harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
v_3\big|_{\partial_-D_1^{(3)}}=\frac{\partial v_3}{\partial x_0}\Big|_{\partial_-D_1^{(3)}}=0, \ \ \v_3\big|_{(\partial'D_1^{(3)}\setminus(\Gamma_0\times(t_1,t_2))}=0, v_3\big|_{\Gamma_0'\times(t_1,t_2)}=f.
W_B \Sigma W_B^* =\frac{1}{4}\begin{bmatrix}I+L I-L \end{bmatrix}\Sigma \begin{bmatrix}I+L I-L \end{bmatrix}^*=\frac{1}{4}\begin{bmatrix}I-L I+L \end{bmatrix}\begin{bmatrix}I+L^* I-L^* \end{bmatrix}&=\frac{1}{4}(( I-L)(I+L^*)+(I+L)( I-L^*))=\frac{1}{4}(2 I-2LL^*)=0.
\det{( e_{z}, e_{\bar{z}}, w )}\! = \! \det \left| \left( \begin{array}{c c c}{1}&{0}&{p}\\{0}&{1}&{\bar{p}}\\{v}&{\bar{v}}&{{\cal H}}\\ \end{array}\right) \right| = \left({\cal H}- p v - \bar{p}\bar{v}\right) = -{\cal L}\neq 0 .
\leq \;\left\langle \prod_{b=1}^N2 \cosh\left( \; 2 \beta^{(b)}\int_{\Lambda}d^2x \;t(x)\; : \cos \left( 2 \sqrt{\pi}\varphi^{(b)}(x)\right) :_{Q^W}\; \right) \;\right\rangle_{Q^\mu}
\psi''(x)= \textstyle \frac{n-1}{2n}\cdot\frac{R^{n-1}}{(R\vee|x|)^n}= \frac{n-1}{2n}\cdot \begin{cases}\frac{1}{R}|x|\leq R \frac{R^{n-1}}{|x|^n}|x|>R, \end{cases}
m_{d+1,1^k}(x_1,x_2,\ldots,x_i) = \begin{cases}m_{d,1^k}(x_1,x_2,\ldots,x_i) x_{i+1}(d\ge2), (k+1) m_{1^{k+1}}(x_1,x_2,\ldots,x_i) x_{i+1}(d=1), \end{cases}
\frac{\partial h_{1}}{\partial p_{1}}=\frac{\partial x_{1}}{\partial p_{1}}+\frac{\partial x_{1}}{\partial w_{1}}x_{1}=-\frac{.7w}{p_{1}^{2}}+\frac{.7}{p_{1}}\frac{.7w}{p_{1}}=-\frac{.21w}{p_{1}^{2}}
\left( \begin{matrix}{0}&{1}\\{1}&{0}\\ \end{matrix}\right) \partial_{0}\left( \begin{matrix}{\psi}\\{\chi}\\ \end{matrix}\right) + \left( \begin{matrix}{- 1}&{0}\\{0}&{- 1}\\ \end{matrix}\right) \partial_{x}\left( \begin{matrix}{\psi}\\{\chi}\\ \end{matrix}\right) = \left( \begin{matrix}{0}\\{0}\\ \end{matrix}\right) .
G_N(1,\xi_n)= e^{i\xi_n}\begin{pmatrix}\beta_N(0,\xi_n) 1\end{pmatrix}\exp\Big(-i \int_0^1\varphi_2(t)\beta_N(t,\xi_n)\,dt\Big)+\frac{S_N(1,\xi_n)}{(2i\xi_n)^N}
\boldsymbol{u}_{\varepsilon ,\lambda}(0,x)=u^{0}(x),\f_{\varepsilon,\lambda}(0,x,v):=f_{\lambda}^{0}(x,v)=\gamma_{\lambda}(v)(f^{0}\ast\Theta_{\lambda})(x,v),\x\in \Omega ,v\in \mathbb{R}^{N},
\mathrm{P}_{t,i}= \mathrm{P}\left( \alpha_1=1, z_t<\max\left\{\frac{\epsilon_1}{\rho\xi_1},\cdots, \frac{\epsilon_i}{\rho\xi_{i}}\right\}\right) + \mathrm{P}\left( \alpha_1<1, z_t<\max\left\{\frac{\epsilon_1}{\rho\xi_1},\cdots, \frac{\epsilon_i}{\rho\xi_{i}}\right\}\right) .
G_Y V(x) = - 2 \mu (1 - \mu/2) x^2 + \delta(1 - \mu ) \mu x + \mu(c_A - \mu) &\leq - 2 \mu (1 - \mu/2) x^2+ \mu \frac{1-\mu}{2}x^2 + \mu\frac{1-\mu}{2}\delta^2 +\mu(c_A - \mu) &= -\frac{1}{2}(3-\mu) \mu x^2 + \mu\frac{1-\mu}{2}\delta^2 +\mu(c_A - \mu) &\leq -\mu x^2 + \mu\frac{1-\mu}{2}\delta^2 +\mu(c_A - \mu),
[y_1, y_1]=y_4+\theta_1y_5, [y_1, y_2]=\frac{\alpha_3}{\gamma_3}y_4+\theta_2y_5, [y_2, y_1]=\frac{\alpha_1\alpha_6}{\gamma_3\gamma_6}y_5, [y_2, y_2]=\frac{\alpha^2_1\beta_2}{\gamma^2_3\gamma_6}y_5, [y_1, y_3]=\frac{\beta_4}{\gamma_6}y_5, [y_2, y_3]=\frac{\gamma_1}{\gamma_3}y_4+\theta_3y_5, [y_3, y_2]=y_4, [y_3, y_3]=y_5.
