harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tau^{(1)}_{j,k}:=\inf\{n\geq 0: i({\bf C}^{(1)}_j(n); Y^n|{\bf C}^{(2)}_k(n)) > \gamma_1\}, \tau^{(2)}_{j,k}:=\inf\{n\geq 0: i({\bf C}^{(2)}_k(n); Y^n|{\bf C}^{(1)}_j(n)) > \gamma_2\}, \tau^{(3)}_{j,k}:=\inf\{n\geq 0: i({\bf C}^{(1)}_j(n),{\bf C}^{(2)}_k(n);Y^n) > \gamma_3\},
n^{-1}\sum_{k=1}^n \zeta_{n.k}^2 = n^{-1}\sum_{k=1}^n ([\mathcal Z_n]_k- Q(u_{n,k}))^2 =\int_0^1 (Q^*_n(u) - Q_n (u))^2 du \le 2 \int_0^1 (Q^*_n(u) - Q (u))^2 du + 2 \int_0^1 (Q (u) - Q_n (u))^2 du.
\times \exp \left( \sum_{b , b^{\prime}= 1}^{N}\sum_{j = 1}^{n_{b}}\sum_{j^{\prime}= 1}^{m_{b^{\prime}}}\frac{1}{N}\Big [ \ln ( x_{j}^{( b )}- y_{j^{\prime}}^{( b^{\prime})})^{2}+ 2 \gamma - \ln ( 4 ) \Big ] \right)
v(\psi\cdot g_{uu}) =(g_v\cdot g_{uu})=(g_v\cdot g_u)_u-g_{uv}\cdot g_u =(g_v\cdot g_u)_u- \frac{1}{2}(g_u\cdot g_u)_v =(f_v\cdot f_u)_u- \frac{1}{2}(f_u\cdot f_u)_v =v (\phi\cdot f_{uu}),
\alpha\left(R(r+s)\right) =\int^{T_2}_{T_1}G\left(\frac{3T_1+T_2}{4},\tau\right)f_\uparrow\left(r(\tau)+s(\tau)\right)d\tau &\ge \int^{T_2}_{\frac{3T_1+T_2}{4}}G\left(\frac{3T_1+T_2}{4},\tau\right)f_\uparrow\left(r(\tau)+s(\tau)\right)d\tau &\ge f_\uparrow\left(a + 4^{-k}d\right)\int^{T_2}_{\frac{3T_1+T_2}{4}}G\left(\frac{3T_1+T_2}{4},\tau\right)d\tau &>a.
f(e^{\gamma},n)= f(e^{\sum_{i=1}^{n}\gamma_{i}},n)=\prod_{i=1}^{n}f(e^{\gamma_{i}},1)=\prod_{i=1}^{n}f(e^{\delta},1)\phi(e^{\delta-\gamma_{i}}) = f(e^{\delta},1)^{n}\phi(e^{n\delta-\sum_{i=1}^{n}\gamma_{i}})=f(e^{\delta},1)^{n}\phi(e^{n\delta-\gamma})
\beta \frac{1- \varepsilon \| b\|}{1+\varepsilon \beta}\le \beta_h:= \inf_{x_h\in X_h \setminus\{0\}}\sup_{y_h\in Y_h\setminus\{0\}}\frac{b(x_h,y_h)}{\| x_h\|_{H}\| y_h\|_H}.
\displaystyle{1 + \sqrt{\frac{2 \pi}{g_{n c}^{2}\theta}}\sum_{l = 1}^{+ \infty}\frac{\exp ( - \frac{\pi l}{g_{n c}^{2}\theta})}{\sqrt{l}}\left( 1 - \exp \left[ i \frac{L x}{\theta}+ \frac{1}{l}\frac{g_{n c}^{2}L^{2}x^{2}}{1 6 \pi^{3}\theta}\right] \right)}.
{\cal L}_{SCM\lambda}=\frac 12 \left( J_\mu^j M^{\mu \nu}_{jk}J_{\nu}^k -2 \varepsilon^{\mu \nu}~J_\nu^j ~\partial_\mu \Phi^j~+i \overline{\chi}^j /\kern-.5em\partial ~\chi^j+i\varepsilon^{jkl}\overline{\chi}^j /\kern-.5em J^k \chi^l+\frac 14 (\bar \chi \chi )^2 \right),
[ \phi_{k}\, , \, \phi_{h}] = -{\frac{i \pi \beta^{2}}{2}}s \left( \frac{k - h}{2 N}\right) \quad , \quad [ \bar{\phi}_{k}\, , \, \bar{\phi}_{h}] ={\frac{i \pi \beta^{2}}{2}}s \left( \frac{k - h}{2 N}\right) \quad , \quad [ \phi_{k}\, , \, \bar{\phi}_{h}] = 0 \quad .
\mathbb{P}\left( b(\mathbf{X}^{z_1, \dots, z_l}, x^{z_1,\dots,z_l})=0 \right) = \prod_{i=0}^l \mathbb{P}\left( \varphi_i(x)=0 \, \forall x\in \mathbf{X}_i\right) &\geq \prod_{i=0}^l \exp\left( -\mathbb{E}\sum_{x\in \mathbf{X}_i}\varphi_i(x)\right) &= \exp\left( -\mathbb{E}b(\mathbf{X}^{z_1, \dots, z_l}, x^{z_1,\dots,z_l}) \right)
\partial_{\vec{z}}^{2}H_{F}+ H_{3}\partial_{\vec{x}}^{2}H_{F}= q_{F}\delta ({\vec{x}- \vec{x}_{0}}) \delta ( \vec{z}- \vec{z}_{0}) , \ \partial_{\vec{z}}^{2}H_{3}={\frac{q_{3}}{V_{y}}}\delta ({\vec{z}- \vec{z}_{0}}) ,
\mathcal{C}_\mathcal{A}(\alpha){\bf u}^{m-1}=\left[\begin{array}{c}-\mathcal{A}_\alpha({\bf u}_\alpha)^{m-1}-\mathcal{A}_{\bar\alpha\alpha}({\bf u}_\alpha)^{m-1}+({\bf u}_{\bar\alpha})^{[m-1]}\end{array}\right].
