Pi}_{V_2})f_2\Vert^2_n\right] &\leq \frac{\varphi^2d_1}{n}\Vert (1-\Pi_{V_2})f_2\Vert^2.
\displaystyle\sup_{t\geq0}I_\infty(tv_{\varepsilon,z}) =\frac{\alpha}{N}\left(\frac{\|v_{\varepsilon,z}\|_{X^\alpha}^2}{\left(\displaystyle\int_{\mathbb{R}^N}b(x)|v_{\varepsilon,z}(x,0)|^{2_\alpha^*}dx\right)^\frac{2}{{2_\alpha^*}}}\right)^\frac{N}{2\alpha}&=\frac{\alpha}{N}S^\frac{N}{2\alpha}+o_\varepsilon(1), \ \ \varepsilon\rightarrow 0.
\frac{\beta(1+(1-2\alpha)r^m)}{(1-r^m)^{2(1-\alpha)+1}}+\frac{(1-\beta)r^m}{(1-r^m)^{2(1-\alpha)}}+\sum_{n=2}^{\infty}\prod_{k=0}^{n-2}\frac{k+2(1-\alpha)}{k+1}\phi_{n}(r) <\frac{1}{4^{1-\alpha}}-\phi_{1}(r).
\delta_{\nu}( s^{\prime}) E_{\mu}[ \xi | s ] = \Phi_{\xi}( s , 0 ) \{\delta_{\nu}( s^{\prime}) F_{\mu}[ \xi | s ] + i g \theta ( s - s^{\prime}) [ F_{\nu}[ \xi | s^{\prime}] , F_{\mu}[ \xi | s ] ] \}\Phi_{\xi}^{- 1}( s , 0 ) ,
\dim_H E_\varphi(\Phi) = \liminf_{\ell\to\infty}\frac{\sum_{j=1}^\ell \alpha (1/c-1) \log j}{d\sum_{j=1}^{\ell+1}\alpha/c \log j - \alpha (1/c-1) \log (\ell+1)}= \frac{1-c}d.
T^{\ast^k}T^{\ast^{(1+n)}}T^{1+n}T^k =M_{|E(uw)|^{2(n+k)}E(|w|^2)}M_{\bar{u}}EM_{u}, T^{\ast^k}TT^{\ast}T^k =M_{|E(uw)|^{2(k-1)}(E(|w|^2))^2E(|u|^2)}M_{\bar{u}}EM_{u}, T^{\ast^k}T^k =M_{|E(uw)|^{2(k-1)}E(|w|^2)}M_{\bar{u}}EM_{u}.
\sigma^{2}=\frac{\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{n}+\frac{2\mu\sum_{i=1}^{n}(x_{i}-\mu)}{n}=\frac{\sum_{i=1}^{n}(x_{i}^{2}-\mu^{2})}{n}=\frac{\sum_{i=1}^{n}x_{i}^{2}}{n}-\mu^{2}
\widetilde{\mathcal{R}}_2^- =T\sum_{m\geq 1}\sum_{n \geq 1}\frac{A(m,n)}{(m^2 n)^{\frac{1}{2}}}g\Big(\frac{m^2 n}{N}\Big)\sum_{c \asymp \frac{\sqrt{N}}{Tm}}\frac{S(n,-1;c)}{c}k^*\Big( \frac{{4\pi \sqrt{n}}/{c}-2T}{2M}\Big).
