harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
a_{n}=\lim_{\varepsilon\downarrow0}\frac{\psi_{\mu_{1}}^{(n)}(\alpha+i\varepsilon)}{n!}=\lim_{\varepsilon\downarrow0}\int_{\mathbb{R}_{+}}\frac{t^{n}}{(1-t(\alpha+i\varepsilon))^{n+1}}\,d\mu_{1}(t)=\int_{\mathbb{R}_{+}}\frac{t^{n}}{(1-\alpha t)^{n+1}}\,d\mu_{1}(t).
s(n,k)=(-1)^{n+k}(n-1)!\sum_{\ell_1=1}^{n-1}\frac1{\ell_1}\sum_{\ell_2=1}^{\ell_1-1}\frac1{\ell_2}\dotsm \sum_{\ell_{k-2}=1}^{\ell_{k-3}-1}\frac1{\ell_{k-2}}\sum_{\ell_{k-1}=1}^{\ell_{k-2}-1}\frac1{\ell_{k-1}}.
\Delta\psi=\Upsilon\quad\hbox{in}\Omega\,, & \psi=0\hbox{on}{\mathcal S}\,, begin{align*}0.1cm] \psi=-m\hbox{on}{\mathcal B}\,, begin{align*}0.1cm] \vert\nabla\psi\vert^{2}+2gY=Q \quad\hbox{on}{\mathcal S}\,.
\big(\Phi^{\mspace{1.5mu}0}_{\mathrm{diag}}\!+\Phi^{\mspace{1.5mu}0}_{\mathrm{off}}\mspace{-1.5mu}+\Phi_{\mathrm{reg}}\big)[f_\beta]=48\pi^2\!\!\int_{\mathbb{R}}\!dx\:\lvert\hat{f}^{\,\sharp}_\beta(x)\rvert^2\,S(x),
{\rm det}^{\prime}\Delta^{(\beta,\alpha)}= \prod_{n_1,n_2}^{\prime}({{4\pi^2}\over{l^2}}) \left [ (n_1+ (\alpha - 1)/2 )^2 +{{l^2}\over{4}}(n_2+ (\beta - 1)/2 )^2 ) \right ] \quad .
\ \tau\phi\big((K^*A^{rs}K)^{\frac{1}{s}}\big) +(1-\tau)\phi\big((K^*B^{rs}K)^{\frac{1}{s}}\big) = \ \tau\phi\big(|G_A(s)|^\frac{2}{s}\big) + (1-\tau)\phi\big(|G_B(s)|^\frac{2}{s}\big) \leq \ \phi\big((M^*M)^\frac{1}{s}\big) = \ \phi\big((K^*C^{rs}K)^{\frac{1}{s}}\big).
:\!|g|^4\!:\, = \tfrac 14 (h_1^4 -6 h_1^2 + 3) + \tfrac 12 (h_1^2 - 1)(h_2^2 - 1)+ \tfrac 14(h_2^4 -6 h_2^2 + 3) & = \tfrac 14(h_1^4 + 2 h_1^2 h_2^2 + h_2^4)- 2 (h_1^2 + h_2^2) + 2 & = |g|^4 - 4 |g|^2 + 2.
N^{-1}\sum_{k=1}^Nq_k^{\varphi}=N^{-1}\sum_{k=1}^Nq_k^{\varphi}\sum_{j=0}^NP_j^{\varphi}= N^{-1}\sum_{j=0}^N\sum_{k=1}^Nq_k^{\varphi}P_j^{\varphi}=N^{-1}\sum_{j=0}^Nj P_j^{\varphi}=\left(\widehat{n^{\varphi}}\right)^{2},
F(\rho_{\theta},\rho_{\theta+d\theta})=1-\frac{1}{4}\sum_{i,j}(F_{Q}^{ij}[\rho(\theta)]+2\sum_{\lambda_{k}(\theta)=0}\partial_{i}\partial_{j}\lambda_{k})d\theta_{i}d\theta_{j}+O(d\theta^{4}),
\int_0^{-x}x^{r_1}y^{r_2}(x+y)^{2n-r_1-r_2-2}\, dy= (-1)^{r_2+1}x^{2n-1}\int_0^{1}t^{r_2}(1-t)^{2n-r_1-r_2-2}\,dt = (-1)^{r_2+1}x^{2n-1}\frac{r_2!(2n-r_1-r_2-2)!}{(2n-r_1-1)!}.