S_{\pm\epsilon}^{Q}\left(v\right)=\left(2\pi\right)^{1-n}\int_{\mathbb{R}^{n-1}}\widehat{S}_{\pm\epsilon}\left(\omega\right)\exp\left(i\left\langle \left(Q\left(v_{1}\right)-Q\left(v_{2}\right),\dots,Q\left(v_{n-1}\right)-Q\left(v_{n}\right)\right),\omega\right\rangle \right)d\omega
N_{1,\varphi}^{\mp}(z) = \frac{1}{2}{M}_{0,\varphi}(\infty) \mp \frac{1}{2 \pi i}\int\limits_{-\infty}^{+\infty}\frac{{M}_{0,\varphi}(t) -{M}_{0,\varphi}(\infty)}{t - z}dt \equiv \frac{1}{2}{M}_{0,\varphi}(\infty) \mp \left({\bf C}_{0}{M}_{0,\varphi}\right)(z).
+\frac{1}{2N^3}\left[ \frac{2 \sin^3 [\widehat{\theta}_t] +\sin [2\widehat{\theta}_t] ( 2 \cos[\widehat{\theta}_t] -(\cos[\theta_i]+\cos[\theta_j]))}{( \cos [\widehat{\theta}_t] - \cos[\theta_i])^2 ( \cos[\widehat{\theta}_t] - \cos[\theta_j])^2}\right]
\int_{0}^{T}\int_{\Omega}u^{-}\partial_{t}\left[ g_{1-\alpha,m}*(u-u_{0}) \right]dxdt & = \int_{0}^{T}\int_{\Omega}u^{-}\partial_{t}\left[ g_{1-\alpha,m}*u\right]dxdt- \int_{0}^{T}\int_{\Omega}u^{-}g_{1-\alpha,m}u_{0}dxdt.
\begin{array}{c}e_{i}= F_{i}^{-}F_{i+1}^{+},~~~ 1 \le i \le n f_{i}= F_{i}^{+}F_{i+1}^{-},~~~ 1 \le i \le n k_{i}= Q^{N_{F_{i}}- N_{F_{i+1}}},~~~ 1 \le i \le n e_{0}= F_{n+1}^{+}F_{1}^{-},~~~ f_{0}= F_{1}^{+}F_{n+1}^{-},~~~ k_{0}= Q^{-N_{F_{1}}+ N_{F_{n+1}}},~~~ \end{array}
2\omega + \frac{m_1^2}{2p_1}- \frac{m_2^2}{2p_2}- \frac{m_3^2}{2p_3}= \delta E_D^{(4)}+ \frac{\xi}{M_D}\left(p_2 p_3 + (p_2 + p_3 )\omega + \omega^2 \right)
Pd_{l_1,k}=e_{l_2,n-2k}P, d_{l_1,k}Q+Qd_{l_1,k}=RP, P^TEe_{l_2,n-2k}=d_{l_1,k}^TFR, Q^TFd_{l_1,k}+d_{l_1,k}^TFQ=P^TEP.
\left\{\mathrm{e}^{-{\frac{A}{2}}}\not{\partial}_{x}+ \Gamma^{r}\left( \partial_{r}+{\frac{D_{1}}{4}}A^{\prime}+{\frac{D_{2}}{4}}B^{\prime}\right) + \lambda \chi + \mathrm{e}^{-{\frac{B}{2}}}\not{\Delta}_{y}\right\}\Psi = 0 .
\partial_{t}\upsilon -i\left[ \Delta \upsilon +A\upsilon +\tilde{V}\upsilon \right] =\eta_{R}^{\prime}\left( t\right) \theta_{M}\left( x\right) \tilde{u}\left( x,t\right) -\left( 2\nabla \theta_{M}.\nabla \tilde{u}+\tilde{u}\Delta \theta_{M}\right) \eta_{R}.
\|u-u_n\|_{X}\leq \|u-p_nu\|_{X}+\|p_nu-u_n\|_{X}\leq \|u-p_nu\|_{X}+ \sup_{m>n_0}\|A_m^{-1}\|_{L(X_m)}\|p_nAp_nu-p_nAu\|_{X}&\leq \|u-p_nu\|_{X}+ \sup_{m>n_0}\|A_m^{-1}\|_{L(X_m)}\|A\|_{L(X)}\|u-p_nu\|_{X}&= \big(1+\sup_{m>n_0}\|A_m^{-1}\|_{L(X_m)}\|A\|_{L(X)}\big) \|u-p_nu\|_{X}.
||\delta^{\ell}(\vec{a})||_{\infty,0}\le C||\delta^{\ell}(\vec{a})||_{s-|\ell|}&= C||\sum_{j,m}m^{\ell}a_{j,m}\prod_{g=1}^nU_g^{m_g}f_j||_{s-|\ell|}&< C||\sum_{j,m}(1+|m_1|^2+\cdots+|m_n|^2)^{|\ell|}a_{j,m}\prod_{g=1}^nU_g^{m_g}f_j||_{s-|\ell|}&= C||P_{\lambda^{|\ell|}}(\vec{a})||_{s-|\ell|}&= C||\vec{a}||_s
\mathbb{P}[ |\Im \log \zeta (1/2 + i UT)| \geq V] \leq \sum_{r = 0}^{p-1}(1 +\log \log T)^r \mathbb{P}[ |I(UT, 2^{-r}H)| \geq (1-\varepsilon) 2^r V ] + (1 +\log \log T)^p \, \mathbb{P}[ | \Im \log \zeta (1/2 + i UT)| \geq 2^p V] + \mathcal{O}_{\varepsilon, \varphi}(T^{-1/3}),
u_{0}(r) =A\left[ \left( r/r_{0}\right)^{1/2+\nu}-c\left(r/r_{0}\right)^{1/2-\nu}\right] ,r>R & =\frac{A}{\sin\sqrt{\lambda}}\left[ \left( R/r_{0}\right)^{1/2+\nu}-c\left( R/r_{0}\right)^{1/2-\nu}\right] \sin(k_{0}r),r<R,
J = 2 \pi \Lambda \int \, d \eta d \beta \, \left[ 1 + \frac{g_{1}}{2}\frac{( \nabla w )^{2}}{1 - w^{2}}+ \frac{g_{2}}{2}\left( \frac{( \nabla w )^{2}}{1 - w^{2}}+ \left( ( \nabla v )^{2}+ \frac{N^{2}\tau^{2}}{a^{2}\sinh^{2}\eta}\right) ( 1 - w^{2}) \right) \right] ( 1 - w^{2}) \, .