\mathcal{V}_\mu(\xi)=\frac{c}{2}\langle \xi,\xi \rangle-2\pi \langle \hat\rho+\mu,\xi\rangle +\sum_{\alpha\in R^+_0}\frac{2}{\langle\alpha,\hat\alpha^\vee\rangle}\int_0^{\langle \xi,\alpha\rangle}v_\alpha() ,
(\tilde\gamma^{-1})^{ki}H_{k\ell}(\tilde\gamma^{-1})^{\ell j}=(\tilde\gamma^{-1})^{ik}H_{k\ell}(\tilde\gamma^{-1})^{j\ell}=\frac{1}{2}(\tilde\gamma^{-1})^{ij}-\frac{1}{2}(\tilde\gamma^{-1})^{ji}
\left\langle \overset{\cdot}{K}_{\alpha}\left( \omega \right) ,\varphi\right\rangle :=-\left\langle K_{\alpha}\left( \omega \right) ,\varphi^{\prime}\right\rangle :=\int_{\mathbb{R}_{+}}K_{\alpha}\left( t,\omega\right) \varphi^{\prime}\left( t\right) dt
a_{i}= \oint_{\alpha_{i}}\frac{dx}{2 \pi i}\left(N - \frac{x F^{\prime}(x)}{F(x)}+ \tilde{q}\frac{x^{2N}(NF - x F^{\prime})}{2F(x)^3}+ \mathcal{O}(\tilde{q}^{2}) \right).
\partial_r(d_{is})+(d_{is}-d_{ir})\Gamma_{rs}^s=\sum_j\phi_{ij}(\partial_r)d_{js}\varphi_j=\phi_{is}(\partial_r)(d_{ss}-1)\varphi_s+\sum_j\phi_{ij}(\partial_r)d_j\varphi_j=\phi_{is}(\partial_r),
\Big[\frac{\partial}{\partial\Omega_1}\,,\frac{\partial}{\partial\Omega_2}\Big]=-\frac{\partial}{\partial\Omega_3}\,, \Big[\frac{\partial}{\partial\Omega_2}\,,\frac{\partial}{\partial\Omega_3}\Big]=-\frac{\partial}{\partial\Omega_1}\,,\Big[\frac{\partial}{\partial\Omega_3}\,,\frac{\partial}{\partial\Omega_1}\Big]= -\frac{\partial}{\partial\Omega_2}\,.
k_\theta \cdot f_{-}= \frac{1}{\sqrt{2}}(s(\theta), c(\theta), s(\theta), c(\theta)), k_\theta \cdot f_{+}= \frac{1}{(108)^{\frac{1}{4}}}(s(3\theta), 3c(3\theta), -3s(3\theta), -c(3\theta)),
f_F=\chi_B(z)B_{I_F^{\psi_k},\omega}\xi_F+\sum_{J,\lambda,\tau}\sum_{j\in J\cap I_k,l\in I_k}\xi_{J\lambda}\overline{z}_j\frac{\partial\chi_B(z)}{\partial\overline{z}_l}(-1)^n\varepsilon(j,J)d\overline{z}_l\wedge dz_N\wedge d\overline{z}_{J\setminus j}\otimes e_\lambda\otimes b_\tau.
w(\mathbf{u})=w(a^{-1}\mathbf{u})=w(\mathbf{y}+a^{-1}\mathbf{z})=d(\mathbf{y},-a^{-1}\mathbf{z})=d(\mathbf{x},-a^{-1}\mathbf{z'})+d(1,0) \geq d+1.
A + B = P([dx_1,d[x_2,x_3]]) + P([dx_2,[dx_1,x_3]]) + P([x_1,[x_2,x_3]]) + P([dx_1,d[x_2,x_3]]) + P([x_2,[x_1,x_3]]) + P([[x_1,x_2],x_3]) = P([x_1,[x_2,x_3]]) + P([x_2,[x_1,x_3]]) + P([dx_2,[dx_1,x_3]]) + P([[x_1,x_2],x_3]) = P([x_1,[x_2,x_3]] + [x_2,[x_1,x_3]] + [dx_2,[dx_1,x_3]] + [[x_1,x_2],x_3]).
2^{\frac{1}{2}-s}\sum_{n=1}^{\infty}\left(2n-1\right)^{s-\frac{1}{2}}\,K_{s-\frac{1}{2}}\left(\pi(2n-1)x\right)=\frac{\Gamma(s)\pi^{-s}x^{-s-\frac{1}{2}}}{4}+\pi\sum_{n=1}^{\infty}(-1)^{n}\,\intop_{0}^{\infty}\,y^{s-\frac{1}{2}}J_{s-\frac{1}{2}}(2\pi xy)\,e^{-2\pi ny}\,dy.