S={1 \over 4G}A_\S -\int_{\Sigma}\left( 8\pi a_1 R+ 4\pi a_2R_{\mu\nu}n^{\mu}_in^{\nu}_i+8\pi a_3 R_{\mu\nu\lambda\rho}n^{\mu}_in^{\lambda}_in^{\nu}_jn^{\rho}_j\right)~~~.
\mathbf P_{so}^{\,i}=1-\sum_{\mathbf D_u\in 2^{\mathcal D}}\varepsilon^{|\mathbf D_u|}\,\left[\vartheta^{m-|\mathbf D_u|}\Theta_{i}(\mathbf D_u,1)+\left(\left(1\!-\!\varepsilon\right)^{m-|\mathbf D_u|}\!-\!\vartheta^{m-|\mathbf D_u|}\right)\Theta_{i}(\mathbf D_u,\beta)\right],
lim_{x\rightarrow+\infty}f_{\kappa}(x)\sim(2\kappa\beta)^{-1/\kappa}(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)}x^{\alpha\nu-1-\alpha/\kappa}
\Omega_L = \frac{[\theta+(L+1)\gamma]}{[\theta+ L \gamma]}\prod_{i=1}^L \frac{[\zeta-\mu_i \,, \theta+\zeta+\mu_i]}{[\zeta+\mu_i \,, \theta+\zeta-\mu_i]}\prod_{K = 0}^{\lfloor L/2 \rfloor}\frac{[\theta-(L-2 K)\gamma]}{[\theta+(L-2 K +1)\gamma]}\ ,
\begin{array}[c]{l}\left\langle \nabla_{b}\varphi,\nabla_{b}\Delta_{b}\varphi\right\rangle=-\left\langle \nabla_{b}\varphi,\nabla_{b}\left\vert \nabla_{b}\varphi\right\vert^{2}\right\rangle -\frac{a\lambda_{1}}{f+a}\left\vert\nabla_{b}\varphi\right\vert^{2}.\end{array}
F(t)=(-1)^{\frac{k}{2}}(it)^{\frac{2-k}{2}+m}(k-1)\sum_{0\leq j\equiv \frac{k-2}{2}+m\;({\rm mod}\;2)}\binom{k}{j}\sum_{s\geq 0}C(j+2s)(it)^{j+2s}
\Phi^{(1)}(N,H,u) = \prod_{p=1}^{m}\{H-[m(N+u)-p][m(N+u)-p+1]\}\times \prod_{r=1}^{n}\left\{[n(N+u)-r][n(N+u)-r+1]-\alpha^{2}+\frac{1}{4}\right\},
\Phi(x,E,u) = (-1)^m m^{2m}n^{2n}\prod_{r=1}^n \left[\left(x+u-\frac{2r-1-2\alpha}{2n}\right) \left(x+u-\frac{2r-1+2\alpha}{2n}\right)\right] \times \prod_{p=1}^m \left[\left(x+u-\frac{2p-1+\sqrt{1+4E}}{2m}\right) \left(x+u-\frac{2p-1-\sqrt{1+4E}}{2m}\right)\right].
\frac{1}{1-itM}=\sum_{n=0}^{N-1}A_{n}\left( t\right) \left( iM\right)^{n}\ ,\ \ \ \exp\left( i\phi M\right) =\sum_{n=0}^{N-1}C_{n}\left(\phi\right) \left( iM\right)^{n}\ ,\ \ \A_{n}\left( t\right) =\int_{0}^{\infty}e^{-s}C_{n}\left( st\right) ds\ .
P^{(BC_n,C_n)}_\lambda(x)=m_\lambda+\mbox{lower terms}, &D^{(BC_n,C_n)}_x P^{(BC_n,C_n)}_\lambda(x)=(ab)^{n/2}t^{n(n-1)/4}\sum_{\sigma_1,\cdots,\sigma_n=\pm 1}s_1^{\sigma_1/2}\cdots s_n^{\sigma_n/2}\cdot P^{(BC_n,C_n)}_\lambda(x),
( 2 \pi )^{3}\, \lambda_{\bf g}^{3 / 2}\, k^{- 3 / 2}\, \tau ( S^{3}, A_{f}\! = \! 0 , \langle \cdot \, , \cdot \rangle_{H^{0}( A_{f}= 0 )})^{1 / 2}\, V ( S U ( 2 ) )^{- 1}
\begin{aligned}M_\beta :=\{Z_{{t}}({t}+2{t}^{2/3}r)>\beta{t}^{2/3}\}E_\beta :=\{L_{(0,0)\to(\beta{t}^{2/3},0)}+L_{(\beta{t}^{2/3},0)\to({t}+2{t}^{2/3}r,{t})}\leq2{t}+2{t}^{2/3}r+s{t}^{1/3}\}N_\beta :=\{L_{(0,0)\to({t}+2{t}^{2/3}r,{t})}\leq2{t}+2{t}^{2/3}r+{t}^{1/3}s\}. \end{aligned}
\begin{cases}\partial_\rho = \frac12(\partial_r + \partial_t), \ \partial_{\sigma}= \frac12(\partial_r - \partial_t), \partial_{\bar{\rho}}= \frac12(\partial_{\bar{r}}+ \partial_s), \ \partial_{\sigma}= \frac12(\partial_{\bar{r}}- \partial_s).\end{cases}
a_{1}a_{2}a_{3}...a_{n}=\sum_{i=0}^{[\frac{n}{2}]}\sum_{\mu\in C}(-1)^{k}Pf(a_{\mu_{1}}\cdot a_{\mu_{2}},...,a_{\mu_{2i-1}}\cdot a_{\mu_{2i}})a_{\mu_{2i+1}}\wedge...\wedge a_{\mu_{n}}
[F_{i}, E_{\xi_{i}}]= \left(\frac{K_{i}- K_{i}^{-1}}{q - q^{-1}}- q^{-1}\frac{q K_{i}- q^{-1}K_{i}^{-1}}{q - q^{-1}}\right) E_{\xi_{i + 1}}= -q^{-1}K_{i}^{-1}E_{\xi_{i + 1}}.