\frac{1}{n}\ln A_{n}=\frac{1}{n}\ln\left( \left\lfloor nx\right\rfloor+1\right) +\frac{1}{n}\ln b_{1}\left( \left\lfloor nx\right\rfloor;n,r\right) & \simeq-D_{1}\left( x\left\Vert r\right. \right) \quad\left(n\gg0\right) .
\frac{a(f_*/R)^{\frac 1{\beta}}}{3 (4f_*/n)^{\frac 1{2\beta+1}}}= \frac{a}{3 R^{\frac 1{\beta}}4^{\frac 1{2\beta+1}}}f_*^{\frac{\beta+1}{\beta(2\beta+1)}}n^{\frac 1{2\beta+1}}\gtrsim \log^5 n,
\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})(20+{\rm ch}(\overline{W_i})-8{\rm ch}({W_i})+B_1\wedge[{\rm ch}({W_i})-8]+B_2)\right\}^{(14)}&=-135432\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(14)}.
\int\limits_0^{\infty}\!dq \, q F_3(c,d,q) F_3(e,t,q) =\int\limits_0^{\infty}\!dq \, q \,\frac{\theta(\Delta_3 (c,d,q))\,\theta(\Delta_3 (e,t,q))}{\Delta_3 (c,d,q) \, \Delta_3 (e,t,q)}\;.
u_{T}^{-4}\alpha_{T}=cd^{2}e^{3}\left( a-24b\right) ,u_{T}^{-6}\beta_{T}=d^{2}e^{4}\left( -a^{2}+36ab-216b^{2}\right) , u_{T}^{-12}\gamma_{T}=d^{4}e^{8}b^{3}\left( a-27b\right) .
\nabla_{\mu}\gamma^{\mu}\psi = f^{-{\frac{1}{2}}}\gamma^{0}\partial_{0}[ \psi ] + ( f g^{p + 2})^{-{\frac{1}{4}}}\gamma^{i}\partial_{i}[ g^{{\frac{p}{4}}}f^{{\frac{1}{4}}}\psi ] = 0
Z = \sum_{i , \bar{\imath}}Z_{i , \bar{\imath}}^{( 9 , 9 )}Z_{i , \bar{\imath}}^{( 6 , 6 )}B_{i}^{( - 2 )}\left( B_{\bar{\imath}}^{( E_{8}^{\prime}\times S O ( 6 ) )}\right)^{*}\, \, .
\begin{aligned}e^{\tau_{L}(L')}_{\pm}= \pm \frac{1}{4\sqrt{\beta}}[\tau_{L}(L')] + \frac{1}{8\beta}[\tau_{L}(L')]^2 = \pm \frac{1}{4\sqrt{\beta}}(\tau_{L})_\ast[L'] + \frac{1}{8\beta}\left( (\tau_{L})_\ast[L']\right)^2 . \end{aligned}
h^{(-)}\stackrel{d}{=}\psi\left(\sum_{k=1}^\infty (\tfrac{\xi_k}{\xi_k+S_k})^2 \,\psi\left( \sum_{j=1}^\infty (\tfrac{\xi^{(k)}_j}{\xi_k+S_k})^2h_k^{(-)}\right) \right),
\rho(L_{0}) \, =\, -\frac{h(z)}{h'(z)}\partial + b(z) \, , \rho(L_i) \, =\, h(z)^i \left(\rho(L_0) +i \lambda \right) P(\rho(L_0) -\lambda-i, i) \, , \ \i\neq 0,
\tilde{C}(p) =-2 \, \gamma \, g^{2}\delta_{a}^{b}\delta_{J}^{I}\int \frac{d^{4}k}{(2\pi)^{4}}d^{2}\theta d^{2}{\overline\theta}\frac{1}{k^{4}}\left\{\Phi^{a\dagger}_{I}(p,\theta,{\overline \theta}){\Phi}_{b}^{J}(-p,\theta,{\overline \theta}) \right\}\, ,
\hat{X}_{t}^{\varepsilon}=x+\int_{0}^{t}A\hat{X}_{s}^{\varepsilon}ds-\int_{0}^{t}B(\hat{X}_{s}^{\varepsilon})ds+\int_{0}^{t}f(X_{s(\delta)}^{\varepsilon},\hat{Y}_{s}^{\varepsilon})ds+\int_{0}^{t}\sigma_1(\hat{X}_{s}^{\varepsilon})dW^{Q_{1}}_s,
Op_{t, h}(a) \phi_{x_0,\xi_0 ,h}=Op_{t, h}(b)\chi(x-x_0)\phi_{x_0,\xi_0 ,h}+Op_{t, h}(b)(1-\chi(x-x_0))\phi_{x_0,\xi_0 ,h}&+Op_{t, h}(c)\chi(hD-\xi_0)\phi_{x_0,\xi_0 ,h}+Op_{t, h}(c)(1-\chi(hD-\xi_0))\phi_{x_0,\xi_0 ,h}.