\mathfrak{p}_s = [\mathfrak{p}_{s-1},\mathfrak{n}] +[J\mathfrak{p}_{s-1},\mathfrak{n}] \subseteq [\mathfrak{d}^{j_0-s},\mathfrak{n}] +[J\mathfrak{d}^{j_0-s},\mathfrak{n}] \subseteq \mathfrak{d}^{j_0-s-1}.
\lim_{n \to \infty}\sup_{\|g\|\leq 1}&\left|\int_{\mathcal{Y}}g(y_{n})P^{\mu}(dy_{n}|y_{[0,n-1]})-\int_{\mathcal{Y}}g(y_{n})P^{\nu}(dy_{n}|y_{0,n-1]}) \right|=0,
\nabla_{\gamma}^{2}=(1-\gamma)\nabla_{5}^{2}+\gamma\nabla_{\times}^{2}=(1-\gamma)[\begin{matrix}0&1&0\\ 1&-4&1\\ 0&1&0\end{matrix}]+\gamma[\begin{matrix}1/2&0&1/2\\ 0&-2&0\\ 1/2&0&1/2\end{matrix}]
s^*(x) =\exp\left[ -\int^x\!\frac{2a^*(y)}{b^*(y)}\,dy \right] &=\exp\left[ -\int^x\!\frac{b'(y)}{b(y)}\,dy+\int^x\!\frac{2a(y)}{b(y)}\,dy-\int^x\!\frac{2m(y)}{M(y)}\,dy \right] &=\frac{1}{b(x)}\frac{1}{s(x)}\frac{1}{M(x)^2}&=\frac{m(x)}{M(x)^2},
|M'_1\cap M'_2|=|M'_1\cap(M'_2\cup\Psi')|=|(M'_1\cup\Psi')\cap M'_2| =|f'(M'_1)\cap f'(M'_2)|, |(M'_1\cup\Psi')\cap(M'_2\cup\Psi')| =|f'(M'_1)\cap f'(M'_2)|+2.
\sum\limits_{Y \in \mathcal Y}\alpha_Y \lambda_{Y,\delta,\ell}=\frac{1}{q^{(k-t)^2}{k \choose t}_q}\sum\limits_{i=0}^{k-t}(-1)^{k-t-i}q^{{k-t-i \choose 2}}{k-t \choose i}_q{k + i -\ell \choose i}_q &= \frac{{k-j \choose \ell}_q}{{k \choose \ell}_q}.
T_{+ 2 - 2}^{+ q}= \frac 1 2 \gamma_{q \dot{p}}^{i}\left( \Omega_{+ 2}^{+ 2 i}\psi_{- 2 \dot{p}}^{1 -}- \Omega_{- 2}^{+ 2 i}\psi_{+ 2 \dot{p}}^{1 -}\right) , \T_{+ p \pm 2}^{+ q}= \frac 1 2 \gamma_{q \dot{p}}^{i}\Omega_{+ p}^{+ 2 i}\psi_{\pm 2 \dot{p}}^{1 -},
(\mathcal{W}_X^{-1})_{i}^j = ~\left(\bar{g}^{jq}+\frac{\langle \bar{g}^{ja}\bar{\nabla}_as,\bar{g}^{qb}\bar{\nabla}_bs\rangle}{1-s^2-|\bar{\nabla}s|^2}\right)\tau_{qi}\sqrt{\frac{1-s^2}{1-s^2-|\bar{\nabla}s|^2}}
W ={{\left( \sqrt{\zeta}\left({{V}_{1}}\cos{{\phi}_{1}}+{{V}_{2}}\cos{{\phi}_{2}}\right)+{{X}_{1}}\right)}^{2}}+{{\left( \sqrt{\zeta}\left({{V}_{1}}\sin{{\phi}_{1}}+{{V}_{2}}\sin{{\phi}_{2}}\right)+{{Y}_{1}}\right)}^{2}}+\sum\limits_{i=2}^{\mu}{X_{i}^{2}+Y_{i}^{2}}.