\begin{cases}\partial_tv + v\cdot\nabla v + \nabla p = \mu\Delta v + \nabla\cdot( \frac{\partial W(F)}{\partial F}F^\top), begin{align*}-4mm] \partial_t F+v\cdot\nabla F=\nabla v F, begin{align*}-4mm] \nabla\cdot v = 0,\nabla\cdot F^\top = 0.\end{cases}
\mathcal{A}_t f(x) = \sigma_{t+1}^2[h]\cdot f'(x) -\{x - \langle \zeta_t, D_t h \rangle \}f(x), \tilde{\mathcal{A}}_t f(x) = \sigma_{t+1}^2[h] \cdot f'(x) -\{x - \mathbb{E}_t\langle \zeta_{t+1}, h \rangle \}f(x).
a_{\beta \gamma}^{\alpha}= -g\big{(}X_{\alpha}, X_{c}\otimes \omega_{\gamma}^{c}(X_{\beta}) \big{)}+g\big{(}X_{\alpha}, X_{c}\otimes \omega_{\beta}^{c}(X_{\gamma}) \big{)}=- \omega_{\gamma}^{\alpha}(X_{\beta})+ \omega_{\beta}^{\alpha}(X_{\gamma}).
Q\bar{Q}Q= \left( \begin{array}{cc}-u_1d_1d_2+u_2 d_1^2 -u_1u_2d_1 + u_1^2 d_2 u_2d_1d_2 -u_1d_2^2 u_1u_2d_2 - u_2^2 d_1 \end{array}\right)
0 = \nabla_{( \alpha}\overline{{{\xi}}}_{\overline{{{\beta}}}) a}= \partial_{( \alpha}\overline{{{\xi}}}_{\overline{{{\beta}}}) a}- H_{\alpha \overline{{{\beta}}}}^{\, \, \, \overline{{{\gamma}}}}\overline{{{\xi}}}_{\overline{{{\gamma}}}a}+ H_{\alpha \overline{{{\beta}}}}^{\, \, \, \gamma}\xi_{\gamma}.
\limsup_\delta y_\lambda\mathbf{d}x_\delta \leq\liminf_n\limsup_\delta(y_\lambda\overline{\mathbf{d}}x_\lambda^n+x_\lambda^n\overline{\mathbf{d}}x_{\gamma_\lambda^n}+x_{\gamma_\lambda^n}\mathbf{d}x_\delta) &\leq\liminf_n(y_\lambda\overline{\mathbf{d}}x_\lambda^n+s_{\gamma_\lambda^n}) &\leq\liminf_n(y_\lambda\overline{\mathbf{d}}x_\lambda^n+2^{1-n}t_\lambda) &=0.
A = \frac{V_{0}^{2}+ g^{2}X_{0}^{2}Y_{0}^{2}}{Y_{0}}, \qquad \varphi_{0}= \arccos \sqrt{\frac{X_{0}^{2}Y_{0}^{2}}{V_{0}^{2}+ g^{2}X_{0}^{2}Y_{0}^{2}}}
c+o(1)= \Phi(u_n)=\big(\frac{1}{2}-\frac{1}{2^*(s_1)}\big)a(u_n)+\big(\frac{1}{2^*(s_1)}-\frac{1}{2^*(s_2)}\big)\lambda_2c(u_n) &+\big(\frac{1}{2^*(s_1)}-\frac{1}{p+1}\big)\lambda_3d(u_n)+o(1)\|u_n\|.
(1-z)^aP^{\lambda, -2\lambda-2a+1}_a\bigg(\frac{3+z}{1-z}\bigg) &=(-1)^{a-1}U_a(\lambda) \bigg((1-A_a(\lambda)z)C^{\lambda+\frac{1}{2}}_{a-1}(z)+B_a(\lambda)(1-z^2)C^{\lambda+\frac{3}{2}}_{a-2}(z) \bigg).
{\rm E}\left({\bf A},{\bf V}\right) =\left\{{\bf A}_{\theta}:\theta \left( x,y\right) = \sum_{i=1}^{s}\theta_{i}\left( x,y\right) e_{i}\ \ \ \ \left\langle \left[ \theta_{1}\right],\dots,\left[ \theta_{s}\right] \right\rangle \in T_{s}({\bf A}) \right\}.
\&\frac{\psi_{x,y-1,z}(a,b)}{\psi_{x,y-1,z-1}(a,b)}\frac{\psi_{x,y,z-1}(a,b)}{\psi_{x,y,z}(a,b)}+\frac{\psi_{x+1,y-1,z-1}(a,b)}{\psi_{x,y-1,z-1}(a,b)}\frac{\psi_{x-1,y,z}(a,b)}{\psi_{x,y,z}(a,b)}= &\frac{(y+z+a+b)(2x+y+z+2a+2b)}{(x+y+z+a+b)(x+y+z+2a+2b)}+\frac{x(x+a+b)}{(x+y+z+a+b)(x+y+z+2a+2b)}=1
G^{JJ'}(x)= \frac{\delta^{JJ'}}{|x|^{2J}}\left(\frac{g_{{\rm YM}}^2N}{8\pi^2}\right)^J \cdot J\cdot \frac{\sinh\left( \frac{1}{2}\frac{J^2}{N}\right)}{\frac{1}{2}\frac{J^2}{N}}
G_L(\cos\vartheta) =-\lim_{\epsilon\to0}\int_0^{1-\epsilon}R^{n+2k}\int_0^Rr^{n+k-2}\cdot\left(\Sigma_n\cdot p_r(\cos\vartheta)-\sum_{l=0}^Lr^l\cdot\frac{\lambda+l}{\lambda}\,\mathcal C_l^\lambda(\cos\vartheta)\right)\,dr\,dR &-\sum_{l=0}^{L-1}\frac{1}{(l-L)(l+n+L-1)}\cdot\frac{\lambda+l}{\lambda}\,\mathcal C_l^\lambda(\cos\vartheta).