\mathcal{B}Q Q^{*}= \Im( P Q^{-1}) Q Q^{*}= \frac{1}{2{\rm i}}( P Q^{-1}- (Q^{*})^{-1}P^{*}) Q Q^{*}= \frac{1}{2{\rm i}}( P Q^{*}- (Q^{*})^{-1}P^{*}Q Q^{*}) = \frac{1}{2{\rm i}}[ P Q^{*}- (Q^{*})^{-1}(Q^{*}P - 2{\rm i}I_{d}) Q^{*}] = I_{d},
S_2 ={\bigcup}_{u=0}^{n-1}{\bigcup}_{z=1}^{e}U(\nu_{u,z})={\bigcup}_{b=1}^c{\bigcup}_{u=n_{b-1}}^{n_b-1}{\bigcup}_{z=q_b}^{e}U(\nu_{u,z}) &={\bigcup}_{z=q_c}^{e}{\bigcup}_{b=b_z}^{c}{\bigcup}_{u=n_{b-1}}^{n_b-1}U(\nu_{u,z})={\bigcup}_{z=q_c}^{e}{\bigcup}_{u=n_{b_z-1}}^{n-1}U(\nu_{u,z}),
\frac{A_m(\lambda)}{W_m(\lambda)}= \frac{\pi}{2}\, \left(i + \frac{\pi \, \cot(|m+\alpha|\, \pi)\, \lambda^{|m+\alpha|}}{4^{|m+\alpha|}\, |m+\alpha|\, \Gamma^2( |m+\alpha|)}\ \left(\frac{a_m'- |m+\alpha|\, a_m}{a_m'+ |m+\alpha|\, a_m}\right) \right)\, \left(1+\mathcal{O}(\lambda)\right),
S^L = \{i_1, \dots, i_p \}\cap \{i \mid \beta_i \in L\}, T^L = \{i \mid \langle \lambda, \beta_i \rangle > 0,\ \beta_i \in L\}, U^L = \{i \mid \langle \lambda, \beta_i \rangle < 0,\ \beta_i \in L\}.
u_nv=0\ \ \ \ \{\rm for}\ \n\ \{\rm sufficiently\large}; & Y({\bf 1},z)=Id_{V}; & Y(v,z){\bf 1}\in V[[z]]\ \ \{\rm and}\ \ \ \lim_{z\to0}Y(v,z){\bf 1}=v; & [L(m),L(n)]=(m-n)L(m+n)+\frac{1}{12}(m^3-m)\delta_{m+n,0}c ; & \frac{d}{dz}Y(v,z)=Y(L(-1)v,z); & L(0)|_{V_n}=n
D_{n}*M_{n,k}=\int_{X_{1}\in=]-\infty,+\infty[}X_{1}^{k}dX_{1}\phi(X_{1})(\int_{X\in=]-\infty,X_{1}]}\phi(X)dX)^{n-1}=\int_{-\infty}^{+\infty}X^{k}\Phi(X)^{n-1}\phi(X)dX
\displaystyle \int_{\Omega}B(x) |\nabla u(x)|^{p(x)-2}\nabla u(x)\nabla v(x)dx + \displaystyle \int_{\Omega}A(x)|u(x)|^{p(x)-2}u(x)v(x)dx + \displaystyle \int_{\Omega}D(x)|u(x)|^{p(x)-1}u(x)v(x)dx+\displaystyle \int_{\Omega}C(x)|u(x)|^{p(x)-3}u(x)v(x)dx &-\int_{\Omega}b(x) |u(x)|^{q(x)-2}u(x) v(x)dx=0,
| \delta e |^{2}= \int d \lambda ({\hat{e}}( \lambda ; l ) )^{- 1}( \delta e ( \lambda ; l ) )^{2}= \int d \lambda [ -{\hat{e}}( \lambda ) ( \delta f ){\frac{d^{2}}{d \lambda^{2}}}( \delta f ) +{\frac{( \delta \rho [ f ( \lambda ) ] )^{2}}{{\hat{e}}}}] ,
G_N(1) =\frac{1}{N}\sum^{N-1}_{n=1}\frac{1-\cos2\phi_n}{(1-\cos2\phi_n)^2}&=\frac{1}{2N}\sum^{N-1}_{n=1}\frac{1}{\sin^2 \phi_n}=\frac{N}{6}-\frac{1}{6 N}
\int_{\Omega_\ell}|v^\varepsilon_\ell|^2 = \int_{-\ell}^\ell\int_{\omega}w_1^2 - 2x_1w_1G_\varepsilon+ \left( x_1G_\varepsilon\right)^2\geq 2\ell \int_{\omega}w_1^2= 2\ell,
\min_u\ \ J(x,u)=\int_0^T \mathcal{L}(t,x(t),u(t))dt,{\rm s.t.}\ \ \dot{x}(t) = f(t,x(t),u(t)), & e(x(0),x(T)) = 0, & g(x(t),u(t))\leq 0,
\mathrm{GW}_{1,J}^{\beta}(\eta_p,\mu_0;\mu)=\langle \psi^p \mu_0 \prod_{j\in J}\mu_j, \prod_{j\not\in J}\psi_j^{a_j}\mathrm{ev}_j^* \mu_j\rangle_{0,1+[k]-J,\beta}.