d_{w}\left(\frac{\log Z_{n}-n\mu}{\sigma\sqrt{n}}\right) = \int_{-\infty}^{\infty}\left|\mathbb{P}\left(\frac{\log Z_{n}-n \mu}{\sigma \sqrt{n}}\leq x\right)-\Phi(x)\right|\mbox{d}x &\leq \frac{C}{n^{\delta / 2}}\int_{-\infty}^{\infty}\frac{1}{1+|x|^{1+\delta^{'}}}\mbox{d}x &\leq \frac{C}{n^{\delta/2}},
\int\psi^{k}f(1+ |\nabla f|^{2})^{\frac{k-2}{2}}|\Delta f| &\qquad\le CMd_x^{\alpha-1}\bigg\{\int\psi^{k-1}(1+ |\nabla f|^{2})^{\frac{k-1}{2}}+(k-2)\int\psi^{k-2}(1+ |\nabla f|^{2})^{\frac{k-2}{2}}\bigg\}.
\left\vert \frac{f^{(j)}(z+c_{i})}{f(z+c_{i})}\right\vert \leq |z|^{j(\rho\left( f+c_{i}\right) -1+\varepsilon )}=|z|^{j(\rho \left( f\right)-1+\varepsilon )},(i,j)\neq (l,0).
y\leq \left(\frac{p+1}{p}x - \frac c p\right)^{\mathrlap{\frac q p}}\cdot x^{1-\frac q p}= \left(\frac{p+1}{p}\right)^{\frac q p}\left[x-\frac{c}{p+1}\right]^{\frac q p}x^{1-\frac q p}.
\left( L2^{m+2}\right)^{-\left( \frac{\varepsilon}{2}+1\right)}+\frac{\left( L2^{m+1}\right)^{-\left( \frac{\varepsilon}{2}+1\right)}}{2^{\left( 1+\frac{2}{\varepsilon}\right)}\left( 2^{\left( 1+\frac{2}{\varepsilon}\right)}-1\right)}
\sum\limits_{k=1}^n \sum\limits_{p=n+1}^{2n}d(p-k)\Big\vert_{n=q}= q^2\Bigl(\sigma^2 + \int\limits_{\mathbb{R}\setminus \{0\}}u^2 \,d\mathcal{M}(u)\Bigr)+o(q^2)\;.
\sup_{\genfrac{}{}{0pt}{}{\scriptstyle{x,\bar x\in \overline{\Omega},\, t,\bar t\in[t_0,t_0+1],}}{\scriptstyle{x \not= \bar x,\, t \not= \bar t,\,{t_0\in{\cal I}}}}}\frac{|h(x,t)-h(\bar x,\bar t)|}{|x-\bar x|^\gamma+|t-\bar t|^{\frac{\gamma}{2}}}< C.
A(\bar{\rho}_m) := \{\eta_i+\underline{\mathbf{C}}(\sigma_{-m-2},m/2)\eta_i\colon \eta_i \in H^1(\partial\overline{X};\bar{\Lambda}(m))_-\colon \underline{\mathbf{C}}(\sigma_{-m-2},m/2)\eta_i\in H^1(\partial\overline{X};\bar{\Lambda}(m))_+ \}.
\begin{array}{lcr}qJ_1J_2-q^{-1}J_2J_1 = (q^2-q^{-2})J_3 qJ_2J_3-q^{-1}J_3J_2 = (q^2-q^{-2})J_1 qJ_3J_1-q^{-1}J_1J_3 = (q^2-q^{-2})J_2 \end{array}
- D_{1}( A_{1}+ 2 A_{2}+ 2 A_{3}+ A_{4}) ( 1 - \frac{1}{N^{2}}) \sum_{\tilde{t}}\frac{( - \lambda ( A_{2}+ A_{3}) )^{\tilde{t}}}{\tilde{t}!}N^{- 2 \tilde{t}}.