nR-n\epsilon_n \leq \sum_{e^n}p(e^n)\sum_{i=1}^{m(e^n)}H(Y_{t_i}^{t_{i+1}-1})-nh_2(\alpha) &\leq\sum_{e^n}p(e^n)\sum_{i=1}^{m(e^n)}C(\ell_i),
+\sin\psi\cdot\left(\frac{F[p,q](\frac{i}{\mu_{0}})w_{2}[p,q](\frac{i}{\mu_{0}})w_{3}[p,q](\frac{i}{\mu_{0}})}{w_{1}[p,q](\frac{i}{\mu_{0}})}\right)^{-\frac{1}{2}}\gamma^{0}\gamma^{1}\gamma^{0}\partial_{\eta}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\lim_{| e \Phi | > > 1}\lim_{m a < < 1}l n d e t = - \frac{| e \Phi |}{4 \pi}\ln \left( \frac{| e \Phi |}{( m a )^{2}}\right) + O ( | e \Phi | , ( m a )^{2}| e \Phi | \ln ( | e \Phi | ) ) .
u =\frac12\ln \left\{\left[(x^1)^2+(x^3)^2\right]\left[(x^2)^2+(x^4)^2\right]\right\}, begin{align*}4pt]v =\arctan \frac{x^1x^4+x^2x^3}{x^3x^4-x^1x^2}+x^0, begin{align*}4pt]w =x^0.
a_N(z):=\sup_{n\in\mathbb N}\mathbb E\left[\int_0^T\|p_N\alpha_s^{\mathcal S_n}\|_{\mathcal H}^zds\right],b_N(z):=\sup_{n\in\mathbb N}\mathbb E\left[\int_0^T\|p_N\sigma_s^{\mathcal S_n}\|_{L_{}(U,H)}^zds\right],
M_{U}^{\scriptstyle \mathrm{m i n}}= \frac{M_{p}g_{U}}{{2 \pi}\left| \eta \left( i \right) \right|^{4}}\times \frac{1}{\sqrt{1 + g_{U}^{2}\frac{Y ( 1 , 1 )}{1 6 \pi^{2}}}}\ .
g_4(x) \geq g_4(-1/\delta) &= a \mu \delta^{-2}- a (1 -\mu) - 2 \mu \delta^{-1}+ a (c_A -\mu) + \delta(1 -\mu) &\geq (a - 2) \mu \delta^{-1}+ (c_A - 1 )a .
\mathrm{P}^o_{1,k}= \sum^{k-1}_{p=0}{k-1 \choose p}\frac{(-1)^p K!\left[\int^{\epsilon_{1,k}^*}_{0}f_{\tilde{x}_k}(y)dy \right]^{K-k+p+1}}{(K-k)!(k-1)!(K-k+p+1)}.
\left[ \bar{P}_{a},\bar{Z}_{b}\right] =\frac{1}{2}\left[ P_{a}+Z_{a},P_{b}-Z_{b}\right] , & =\frac{1}{2}\left( \left[ P_{a},P_{b}\right] -\left[ P_{a},Z_{b}\right] +\left[ Z_{a},P_{b}\right] -\left[ Z_{a},Z_{b}\right]\right) , & =\frac{1}{2}\left( J_{ab}-Z_{ab}-Z_{ba}+J_{ab}\right) , & =J_{ab},
K_{b}^{a}K_{a}^{b}= \frac{2}{r^{2}( 1 - \dot{r}^{2})}= \frac{2 ( E^{N - G})^{2}}{r^{4}}, \; \; K_{c}^{a}K_{a}^{b}K_{b}^{c}= 0 .
h_{0,1}(x)=\frac{1}{2}\left((1+\sqrt{x})^m+(1-\sqrt{x})^m\right)\equiv \frac{1}{2}\left(1+x^{\frac{m}{2}}+1-x^{\frac{m}{2}}\right)=1\pmod{p}, &h_{1,1}(x)=\frac{1}{2\sqrt{x}}\left((1+\sqrt{x})^m-(1-\sqrt{x})^m\right)\equiv \frac{1}{2\sqrt{x}}\left(1+x^{\frac{m}{2}}-1+x^{\frac{m}{2}}\right)=x^{\frac{m-1}{2}}\pmod{p},
\begin{cases}\begin{aligned}u_{\epsilon,M}\to u \mbox{in $L^2_{loc}\cap L^{1+m}_{loc}$}U^{\frac{m-1}{2}}\left|\nabla u_{\epsilon,M}\right| \to U^{\frac{m-1}{2}}\left|\nabla u\right| \mbox{in $L^2_{loc}$}\epsilon\left|\nabla u_{\epsilon,M}\right| \to 0 \mbox{in $L^2_{loc}$}\end{aligned}\end{cases}
M = \begin{bmatrix}\tfrac{1}{2}I_n 0 \tfrac{1}{2}I_n\cr \ddots \ddots \ddots \cr \tfrac{1}{2}I_n 0 \tfrac{1}{2}I_n\cr I_n 0\cr \end{bmatrix}\in \mathbb{R}^{nk\times n(k+1)}.
\mathfrak{b}_{i,\rho,t}< (Z_{D_{i,\rho}(t)})^* u_i \Omega t \in (\tau_i, \delta], \mathfrak{b}_{i,\rho,\tau_i}(Q_i^\star) = [(Z_{D_{i,\rho}(\tau_i)})^* u](Q_i^\star).