T^{k}g_{C}=(-1)^{k}\begin{bmatrix}z^{k}0 z^{-k}C(z) z^{-k}\end{bmatrix}=\begin{bmatrix}z^{k}0 \big(z^{-k}C(z)\big)_{-}z^{-k}\end{bmatrix}\left((-1)^{k}\begin{bmatrix}1 0 \big(z^{-k}C(z)\big)_{0+}z^{k}1 \end{bmatrix}\right).
\partial_{t}\left([z_{t},H]\frac{f_{\alpha}}{z_{\alpha}}\right)=[z_{tt},H]\left(\frac{f_{\alpha}}{z_{\alpha}}\right)+[z_{t},H]\frac{f_{t\alpha}}{z_{\alpha}}+\frac{1}{\pi i}\int_{0}^{2\pi}\left(\frac{z_{t}(\beta)-z_{t}(\alpha)}{z(\beta)-z(\alpha)}\right)^{2}f_{\beta}(\beta)d\beta.
\mathcal{S}[F_{S_{n: n_{1},...,n_{k-1}}}; n](z)=\prod_{i=1}^{k}\mathcal{S}[F_{i}; r_i](z),z\in \mathbb{C}{\}{\bigcap_{i=1}^{k}}(\mbox{supp}\mathit{F_{i}})^{c}.
\rho_f(x) = \int_{-\infty}^\infty f(x,v) \mathrm{d}v, \rho_f(x)u_f(x) = \int_{-\infty}^\infty v f(x,v)\mathrm{d}v ,\quad P_f(x) = \int_{-\infty}^\infty v^2 f(x,v) \mathrm{d}v = \rho_f(x) [T_f(x)+ u_f(x)^2 ]
b_{sj}= \frac{1}{\omega}\int_0^\omega q(t)e_s(t)\overline{e_j(t)}\,dt=\frac{1}{\omega}\sum_{l\in\mathbb{Z}}q_l\int_0^\omega e^{i2\pi lt/\omega}e^{-i\pi(2s+\theta)t/\omega}e^{i\pi(2j+\theta)t/\omega}\,dt=q_{s-j}.
0 \mkern5mu=\mkern5mu y_-\mkern-3mu- y_+\mkern1mu=\mkern5mu \max_{k_1>0}y_-(k_1) - \min_{k_2>0}y_+(k_2)\mkern5mu=\mkern1mu \max_{k_1, k_2 >0}\! \left(y_-(k_1)\mkern-.5mu - y_+(k_2)\right),
V = -{{N_f}\over 2}{{F_\Phi^{\dag}F_\Phi}\over \sqrt{A_\Phi^{\dag}A_\Phi}}+ 2 N_f m^2 \sqrt{A_\Phi^{\dag}A_\Phi}+ \left\{N_f{{F_\Phi}\over{A_\Phi}}\left({{\Lambda^{3N_c-N_f}}\over{A_\Phi^{N_f}}}\right)^{1 \over{N_c-N_f}}+{\rm h.c.}\right\}.
\phi_X(3\pi)=\left(1-p+pe^{i\frac{3\pi}{2}}\right) \left(1-p+pe^{i\frac{3\pi}{4}}\right) \left(1-p+pe^{i\frac{3\pi}{8}}\right) \left(1-p+pe^{i\frac{3\pi}{16}}\right) \phi_X(\frac{3\pi}{16})
\xi_1=\xi_0+\frac{2\pi}{L}k, \xi_2=\xi_1+\frac{2\pi}{L}l, \xi_3=\xi_2+\frac{2\pi}{L}m, \xi_4=\xi_3+\frac{2\pi}{L}n \xi_5=\xi_4+\frac{2\pi}{L}s,
L_{e f f}^{( 1 )}= 2 \pi \kappa \frac{\sigma_{1}\sigma_{2}}{q_{1}q_{2}}\sum_{p_{1}= 1}^{n_{1}}\sum_{p_{2}= 1}^{n_{2}}\frac{d}{d t}A r g ( \vec{R}_{p_{1}}^{( 1 )}- \vec{R}_{p_{2}}^{( 2 )}) \; \; .
\frac{1}{c_2\,|x-y|^N}\left(|x-y|^2\,\wedge\,\delta(x)\delta(y)\right)^s \leq\G_\Omega(x,y)\ \leq &\leq\ \frac{c_2}{|x-y|^N}\left(|x-y|^2\,\wedge\,\delta(x)\delta(y)\right)^sx,y\in\Omega.
\textstyle \sup_{R>0}\int_{\Omega}I_{v}\ge (1 -\gamma_{1}-c(n)\gamma_{2}-c(n)\|C_{a}\|_{L^{\infty}}) \|v\|_{\dot X}^{2}+ (\lambda-\Gamma_{3}-c(n)\Gamma_{4}) \|v\|_{\dot Y}^{2}-(\gamma_{2}+\delta)\|\nabla^{b}v\|_{\dot Y}^{2}
C(x_1x_4+x_4x_1) = Ax_0^2, C(x_2x_3+x_3x_2) = Bx_0^2, C(x_2x_0+x_0x_2) = Ax_1^2, C(x_3x_4+x_4x_3) = Bx_1^2, C(x_3x_1+x_1x_3) = Ax_2^2, C(x_4x_0+x_0x_4) = Bx_2^2, C(x_4x_2+x_2x_4) = Ax_3^2, C(x_0x_1+x_1x_0) = Bx_3^2, C(x_0x_3+x_3x_0) = Ax_4^2, C(x_1x_2+x_2x_1) = Bx_4^2.