\psi^C_{i,i}= \partial_1+\cdots+\partial_i \ \ \ \ \ \ \ \ \ \ \ \ \ \ \{\rm for}\i=1,\ldots, n, &\psi^C_{i,j}= \psi^C_{i-1,j-1}+\alpha_{i,j}\psi^C_{i,j-1}\ \ \ \ \ \{\rm for}\ (i,j)\in \triangledown_n,
k'' \otimes_{k}\overline{\Lambda}= k'' \otimes_{k}(k \otimes_{S}\Lambda) \simeq k'' \otimes_{S}\Lambda\simeq k'' \otimes_{S''}(S'' \otimes_{S}\Lambda) \simeq k'' \otimes_{S''}\Lambda'' \simeq M_{\chi(1) \times \chi(1)}(k'').
\lim_{t \rightarrow -\infty}\| u(t) - S(t)(u_{0}^{-}, u_{1}^{-}) \|_{\dot{H}^{1/2}(\mathbf{R}^{3})}+ \| u_{t}(t) - \partial_{t}S(t)(u_{0}^{-}, u_{1}^{-}) \|_{\dot{H}^{-1/2}(\mathbf{R}^{3})}= 0.
E(x) = 1 + xE(x) + xS_p(x,0) S_p(x,u) = x(S_n(x,u) + S_p(x,u)) + \frac{x}{u}\left(S_p(x,u) - S_p(x,0)\right) + x\left(D_p(x,0)\right) S_n(x,u) = E(x) + u(S_n(x,u) + S_p(x,u)) + u^2(D_n(x,u) + D_p(x,u)) D_p(x,u) = x(D_n(x,u) + D_p(x,u)) + \frac{x}{u}\left(D_p(x,u) - D_p(x,0)\right) D_n(x,u) = S_n(x,u) + S_p(x,u)
G_{\mu\nu}^{(p)}= \eta_{\mu\nu}+ \partial_\mu \phi^i\partial_\nu \phi^i - \bar\lambda (\Gamma_\mu + \Gamma_i \partial_\mu\phi^i)\partial_\nu \lambda- \bar\lambda (\Gamma_\nu + \Gamma_i \partial_\nu\phi^i)\partial_\mu \lambda+ \bar\lambda \Gamma^m \partial_\mu \lambda \bar\lambda \Gamma_m \partial_\nu \lambda,
\begin{aligned}\frac{\mathrm{d}}{\mathrm{d}t}\hat{q}(0,t) =-\lambda \hat{q}(0,t), \frac{\mathrm{d}}{\mathrm{d}t}\hat{q}(n,t) =-\lambda \hat{q}(n,t)-\frac{\lambda}{\ln p}\sum_{j=1}^{n}\frac{(1-p)^{j}}{j}\hat{q}(n-j,t),\n\ge 1,\end{aligned}
^{\star}({\hat{F}}+ \Phi )^{i j}\equiv \frac{1}{2}\epsilon^{i j k l}({\hat{F}}+ \Phi )_{k l}^{\prime}= ( \det A ) ( ( A^{- 1})^{t}^{\star}( \hat{F}+ \Phi ) A^{- 1})^{i j}.
\mathcal{U}[N_1,N_2]:=\Big\{q\in C^2(\overline{\Omega})\,\Big|\, \|q-\theta_L\|_1\leq N_1\delta,\quad\|q_{xx}\|\leq N_2\delta,q(0)=\theta_l,\q(1)=\theta_r \Big\}.