K^{\alpha \gamma}|_{\beta \delta}(x,y) = K^{\gamma \alpha}|_{\delta \beta}(x,y) K^{\beta \delta}|_{\alpha \gamma}(y,x) = K^{\delta \beta}|_{\gamma \alpha}(y,x) K^\alpha_{\;\;\,\gamma}|_\beta^{\;\;\,\delta}(x,y) = K^\delta_{\;\;\,\beta}|_\gamma^{\;\;\,\alpha}(y,x) = \overline{K^\gamma_{\;\;\,\alpha}|_\delta^{\;\;\,\beta}(x,y)}\:.
\begin{array}{l}x^N_1=-(u_1u_3)^{N/2}\sqrt{\frac{1-(qu_2)^N}{1-(q^{-1}u_2)^N}}exp(ia_2), begin{align*}0.4cm]x^N_2=-(u_1u_3)^{-N/2}\sqrt{\frac{1-(qu_2)^N}{1-(q^{-1}u_2)^N}}exp(ia_2), begin{align*}0.4cm]x^N_5=-q^{-N}(u_1u_3^{-1})^{N/2}\sqrt{\frac{1-(qu_2)^N}{1-(q^{-1}u_2)^N}}exp(ia_2), begin{align*}0.4cm]x^N_6=-q^{-N}(u_1^{-1}u_3)^{N/2}\sqrt{\frac{1-(qu_2)^N}{1-(q^{-1}u_2)^N}}exp(ia_2),\end{array}
\left( \begin{array}{c}-p_{0}au_{3}i+p_{x}jau_{3}i p_{y}au_{3}i+p_{z}jau_{3}i p_{0}au_{1}i-p_{x}jdu_{1}i+p_{y}au_{2}i-p_{z}jdu_{1}i\end{array}\right) = m \; \left( \begin{array}{c}u_{1}u_{2}u_{3}\end{array}\right)
\frac 1 2 ( 2 \omega + q ) ( 2 \rho + q ) C_{i_{1}, . . . . i_{n - q}}^{\omega , \rho}+ ( n - q + 2 ) ( n - q + 1 ) C_{j , j , i_{1}, . . . i_{n - q}}^{\omega , \rho}= 0 ~ ~ ~
L=\sqrt{-g}\sum_{n=0}^{t}\alpha_{n}R^{n},R^{n}=\frac{1}{2^{n}}\delta_{\alpha_{1}\beta_{1}...\alpha_{n}\beta_{n}}^{\mu_{1}\nu_{1}...\mu_{n}\nu_{n}}\prod_{r=1}^{n}R_{\mu_{r}\nu_{r}}^{\alpha_{r}\beta_{r}}
\lim\limits_{x\to\infty}\frac{1}{x^{\frac{1}{\beta}}\ell^{\frac{1}{\beta}}(x^{\frac{1}{\beta}})}\eta_K(x)=\int^{\infty}_0 \big(K_{\infty}(t^{-\beta})-K_{\infty}(0)\big)dt.
m \frac{d^{2}}{d t^{2}}R_{a}^{i}= k \sum_{j , b}M_{\left( i a \right) \left( j b \right)}R_{b}^{j}, M_{\left( i a \right) \left( j b \right)}= \varepsilon^{i j}\sum_{c = 1 , 2 , 3}\varepsilon^{a c b}N_{c}N_{b}
I(z) \leq |\partial^2 u_{k}(z) - \partial^2 u_{k}(0)| + |\partial^2 u_{k}(0) - \partial^2 u(0)| + &\qquad\qquad\quad+|\partial^2 u(z) - \partial^2 u_{k}(z)| =: I_{1}(z)+I_{2}(z)+I_{3}(z).
\begin{aligned}a_h(u_h, v) = \sum_{K\in \mathcal{T}_h}(\nabla \Pi_K u, \nabla \Pi_K v)_K + \sum_{K\in \mathcal{T}_h}\langle \nabla( u - \Pi_K u)\cdot n, Q_K v - Q_F v \rangle_{\partial K}. \end{aligned}
\sup_{\theta\in\mathcal{E}(\alpha_{0},B)}R(\theta,\hat{\theta}_{(d)})\geq\sup_{\theta\in\mathcal{E}(\alpha_{0},B)}\sum_{i=d+1}^{\infty}\theta^{2}_{i}/B^2\geq (d +1 )^{-2\alpha_{0}}, \ \varepsilon>0.
F = \sum_{j=1}^{\infty}\lambda_{j}^{(1)}a_j^{(1)}= \sum_{j=1}^{\infty}\lambda_{j}^{(1)}\,\Pi_l(G_j^{(1)}, \tilde G_j^{(1)})+\sum_{j=1}^{\infty}\lambda_{j}^{(1)}\left(a_j^{(1)}-\Pi_l(G_j^{(1)}, \tilde G_j^{(1)})\right) &:= M_1+E_1.
\sigma_{W_{\rm s, id}}^2=\left(\psi_{1}\kappa_{\rm b}+\psi_{2}\right)T_{\rm s}-\psi_{1}\kappa_{\rm b}Q(\kappa_{\rm b})T_{\rm s}+\psi_{1}\kappa_{\rm t}Q(\kappa_{\rm t})T_{\rm s}+\psi_{1}T_{\rm s}f(\kappa_{\rm b})-\psi_{1}T_{\rm s}f(\kappa_{\rm t}).
\delta_{K}j_{T}^{0 M_{1}\cdots M_{p}}= \sum_{k = 1}^{p}\partial_{\mu_{k}}\left[ \frac 1{p !}\epsilon^{0 \mu_{1}\cdots \mu_{k}\cdots \mu_{p}}\cdot \phi_{, \mu_{1}}^{M_{1}}\cdots \delta_{K}\phi^{M_{k}}\cdots \phi_{, \mu_{p}}^{M_{p}}\right] \quad .
N \left( \frac{|\nabla H - h^S_{\nu \nu}H\nu_S^T|^2}{H}\right) \le C(S) \frac{|\nabla H - h^S_{\nu \nu}H\nu_S^T|^2}{H}+C(S)H|\nabla H| \le C(S) \frac{|\nabla H - h^S_{\nu \nu}H\nu_S^T|^2}{H}+C(S)H|\nabla H - h^S_{\nu \nu}H\nu_S^T|+C(S)H^2 \le C(S) \frac{|\nabla H - h^S_{\nu \nu}H\nu_S^T|^2}{H}+\frac{1}{4}H^3 + C(S).