\left .\frac{d}{d t}(X,\delta_X Df \wedge \delta_X Dg) \right |_{t=0}= (\dot X, \delta_X Df \wedge \delta_X Dg) + (X, \dot{\delta_X}Df \wedge \delta_X Dg) +( X, \delta_X \dot{Df}\wedge \delta_X Dg)+( X, \delta_X{Df}\wedge \dot{\delta_X}Dg)+( X, \delta_X{Df}\wedge \delta_X \dot{Dg}).
{\cal{G}}_{k_{\perp}}^{\pm ( \eta )}( t , t^{\prime}) = -{\frac{\Gamma ( - \nu )}{2 \sqrt{\pi}}}e^{-{\frac{i \pi}{4}}}{\frac{1}{S^{\prime}}}D_{\nu}\left({\frac{1 + i}{\sqrt{\hbar}}}S ( t ) \right) D_{\nu}\left( -{\frac{1 + i}{\sqrt{\hbar}}}S ( t^{\prime}) \right)
SP_{u}\:=\: S \begin{pmatrix}\vec{u}\vec{u}^{\,*}\overline{u}_{n}\vec{u}u_{n}\vec{u}^{\,*}|u_{n}|^{2}\end{pmatrix}\: = \: \begin{pmatrix}\hat{S}_{0}(\vec{u}\vec{u}^{*}) \overline{u}_{n}S_{1}\vec{u}u_{n}( S_{1}\vec{u})^{*}|u_{n}|^{2}\end{pmatrix}.
\tau_{k}^{i,N}:= \inf \biggl\{t > \tau_{k-1}^{i,N}: X_{t-}^{i,N}+ \frac{\alpha}{N}\sum_{j =1}^N \bigl(M^{j,N}_{t}- M^{j,N}_{t-}\bigr) \geq 1 \biggr\},k\geq1,
SG_{true}=\frac{\rho_{sample}}{\rho_{H_{2}O}}=\frac{\frac{m_{sample}}{V}}{\frac{m_{H_{2}O}}{V}}=\frac{m_{sample}}{m_{H_{2}O}}\frac{g}{g}=\frac{W_{V,sample}}{W_{V,H_{2}O}},
\begin{array}{rl}\displaystyle \int\frac{d^4k_1}{(2\pi)^4}\int\frac{d^4k_2}{(2\pi)^4}D(k_1)D(k_1+k_2)\overline{D(k_2+p)}K_1 \approx \displaystyle 2^{-1}(4\pi)^{-4}p^2\ln(p^2\tilde{p}^2).\end{array}
p_{M}\to \overline{{{a}}}_{M}^{N}\circ p_{N}= \overline{{{a}}}_{M}^{N}p_{N}+ \frac{1}{2}[ p_{N}, \overline{{{a}}}_{M}^{N}] = \overline{{{a}}}_{M}^{N}p_{N}- \frac{i \hbar}{2}( \partial_{N}\overline{{{a}}}_{M}^{N}+ b_{M}^{P}\partial_{P}\overline{{{a}}}_{M}^{N}) \, ,
\mathbf{Q}^k = \frac{q^n}{3}\left[ \begin{array}{ccc}2^k + 2(-1)^k 2^n -(-1)^n 2^k -(-1)^k 2^k -(-1)^n 2^k + 2(-1)^n 2^k -(-1)^k 2^k -(-1)^n 2^k -(-1)^n 2^k + 2(-1)^k\end{array}\right].
\begin{aligned}\Re\Bigg(\sum\limits_{\substack{\rho |\Im(\rho)|\le N}}\left(1-\left(\frac{\rho}{\rho-\tau}\right)^n \right)\Bigg)<\sum\limits_{\substack{\rho |\Im(\rho)|\le N}}1-\frac{1}{20}R^n\end{aligned}
{}~ [ \kern - 0 . 4 5 e m \llap ~ [ ~ J_{1}( x ) , \phi^{a b}( y ) ~ ] \kern - 0 . 4 5 e m \llap ~ ] ~ = - \delta ( x - y ) ~ M^{a b}= - ~ [ \kern - 0 . 4 5 e m \llap ~ [ ~ \phi ( x ) , J_{1}^{a b}( y ) ~ ] \kern - 0 . 4 5 e m \llap ~
W^h_{\ijk}= R^h_{\ijk}- \tfrac{1}{n-1}\left(\delta^h_{\k}R_{ij}- \delta^h_{\j}R_{ik}\right)+ \tfrac{1}{n+1}\left(\delta^h_{\i}R_{{[jk]}}- \tfrac{1}{n-1}\left( \delta^h_{\k}R_{{[ji]}}- \delta^h_{\j}R_{{[ki]}}\right) \right).