\begin{array}{c c c}{{\chi^{a}( \sigma ) = \frac 1{\sqrt{2 \pi}}\sum_{r \in Z_{0}+ \phi}\left( c_{r}\cos{r \sigma}+ d_{r}\sin{r \sigma}\right)}}&{{}}&{{}}\\{{\overline{{{\chi}}}^{a}( \sigma ) = \frac 1{\sqrt{2 \pi}}\sum_{r \in Z_{0}+ \phi}\left( \overline{{{c}}}_{r}\cos{r \sigma}+ \overline{{{d}}}_{r}\sin{r \sigma}\right)}}&{{}}&{{}}\\ \end{array}
P_{\textrm{out,N}}^{\textrm{CSANC}}=\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{NDN=0}\right\rbrace=\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{NDF=0}\right\rbrace+\textrm{Pr}^{\textrm{NOMA}}\left\lbrace \textrm{NDF=1,NDN=0}\right\rbrace.
h_z(a_1,a_2,\ldots,a_n) V(a_1,a_2,\ldots,a_n) = \det\begin{bmatrix}1 a_1 a_1^2 \cdots a_1^{n-2}a_1^{z+n-1}1 a_2 a_2^2 \cdots a_2^{n-2}a_2^{z+n-1}\vdots \vdots \vdots \ddots \vdots \vdots 1 a_n a_n^2 \cdots a_n^{n-2}a_n^{z+n-1}\end{bmatrix},
\frac{d^2\mathcal{T}(\alpha)}{d\alpha^2}=-\left. \frac{\eta_{\alpha\alpha}\eta_T^2 - 2 \eta_{\alpha T}\eta_{\alpha}\eta_T + \eta_{TT}\eta_{\alpha}^2}{\eta_T^3}\right|{}_{\left(\alpha,\mathcal{T}(\alpha)\right)},
{}~ \times \Bigg \{\sum_{\sigma \in Z_{N}}\frac{w ( x_{1 3}, x_{1}u , x_{5 8}| \sigma + j_{2}+ j_{3}) w ( x_{2}x_{6 7}, x_{2}v , x_{6}x_{2 4}| \sigma ) s ( \sigma , j_{1})}{w ( x_{5}x_{1 3}, \omega x_{1}u , \omega x_{1}x_{5 8}| \sigma + j_{2}) w ( x_{6 7}, x_{2}v , x_{2 4}| \sigma + i_{3})}\Bigg \}_{0}~ ~
w_{k l}^{+}{'^{\prime}}- \left[ \frac{l \left( l + 1 \right)}{r^{2}}- \frac{\kappa}{r}\frac{V^{\prime}}{2 m - V}+ \frac{1}{2}\frac{V^{\prime \prime}}{2 m - V}+ \frac{3}{4}\left( \frac{V^{\prime}}{2 m - V}\right)^{2}- V^{2}+ 2 m V - k^{2}\right] w_{k l}^{+}= 0 \, ,
\sigma_{N - 2 , 3}= \left( \begin{array}{c c c c c c c c c c}{{1}}&{{0}}&{{0}}&{{0}}&{{\ldots}}&{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\star}}&{{\star}}&{{\star}}&{{\ldots}}&{{\star}}&{{1}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right)
d s_{( \; , 1 )}^{2}= - \int_{s_{2}}^{s_{3}}\sum_{a , b = 1}^{8}\sum_{\mu}{y}_{\mu}^{a}{y}_{\mu}^{b}d s \, \rho_{a}\rho_{b}= - \int_{s_{2}}^{s_{3}}\sum_{\mu}d{t}_{\mu}d{t}_{\mu}.
\begin{gathered}\int_x^\infty \rho^{-2\alpha-\nu+1}({x^2}/{\rho^2}; q^2)_{-\alpha}\int_0^\infty C_2(u) J_\nu (u\rho; q^2) \,d_q u \,d_q \rho = \int_x^\infty g_2 (\rho) \rho^{-2\alpha-\nu+1}({x^2}/{\rho^2}; q^2)_{-\alpha}\,d_q\rho. \end{gathered}
f(v_1,v_1,v_2)=0, f(v_1,v_2,v_3)+f(v_2,v_3,v_1)+f(v_3,v_1,v_2)=0, f(v_1,v_2,[u_1,u_2,u_3]_T)+D_T(v_1,v_2)f(u_1,u_2,u_3) = f([v_1,v_2,u_1]_T,u_2,u_3)+f(u_1,[v_1,v_2,u_2]_T,u_3)+f(u_1,u_2,[v_1,v_2,u_3]_T) +\theta_T(u_2,u_3)f(v_1,v_2,u_1)-\theta_T(u_1,u_3)f(v_1,v_2,u_2)+D_T(u_1,u_2)f(v_1,v_2,u_3).