T_{F,G,N}^L(f_1,\dots,f_d) = \sum\limits_{\mathbf{r}\in R}\sum\limits_{\substack{\mathbf{n}\in [N]^d S\mathbf{n}= \mathbf{r}}}\prod\limits_{j=1}^d f_j(n_j) \ll_{c,C,\varepsilon}\rho^{\Omega(1)}+ o_{\rho}(1)
t_{j}= \frac{1}{2 \cos^{2}( \frac{\pi j}{2 H})}\prod_{i = 1}^{j}\frac{\tan ( \frac{\pi ( 2 w_{r}+ 2 i - 1 )}{4 H}) \tan ( \frac{\pi ( 2 w_{r}- 2 i + 1 )}{4 H})}{\tan^{2}\left( \frac{\pi i}{2 H}\right)}.
G^{{\mathrm{ZF - CB}}}= \frac{{{\mathrm{SINR}}^{{\mathrm{ZF,non-ideal}}}}}{{{\mathrm{SINR}}^{{}}}}= \frac{{\left({1 - z}\right)^2 \left({z + K\gamma}\right)}}{{\left({1 - \alpha^2 - K\gamma}\right)z + K\gamma}}
\left( i \hbar ( 1 + \beta p^{2}) \partial_{p}- \langle{\bf{x}}\rangle + i \hbar \frac{1 + \beta ( \Delta p )^{2}+ \beta \langle{\bf{p}}\rangle^{2}}{2 ( \Delta p )^{2}}( p - \langle{\bf{p}}\rangle ) \right) \psi ( p ) = 0
\sum_{d>x/2}\frac{|h_{1/\tau}(d)|}{d}(\log d)^{\ell}=\sum_{d=2^{\nu}> x/2}\frac{|h_{1/\tau}(2^{\nu})|}{2^\nu}(\log 2^\nu)^{\ell}\ll \sum_{\nu > \log x/\log 2}\frac{\nu^{\ell-1}}{2^{\nu}}\ll\frac{(\log x)^{\ell-1}}{x}.
\sum_{i = 2}^{\infty}\mathbb{E}\exp\left(-\frac{x W_i}{T_i}\right) = \int_0^1 \mathbb{E}\exp \left(-\frac{xw}{T_{i-1}(1-w)}\right) \cdot \theta(1-w)^{\theta-1}dw & = \int_0^1 \int_0^1 \exp \left(-\frac{xw}{t (1-w)}\right) \cdot \frac{\theta}{u}\cdot \theta(1-w)^{\theta-1}dt dw,
\hat{H}^{\mathsf{Bayes}}\leq \psi(Sa + n + 1) - \sum_{i = 1}^S \frac{a+X_i}{Sa + n}\psi(a+1) & = \psi(Sa + n+1) - \psi(a+1) & \leq \ln \left( \frac{Sa + n + e^{-\gamma}}{a + \frac{1}{2}}\right).
\left(-ig^{-1}\xi\cdot\eta+L\right)\left(g\cdot e_{j}\right) =\left(-i\xi\cdot g\eta+L\right)\left(g\cdot e_{j}\right) =\left(-i\xi_{1}\left|\eta\right|+L\right)\left(g\cdot e_{j}\right) =\varrho_{j}\left(\eta\right)\left(g\cdot e_{j}\right).
\mathfrak{C}(n;q_1,q_2,q_2')=\sum_{\alpha\bmod{q_1q_2q_2'}}K_{q_1}(\alpha\ell\bar{q}_2^3)K_{q_2}(\alpha\ell\bar{q}_1^3)K_{q_1}(-\alpha\ell\bar{q_2'}^3)K_{q_2'}(-\alpha\ell\bar{q}_1^3)e_{q_1q_2q_2'}(\alpha n)
ET(k) := 2 \sum_{i=1}^{k-1}\sum_{j=i+1}^{k-1}\frac{1}{((x_k-x_i)+\epsilon^2)((x_k-x_j)+\epsilon^2)}- 2 \sum_{i=1}^{k-1}\sum_{j=k+1}^{d}\frac{1}{((x_k-x_i)+\epsilon^2)((x_j-x_k)+\epsilon^2)}+ 2 \sum_{i=k+1}^{d}\sum_{j=i+1}^{d}\frac{1}{((x_i-x_k)+\epsilon^2)((x_j-x_k)+\epsilon^2)}.
\lim_{h\to0}\int_{U^c}\frac{v(t+h,x\pm y) -v(t,x\pm y)}{h|y|^{N+\alpha}}\dy = \int_{U^c}\lim_{h\to0}\frac{v(t+h,x\pm y) -v(t,x\pm y)}{h}\frac{1}{|y|^{N+\alpha}}\dy &=\int_{U^c}\frac{\partial_tv(t,x\pm y)}{|y|^{N+\alpha}}\dy
{\frac{i}{4 \pi}}\delta_{a b}\epsilon^{\mu \nu \rho}{\frac{( x - y )_{\rho}}{| x - y |^{3}}}, \, \, \, \, \, \, \, \, - i g f_{a b c}\epsilon_{\mu \nu \rho}\int d^{3}x , \, \, \, \, \, \, \, \, g ( T^{a})_{i}^{j}.