\chi_{\{\xi^{1}_{1}\neq\xi^{1}_{2}\}}=\sum_{Q'\in\mathbb{Q}'}\phi_{Q'}(\xi^{1}_{1},\xi^{1}_{2})= \{\sum_{Q'\in\mathbb{Q}'_{\mathbb{I}}}+\sum_{Q'\in\mathbb{Q}'_{\mathbb{II}}}\}\phi_{Q'}(\xi^{1}_{1},\xi^{1}_{2}) =\chi_{\{\xi^{1}_{1}<\xi^{1}_{2}\}}+\chi_{\{\xi^{1}_{1}>\xi^{1}_{2}\}},
\langle\mathbf{1}|\tilde{a}(\mathrm{m})\varphi\rangle=-\frac{\langle\mathbf{1}|\tilde{a}(\mathrm{m})^{-1}\psi\rangle}{\langle\mathbf{1}|\tilde{a}(\mathrm{m})^{-1}\mathbf{1}\rangle},
\lim_{t \to 0^+}\frac{F(t)}{t}= \lim_{t \to 0^+}\frac{t}{F^{-1}(t)}= \lim_{t \to 0^+}\frac{t\, \psi\left( \left(\frac1t \right)^{\frac1n}\right )^{n-1}}{\left( \frac1t\right)^{1- \frac1p}}= \lim_{t \to 0^+}\phi(1)^{n-1}t^{1-\frac1p + 1 - \frac{n-1}{n}}=0
\omega_{-}= \left( \begin{array}{c c}{\Sigma_{-}^{( 1 )}}&{0}\\{I_{n}}&{\Sigma_{-}^{( 2 )}}\\ \end{array}\right) , \qquad \omega_{+}= \left( \begin{array}{c c}{0}&{\Gamma^{( 1 ) - 1}\Gamma^{( 2 )}}\\{0}&{0}\\ \end{array}\right) ,
\sum_{i+j=n}\mu_i(x_1,x_2, \mu_j(x_3,x_4, x_5)) = \sum_{i+j=n}\mu_i(\mu_j(x_1,x_2, x_3),x_4, x_5)+\mu_i(x_3,\mu_j(x_1, x_2, x_4), x_5)+\mu_i(x_3,x_4, \mu_j(x_1,x_2, x_5)),
H^{2}= \frac{\rho}{3}\left( 1 + \frac{\rho}{2 \lambda}\right) + \frac{1}{6}\left( \sqrt{\frac{6}{\lambda}}\Lambda_{( 5 )}+ \frac{1}{2}\lambda \right) + \frac{\sigma_{0}^{2}}{6}a^{\- 6}- k a^{- 2}+ \frac{2{\cal U}_{0}}{\lambda}a^{- 4}.
M_{pl}^2 ={{M_X^3}\over k}e^{-2 k y_1-2 c}\left({1\over 2}e^{-2k (y_2-2y_1)}-e^{-2k(y_2-y_1)}+1-{1\over 2}e^{-2k (\pi \rho - y_2)}\right)~.~\,
\frac{dN_i}{dv}=v\frac{d\hat{B}_i}{dv}-vA\hat{B}_i+\left\{\frac{2}{v^3}+\frac{2A}{v^2}-\frac{1}{v}\frac{dA}{dv}+\frac{A}{v^2}\right\}\int_0^vv'^2\hat{B}_i(v')e^{-\int_{v'}^vA(v'')dv''}dv'.
S_x(u-P_{\tilde{y}}(\cdot-\tilde{y}))(z) = (u-P_{\tilde{y}}(\cdot-\tilde{y}))(\Psi_x^{-1}(z)) &= (u-P_{\tilde{y}}(\cdot-\tilde{y}))(\Psi_y^{-1}\circ \Psi_y\circ \Psi_x^{-1}(z)) &= S_y(u-P_{\tilde{y}}(\cdot-\tilde{y})) (\Psi_y\circ \Psi_x^{-1}(z)).
\frac{\cal L}{N}= \frac{1}{2 N}(\partial_\mu \phi_i)^2 -\frac{1}{2}\mu^2 \rho -\frac{\lambda_0}{4}\rho^2-\frac{\eta}{6}\rho^3 + \frac{\chi}{2}(\rho - \frac{\vec \phi \cdot \vec \phi}{N}).
\begin{array}{ll}a^\dagger=\frac{x/m - i p_x}{\sqrt{2}}, a =\frac{x/m + i p_x}{\sqrt{2}}, begin{align*}0.24in]b^\dagger=\frac{y/n - i p_y}{\sqrt{2}}, b=\frac{y/n + i p_y}{\sqrt{2}}.\end{array}
\chi_{1 2}( p_{3 0}) = \beta_{1}\beta_{2}T_{1 2}^{0}( p_{3 0}) \int d p_{1 2 0}G_{0 1}G_{0 2}\beta_{1}\beta_{2}[ \chi_{2 3}( p_{1 0}) + \chi_{3 1}( p_{2 0}) ]
\theta(\varepsilon_N )^2 > \sup\left\{\frac{\|w\|^2}{\|(w,\zeta)\|^2}\ \Bigg|\w\in \left(B^{F}(T_{N,l})\right)^{l}, \zeta \in A^{(N-l)^{F}}_{\varepsilon_{N,l}}\right\}&=\frac{lT_{N,l}^2}{l T_{N,l}^2+ (1-\varepsilon_{N,l})^2((N-l)^F-1)}=\left(1+\frac{(1-\varepsilon_{N,l})^2((N-l)^F-1)}{lT_{N,l}^2}\right)^{-1}.
\overline w_{M_n,n}(x)= \dfrac{1}{n}\sum_{k=1}^n \widetilde u_T(x_k)+ \sum_{p=1}^{M_n}\Bigg[\dfrac{\pi}{n}\sum_{k=1}^n \widetilde u_T(x_k) \phi_p(x_k) \Bigg] \phi_p(x).
{{\mathcal{L}}}_m({{\mathfrak{A}}}(\boldsymbol{t}))=E\cdot{\rm{det}}\left(\varphi_{\zeta_j,s_j}(t^{u+\ell}(t-1)^{rn})\right)_{\substack{0\le \ell \le r-1 1\le j \le d, 1\le s_j \le r_j}}\enspace.
\sigma_{\mathbf{a},\mathbf{B}}^{2}\left( P\right) -\sigma_{\mathbf{a},\mathbf{G},\mathbf{B}}^{2}\left( P\right) =E_{P}\left[ \left\{\frac{1}{\pi_{\mathbf{a}}\left( \mathbf{B};P\right)}-1\right\}var_{P}\left[ \left. b_{\mathbf{a}}(\mathbf{G,B};P)\right\vert \mathbf{B}\right] \right] \geq 0.