\lambda = \frac{\mathrm{d}x}{y}={R}\, \frac{\mathrm{d}a}{\mathrm{d}u}\,{\cal W}={R}\left( \frac{\mathrm{d}a}{\mathrm{d}u}\mathrm{d}B - \frac{\mathrm{d}a_{D}}{\mathrm{d}u}\mathrm{d}A \right) \ ,
\left( 1-\frac{x_{j}}{x_{i}}\right) \sum_{\gamma\in\boldsymbol{Z}_{N}}B_{\gamma}^{(i,j)}x^{\gamma}=\sum_{\gamma\in\boldsymbol{Z}_{N}}\big( B_{\gamma}^{(i,j)}-B_{\gamma+\varepsilon_{i}-\varepsilon_{j}}^{(i,j)}\big) x^{\gamma}=\sum_{\gamma\in\boldsymbol{Z}_{N}}\widehat{\mu}_{\gamma}x^{\gamma}
P^{0,1}{\rm ad}\nu_i\Delta_0^{-1}\bar\partial^*\mu_1P^{1,0}{\rm ad}\bar\nu_j^T\Delta_0^{-1}\partial^*\bar\mu_2P^{0,1}\nu_i =i P^{0,1}{\rm ad}\nu_i\Delta_0^{-1}\bar\partial^*\mu_1P^{1,0}{\rm ad}(\Delta_0^{-1}\partial^*\bar\mu_2 \nu_i)\star\nu_j
\sum_{i\leq k_n}\frac{\kappa_i^2f_{0,i}^2}{(1+n\lambda_i\kappa_i^2)^2}\leq n^{-2}\sum_{i\leq k_n}\lambda_i^{-2}\kappa_i^{-2}i^{-2\beta}i^{2\beta}f_{0,i}^2 &= n^{-2}\sum_{i\leq k_n}i^{2(\alpha-\beta)}\exp(2(\xi+\gamma)i^p)i^{2\beta}f_{0,i}^2,
H ( x_{1}, x_{2}) = \frac{1}{x_{1}^{2}- x_{2}^{2}}\log \left( \frac{\mu^{2}- x_{1}^{2}}{\mu^{2}- x_{2}^{2}}\right) + \frac{\mu^{2}- x_{1}x_{2}}{( \mu^{2}- x_{1}^{2}) ( \mu^{2}- x_{2}^{2})}\, .
f(t) =\left|(\Phi_t(x)-x)-(\Phi_t(y)-y)\right| &\leq \int_0^t\left|X(\Phi_s(x))-X(\Phi_s(y))\right|\,ds &\leq \int_0^t M\left|\Phi_s(x)-\Phi_s(y)\right|\,ds &\leq M\int_0^t |x-y|+\left|(\Phi_s(x)-x)-(\Phi_s(y)-y)\right|\,ds &=Mt|x-y|+\int_0^t Mf(s)\,ds\ .
\mathcal K =-\frac{4^{d+2}}{\Gamma(2d+4)}\cdot\int_0^\infty\sum_{l=0}^\infty\frac{(\rho l)^{2d+3}}{n+l-1}\cdot e^{-2\rho l}\cdot\frac{\lambda+l}{\lambda}\,\mathcal C_l^\lambda\,d\rho &=\sum_{l=1}^\infty\frac{-1}{l(n+l-1)}\cdot\frac{\lambda+l}{\lambda}\,\mathcal C_l^\lambda.
\partial_\mu \left( \sqrt{-k}(k^{-1})^{[ \mu \nu ]}e^{-\phi}-\frac{2 T_p}{T'_p}\frac{1}{(n-2)!}\epsilon^{\mu \nu \gamma_1 \ldots \gamma_{n-2}}(C^{(n-2)})_{\gamma_1 \ldots \gamma_{n-2}}\right) = 0 .
\o_{M,K}(u,u') \ge K | \pi^* \o_B (w,w')| - || \mu || \cdot |w| \, |w'| \ge K c_0 |\pi_* w| |\pi_* w'| - || \mu || \cdot |w|\, |w'| \ge K c_0 c^2_1 |w| |w'| - || \mu || \cdot |w| \, |w'| = |w|( K c^2_1 c_0 - || \mu || ) >0 \, ,