\mathbb{E}_{{n''},d,\theta_{n''}}(\varphi_{n''}) = \mathbb{P}_{{n''},d,\theta_{n''}}\left( \| \hat{\theta}_{n''}\| \geq s_{n''}^{-1}C \right) \geq \mathbb{P}_{{n''},d,\theta_{n''}}\left( \| \hat{\theta}_{n''}\| \geq s_{n''}^{-1}C, \| \hat{\theta}_{n''}- \theta_{n''}\| < \delta/2 \right)
\frac{1}{2}\vert \left({}_q\langle J_1,\gamma_1; J_2,\gamma_2\vert \left[\hat P_{1,q},\hat P_{2,q}\right] \vert J_1,\gamma_1; J_2,\gamma_2\rangle_q\right)\vert = \frac{1}{4(\lambda_1 +\lambda_2)^2}\vert \lambda_2^2K_1(1 -(1-q^2)J_1) - \lambda_1^2 K_2(1 -(1-q^2)J_2)\vert.
\hat{H}_{2}=-\sum_{i>j}\frac{q_{i}q_{j}}{2r_{ij}m_{i}m_{j}c^{2}}[\hat{p}_{i}\cdot\hat{p}_{j}+\frac{(r_{ij}\cdot\hat{p}_{i})(r_{ij}\cdot\hat{p}_{j})}{r_{ij}^{2}}]
f^{[i]}_{\ell,m-j}=\frac{\tilde{f}^{[i]}_{\ell,m-j}}{w_j}=\frac{\sum_{h=1}^{m-j}\tilde{b}^{[i]}_{h,\ell,m-j}D g^{[i]}_{h+1,m-j+1}}{\sum_{h=1}^{m-j}\tilde{b}^{[i]}_{h,1,m-j}D g^{[i]}_{h+1,m-j+1}}=\sum_{h=1}^{m-j}\frac{\tilde{b}^{[i]}_{h,\ell,m-j}}{\tilde{b}^{[i]}_{h,1,m-j}}B^{[i]}_{h,m-j}=\sum_{h=1}^{m-j}b^{[i]}_{h,\ell,m-j}B^{[i]}_{h,m-j},
A=x_0 \left[\begin{array}{ccc}0 0 1 0 1 0 1 0 0 \end{array}\right]+x_2 \left[\begin{array}{ccc}0 1 0 1 0 0 0 0 0 \end{array}\right]+x_1 \left[\begin{array}{ccc}\alpha+\frac{3}{4}t^2 s \frac{t}{2}-s -t 0 \frac{t}{2}0 -1 \end{array}\right]
m(x)\,u_{tt}- (p(x)\,u_{x}+ 2\,\delta(x)\,u_{xt})_{x}+ \eta\,\theta_{x}= 0, & \theta_{t}+ \kappa\,q_{x}+ \eta\,u_{xt}= 0, & \tau\,q_{t}+ \beta\,q + \kappa\,\theta_{x}=0
t_{2}(N)=-t_{2}(N') \Longleftrightarrow t_{2}(N_{1}-1)+t_{2}(N_{1}+1)=-2t_{2}(N_{1}) \Longleftrightarrow -2t_{2}\left(\frac{N_{1}}{2}-1\right)-2t_{2}\left(\frac{N_{1}}{2}\right)=-2t_{2}(N_{1}) \Longleftrightarrow t_{2}(N_{1})=t_{2}\left(\frac{N_{1}}{2}\right)+t_{2}\left(\frac{N_{1}}{2}-1\right).
I_5= \frac{1}{16 \pi G_5}\int d^5x \sqrt{-g^{(5)}}\left\{e^{-2\phi_1}\left[ R^{(5)}+20 e^{2\psi}+4(D\phi_1)^2 -5(D\psi)^2 \right]-\frac{1}{2}e^{5\psi}(F^{(5)2}_1+F^{(5)2}_2) \right\}
(\partial^h\Psi)_{A_{p+1}\overline{B}_q}= \sum_{i=1}^{p+1}(-1)^{i-1}\nabla_{\mathcal{X}_{\alpha_i}}\psi_{\alpha_1\dots\hat{\alpha}_i\dots\alpha_{p+1}\overline{B}_q}, (\bar{\partial}^h\Psi)_{A_p\overline{B}_{q+1}}= (-1)^p\sum_{i=1}^{q+1}(-1)^{i-1}\nabla_{\mathcal{X}_{\bar{\beta}_i}}\psi_{A_p\bar{\beta}_1\dots\hat{\bar{\beta}}_i\dots\bar{\beta}_{q+1}},
\tag*{${\bf (B_1)}$}\mu: L_p(J,E_0)\rightarrow L_\infty\big(J,\mathcal{L}(E_0)\big)\,,\u\mapsto \mu(u,\cdot)\ &\ \|\mu(u,\cdot)\|_{L_\infty(J,\mathcal{L}(E_0))}\le f(\|u\|_{L_p(J,E_0)})\,,\u\in L_p(J,E_0)\,,
\begin{array}{lll}\nabla_x \xi=\phi x, (\nabla_x\eta)y=g(x,\phi y), R(x,y)\xi=-\eta(y)x+\eta(x)y, (x,\xi)=-2n\ \eta(x), R(\xi,y)\xi=\phi^2 y, (x,\xi)(\xi,\xi)=-2n,\end{array}
\begin{array}{c c c c c}{\alpha + \beta + \gamma}&{=}&{\bar{\alpha}+ \bar{\beta}+ \bar{\gamma}}&{=}&{k n - m - n - 1 ,}\\{\alpha^{\prime}+ \beta^{\prime}+ \gamma^{\prime}}&{=}&{\bar{\alpha}^{\prime}+ \bar{\beta}^{\prime}+ \bar{\gamma}^{\prime}}&{=}&{k m - m - n - 1 ,}\\ \end{array}
\int_U(-\det D^2v) v^2\psi\,dx =- \int_U D^2v Dv \cdot \big[vDv \psi +\frac12 v^2 D\psi\big]\,dx+ \int_U \Delta v Dv\cdot \big[vDv \psi+\frac12 v^2 D\psi\big]\,dx.