\mbox{if}\ \alpha=(\alpha^1,\alpha^2,\alpha^3),\ \mbox{then}\ \check{\alpha}=\alpha^\vee= \left[\begin{array}{ccc}0 -\alpha^3 \alpha^2 \alpha^3 0 -\alpha^1 -\alpha^2 \alpha^1 0 \end{array}\right]\ .
{w}_{\beta}^{\alpha}= - 2 d \theta^{\alpha}D_{\beta}\Phi - 2 d \theta_{\beta}D^{\alpha}\Phi + \Pi^{\gamma \dot{\gamma}}w_{\gamma \dot{\gamma}\beta}^{\alpha}, \qquad w_{\dot{\beta}}^{\dot{\alpha}}= ( w_{\beta}^{\alpha})^{*}.
S_{l 2}\left( \theta + \theta_{h}+ 3 \theta_{H}\right) S_{l 2}\left( \theta - \theta_{h}- 3 \theta_{H}\right) \! \! = \! \! S_{l 1}\left( \theta \right) S_{l 1}\left( \theta + 2 \theta_{H}\right) S_{l 1}\left( \theta - 2 \theta_{H}\right) ,
I_{\varphi}^3(\phi(x)) =u_{\varphi}(x)I_{\varphi}(\phi(x)) =I_{\varphi}(u_{\varphi}(x)\phi(x)) =\iota^{\ast}\circ \varphi (\overline{u_{\varphi}(x)}c(\phi(x))) =\iota^{\ast}(\overline{u_{\varphi}(x)}\varphi \circ c(\phi(x))) =\overline{u_{\varphi}(\iota^{-1}(x))}I_{\varphi}(\phi(x)).
\begin{array}[c]{rl}m(s)= W_{x}(s,\bar{X}^{t,x;\bar{u}}(s))h(s), n(s)= \left( 1-W_{x}(s,\bar{X}^{t,x;\bar{u}}(s))\sigma_{z}\left( s\right)\right)^{-1}b_{z}(s)(W_{x}(s,\bar{X}^{t,x;\bar{u}}(s)))^{2}& +g_{z}(s)W_{x}(s,\bar{X}^{t,x;\bar{u}})+W_{xx}(s,\bar{X}^{t,x;\bar{u}})\sigma(s)h(s),\end{array}
\tilde S^i(t,q)-\frac{1}{2}\sum_\xi (\tilde\gamma^{-1}\sigma)^i_\xi\partial_j(\tilde\gamma^{-1}\sigma)^j_\xi = -(\tilde\gamma^{-1})^{ik}\partial_k\beta^{-1}-\partial_k(\beta^{-1}(\tilde\gamma^{-1})^{k\ell}H_{\ell j}(\tilde\gamma^{-1})^{ij})
P=P(x,l,m,n) &=\frac{\Gamma\left(\frac 12(x+l+m+n+1)\right)\Gamma\left(\frac 12(x+l-m-n+1)\right)\Gamma\left(\frac 12(x-l+m-n+1)\right)\Gamma\left(\frac 12(x-l-m+n+1)\right)}{\Gamma\left(\frac 12(x-l-m-n+1)\right)\Gamma\left(\frac 12(x-l+m+n+1)\right)\Gamma\left(\frac 12(x+l-m+n+1)\right)\Gamma\left(\frac 12(x+l+m-n+1)\right)}.
(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)}x^{\alpha\nu-1}exp_{\kappa}(-\beta x^{\alpha})
S \left( \begin{array}{c c}{{a}}&{{d}}\\{{b}}&{{c}}\\ \end{array}\right) ( \theta_{1 2}) ( \pi_{a , b}( \theta_{1}) \otimes \pi_{b , c}( \theta_{2}) ) \Delta ( g ) = ( \pi_{a , d}( \theta_{2}) \otimes \pi_{d , c}( \theta_{1}) ) \Delta ( g ) S \left( \begin{array}{c c}{{a}}&{{d}}\\{{b}}&{{c}}\\ \end{array}\right) ( \theta_{1 2})
\int_{0}^{\infty}\!\!dr\,\exp\left[-v_{_{F}}(r;x)\right]\simeq(2\pi)^{{1\over2}}\exp\left[-\left\{v_{_{F}}(r_{0}(x);x)+{1\over2}\ln\left[v''_{_{F}}(r_{0}(x);x)\right]\right\}\right],
\begin{aligned}|D_t^{(0)}-D_s^{(\ell(\theta))}|^2 = \beta^{\frac{1}{2}}N^{\frac{1}{2}}\left(t^{\frac{3}{4}}-s^{\frac{3}{4}}\cos\theta\right)^2 + \beta^{\frac{1}{2}}N^{\frac{1}{2}}\left(s^{\frac{3}{4}}\sin\theta\right)^2. \end{aligned}
U_{j}=\left[\frac{a-1}{a-b}B^{a-2}-\frac{b-1}{a-b}B^{b-2}\right]p_{j}q^{\beta-2}_{j}&V_{j}=\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]q^{\beta-1}_{j}
f(z)= f(z_*)+Df(z_*)(z-z_*)+\frac{1}{2}\langle D^2f(z_*)(z-z_*) , (z-z_*) \rangle +\frac{1}{6}\sum \limits_{i,j,k=1}^{2n}D_{ijk}f((1-\theta_z) z_*+\theta_z z)(z-z_*)_i(z-z_*)_j(z-z_*)_k,
\theta (\theta(\rho_{i}, \rho_{j}), \theta(\rho_{i}, \rho_{\ell})) =\theta\left(\rho_n^{qi+B(1,n-i+1)}\rho_{{j-i}}, \rho_n^{qi+B(1,n-i+1)}\rho_{\ell-i}\right)=\theta(\rho_{{j-i}}, \rho_{{\ell-i}}) &=\rho_n^{q(j-i)+B(1,n-j+i+1)}\rho_{\ell-j},
R ( \lambda , \mu ) = \left[ \begin{array}{c c c c}{{f ( \mu , \lambda )}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{g ( \mu , \lambda )}}&{{0}}\\{{0}}&{{g ( \mu , \lambda )}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{f ( \mu , \lambda )}}\\ \end{array}\right]
\sum_{\substack{n,n'\in\{1,\ldots,2N\}, n>n'}}\sum_{j,j'\in\{1,\ldots,N^K\}}(jj')^{-1}\mathbf{1}_{|jM_{n,1}-j'M_{n',1}|<1/2\cdot q^{-CN^{1-\varepsilon}}M_{n',1}}=o(N)
s_4 s_7 s_6 s_5 (s_2 s_4 s_3 s_8 s_1 s_3 s_4 s_2 )s_5 s_6 s_7 s_4 = s_4 s_7 s_6 s_5 (s_8s_{\alpha_1 + \alpha_3 + \alpha_4 + \alpha_2})s_5 s_6 s_7 s_4 = s_{\alpha_7+\alpha_8}s_{\alpha_1 + \alpha_3 + 2\alpha_4 + \alpha_2 + \alpha_5 + \alpha_6 + \alpha_7}.
\bigg\|\Big(\sum_j |[T, \textbf{b}]_\alpha (f_1^j, f_2^j,\dots, f_m^j)|^s\Big)^\frac1s\bigg\|_{L^{q}(v)}\lesssim\prod_{i=1}^m \|b_j\|^{\alpha_i}_{{\rm BMO}}\bigg\|\Big(\sum_j |f_i^j|^{s_i}\Big)^\frac1{s_i}\bigg\|_{L^{q_i}(v_i)},
W ( z ) = \frac 1 2 \bigg ( Q^{\prime}- \frac{P^{\prime \prime}}2 \bigg ) + \frac 1{4 P}\bigg ( Q - \frac{P^{\prime}}2 \bigg ) \bigg ( Q - \frac{3 P^{\prime}}2 \bigg ) + c \, z^{2}\, ,
\frac{\lambda_v^{K_{\ell}}}{\lambda_v^L}=\Bigl\{q_v^{(\dim M_v^{K_{\ell}}-m)/2}\cdot|M_v^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}\Bigl\{q_v^{(\dim M_v^L-m)/2}\cdot|M_v^L|^{-1}\cdot\prod_{i=1}^{m}(q_v^{2i}-1)\Bigr\}^{-1}\leq q_v^{-1/2}(q_v^{n_{v,_nu_v}}-1).
E_{2}(\Gamma_{0}(20)) = \mbox{span}_{\mathbb{C}}\left\{2P(q^{2})-P(q),\,4P(q^4)-P(q),\,5P(q^5)-P(q),\atop10P(q^{10})-P(q),\,20P(q^{20})-P(q)\right\}&=\left\{\left.c_{1}P_{1}+c_2P_2+c_4P_4+c_{5}P_{5}\atop+c_{10}P_{10}+c_{20}P_{20}\; \right| \;{20c_{1}+10c_{2}+5c_{4}+4c_{5}\atop+2c_{10}+c_{20}=0}\right\}
\ln Z = \frac{V}{( 2 \pi )^{d}}\: \sqrt{\frac{\alpha^{\prime}c}{\hbar}}\: \int_{m_{0}}^{\infty}\: d m \: \rho ( m ) \: \int d^{d}k \ln \left\lbrace \frac{1 + \exp \left\lbrack - \beta_{H}\left( m^{2}c^{4}+ k^{2}\hbar^{2}c^{2}\right)^{\frac{1}{2}}\right\rbrack}{1 - \exp \left\lbrack - \beta_{H}\left( m^{2}c^{4}+ k^{2}\hbar^{2}c^{2}\right)^{\frac{1}{2}}\right\rbrack}\right\rbrace
\begin{aligned}F_{(1)}\left(z,G_{(1)}^Y(z)\right) =0\ , F_{(1)}\left(G_{(2)}^Y(z)-z,G_{(2)}^Y(z)\right)=0\ , &F_{(3)}\left(z,G_{(3)}^Y(z)\right) =0\ ,F_{(3)}\left(G_{(0)}^Y(z)-z,G_{(0)}^Y(z)\right)=0\ .\end{aligned}
\begin{cases}\partial_t W + (U\cdot \nabla )W = (W\cdot \nabla) U, U=-\Delta^{-1}\nabla \times W, W\Bigr|_{t=0}=W_0 \in H_x^{\infty}(\mathbb R^3) =\bigcap_{l=0}^{\infty}H_x^l(\mathbb R^3).\end{cases}
(\mathfrak{B}_1)_j^{j'}(l):=\begin{cases}-2 i j\,c_2\,(j-j')^2\,\sqrt{\lvert j-j'\rvert \xi_{j-j'}}-6 i j\,c_3\,\sqrt{\lvert j-j'\rvert \xi_{j-j'}}\mbox{if}\,\,\,j-j'\in S,l=\mathtt{l}(j-j') 0 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\mbox{otherwise}.\end{cases}
\mathbf{u}^{[i,j]}= \arg \min_{\mathbf{u}}\eta^{[i,j]}= \arg\min_{\mathbf{u}}\sum_{k=1, k\neq i}^{K}\left\|\mathbf{u}^{\mathsf{H}}\mathbf{H}_{k}^{[i,j]}\mathbf{P}_k\right\|^2.