A(z) = Q_{n}^{(N,M)}(z) -CQ_{n-1}^{( N+1,M+1)}(z) \hphantom{A(z)}{}= \frac{1}{2}\int_{-1}^{z}d\zeta \Big[ C(1-\zeta)^{N+1}(1+\zeta)^{M+1}\big( P_{n-1}^{(N+1,M+1)}(\zeta) \big)^{2}\hphantom{A(z)=}{}-(1-\zeta)^{N}( 1+\zeta )^{M}\big( P_{n}^{(N,M)}(\zeta) \big)^{2}\Big]
\frac{\langle s^{\prime}| e^{- \hat{H}_{s}\tau_{f}}\hat{O}_{1}^{s}( \tau_{1}) \hat{O}_{2}^{s}( \tau_{2}) \ldots \hat{O}_{n}^{s}( \tau_{n}) e^{\hat{H}_{s}\tau_{i}}| s \rangle}{\langle s^{\prime}| e^{- \hat{H}_{s}( \tau_{f}- \tau_{i})}| s \rangle}.
T_2(n,d) = d \Psi(\mathbf{m}^*) + \frac{\sqrt{d}}{n}\Big[ \max_{\underline{\sigma}\in \Sigma(\mathbf{m}^*)}H_f(\underline{\sigma}) \Big] |T_2(n,d)| \leq d \| f \|_{\infty}+ \sqrt{d}\Big| \frac{1}{n}\max_{\underline{\sigma}\in \Sigma(\mathbf{m^*})}H_f(\underline{\sigma}) \Big|.
y_\pm \simeq \frac{a}{t}\, \sum_{n=-\infty}^{\infty}\left[ (n\pm b)^2+c^2 \right]^{1/2}, \ \b=\frac{\beta t}{4\pi}\sqrt{\rho(\lambda^{11}- \lambda^{22})}, \ \c^2= \frac{\rho \beta^2 t^2(\lambda^{11}+ \lambda^{22})}{16\pi^2}\simeq\frac{\alpha_0^2}{16\pi^2}.
T_j:=\big\{x\in X~\big|~\langle u_i^*,x\rangle\le 0,\;i\in I_j\big\},\;\;N_j:=\Big\{\sum\limits_{i\in I_j}\lambda_i u_i^*~\Big|~\lambda_i\ge 0\Big\},\;j=1,\ldots,r.
\begin{matrix}{\zeta^{+}\wedge E^{z}= 0 ,}&{\quad \zeta^{-}\wedge E^{z}= 0 ,}&{\quad \tilde{\zeta}_{+}\wedge E^{\bar{z}}= 0 ,}&{\quad \tilde{\zeta}_{-}\wedge E^{\bar{z}}= 0 ,}\\ \end{matrix}
S_{2,2}=\left\{\begin{array}{c}A_{1}B_{1}B_{2}A_{2},\,A_{1}B_{1}A_{1}B_{2}A_{2},\,A_{1}B_{1}B_{2}A_{1}A_{2},\,A_{1}B_{1}A_{2}B_{2}A_{2},\,A_{1}A_{2}B_{1}B_{2}A_{2}, A_{1}B_{1}A_{1}B_{2}A_{1}A_{2},\,A_{1}B_{1}A_{1}A_{2}B_{2}A_{2},\,A_{1}A_{2}B_{1}A_{2}B_{2}A_{2}\end{array}\right\}.
\mathcal{R}_1( \boldsymbol{\eta},\boldsymbol{\tau}) = \boldsymbol{u}- \sigma \partial_t \boldsymbol{\eta}- \textbf{curl}\, \boldsymbol{\tau}\mathcal{R}_2( \boldsymbol{\eta},\boldsymbol{\tau}) = \boldsymbol{\tau}- \nu \, \textbf{curl}\, \boldsymbol{\eta}.
V^{\alpha\beta}= \Big[(\eta +{\hat{F}})^{-1}\Big]^{\alpha \beta}=\left( \begin{array}{ccccc}-\frac{\Delta_{m,n}}{n^2}\frac{m\,\Delta_{m,n}^{1/2}}{n^2}-\frac{m\,\Delta_{m,n}^{1/2}}{n^2}\frac{\Delta_{m,n}}{n^2}1 \ddots 1 \end{array}\right)
[x_{1}^{+}\otimes a_{1}^{-},x_{2}^{+}\otimes a_{2}^{-}]=[\left[\begin{array}{cc}x_{1}^{+}\otimes a_{1}^{-}0 0 0\end{array}\right],\left[\begin{array}{cc}x_{2}^{+}\otimes a_{2}^{-}0 0 0\end{array}\right]]=\left[\begin{array}{cc}x_{1}^{+}x_{2}^{+}\otimes a_{1}^{-}a_{2}^{-}-x_{2}^{+}x_{1}^{+}\otimes a_{2}^{-}a_{1}^{-}0 0 0\end{array}\right]
\gamma_{\theta}\gamma_{S_{1}}\gamma_{\theta}^{- 1}= \gamma_{S_{2}}~ , \gamma_{\theta}\gamma_{S_{2}}\gamma_{\theta}^{- 1}= \gamma_{S_{3}}~ , \gamma_{\theta}\gamma_{S_{3}}\gamma_{\theta}^{- 1}= \gamma_{S_{1}}~ .
\Sigma_{m}( \zeta ) = \lim_{N \rightarrow \infty}\sum_{r = 1}^{N}( \Delta_{m}( \zeta ) )^{r}= \lim_{N \rightarrow \infty}{\frac{1 - \Delta_{m}( \zeta )^{N}}{1 - \Delta_{m}( \zeta )}}.
x_{0}^{(k+4)}x_{0}^{(k)}=v^{\bullet}\bigg[ e_{0,2}e_{0,1}e_{1,2}\left(\prod_{j=2}^{k+1}(e_{j,j+1})^2\right)y_{k+2}^{(3)}\bigg]+v^{\bullet}(x_{2}^{(k+2)})^2.