X^{(k),\pm}_{r+1,n}:=\sum_{s_0,\ldots,s_{r}\in \{0,1\}}^{s_0+\ldots+s_r=k}[e_{n\pm s_0},[e_{n+2\pm s_1},\cdots,[e_{n+2r-2\pm s_{r-1}},e_{n+2r \pm s_r}]_{q^{-4}}\cdots]_{q^{-2r}}]_{q^{-2r-2}}\,.
H_{0}^{\prime}= \frac{1}{2}\int d^{3}x \left( B_{a}^{i j}F_{i j}^{a}- A_{i}^{a}A_{a}^{i}+ \left( \partial^{i}B_{0 i}^{a}\right) \left( \partial_{j}B_{a}^{0 j}\right) \right) \equiv \int d^{3}x h_{0}^{\prime},
Y_1 =ha_{11}f(Y_1)+ha_{12}f(Y_2)+u_{11}y_1^{[m-1]}+u_{12}y_2^{[m-1]}Y_2 =ha_{21}f(Y_1)+ha_{22}f(Y_2)+u_{21}y_1^{[m-1]}+u_{22}y_2^{[m-1]}y_1^{[m]}=hb_{11}f(Y_1)+hb_{12}f(Y_2)+y_1^{[m-1]}y_2^{[m]}=hb_{21}f(Y_1)+hb_{22}f(Y_2)-y_2^{[m-1]}
0 = \int^a_b \left[{Z^{\rm C}_\alpha}(L) w^\alpha +{Z^{\rm V}_\alpha}(L) (\delta{\rm v}^\alpha + R^\alpha_{\beta\gamma}{\rm v}^\gamma w^\beta) + \delta{\rm p}_\alpha({\rm u}^\alpha -{\rm v}^\alpha) +{\rm p}_\alpha \delta{\rm u}^\alpha -{\rm p}_\alpha \delta{\rm v}^\alpha \right]dt.
\mathbb{I}(a_0;a_1,\ldots,a_N;a_{N+N'+1})\mathbb{I}(a_0;a_{N+1},\ldots,a_{N+N'};a_{N+N'+1}) &= \sum_{\sigma\in\Sigma(N,N')}\mathbb{I}(a_0;a_{\sigma^{-1}(1)},\ldots,a_{\sigma^{-1}(N+N')};a_{N+N'+1}),
q^{\tfrac{p-1}{2}}(-1)^{\tfrac{s(p-1)}{2}}\sum_i (-1)^{\tfrac{r-i}{2}}\binom{(i-s)\tfrac{(p-1)}{2}-1}{\tfrac{pi-r-s(p-1)}{2}}Q_{(r+ps-pi)(p-1)}Q_{i(p-1)}(x).
\frac{1+a}{1+\phi_a}=1+\frac{a-\phi_a}{1+\phi_a},\,\,\,\mathrm{where}\,\,\, \left|\frac{a-\phi_a}{1+\phi_a}\right|\leq \frac{|\psi_a|\cdot\Delta t}{\frac{\Delta t}{\eta_4}}\leq\frac{\eta_4\cdot\epsilon^{-1}}{\eta_1\cdot |I|}
\exp\Big{\{}-M\eta\sum_{j=1}^W Tr (\Re B_j)J\Big{\}}\leq \exp\Big{\{}-2M\eta\sum_{j=1}^W \frac{\Re b_{j,1}+\Re b_{j,2}}{1+2t_j^2}(1+2t^2)\Big{\}}&\leq \exp\Big{\{}-cM\eta \sum_{j=1}^W \frac{r_{j,1}+r_{j,2}}{1+2t_j^2}(1+2t^2)\Big{\}},
\vert\vert E(\lambda_N)-E(1)\vert\vert^2_2 \ge \sum_{n=N}^{3N/2}\frac{\vert \lambda_N^n -1\vert^2}{n^{2\kappa}}&\ge C^2\sum_{n=N}^{3N/2}\frac{1}{n^{2\alpha+1}}&\ge C^2\int_{N}^{3N/2+1}\frac{dt}{t^{2\alpha+1}}=\frac{C^2}{2\alpha}\bigg(\frac{1}{N^{2\alpha}}-\frac{1}{(3N/2+1)^{2\alpha}}\bigg)
\mathrm{RUC}((x_n)_n):= \inf\{C>0: \|\sum_{i=1}^n a_i e_i\|\ge \frac 1 C \mathbb{E}\Big(\Big\|\sum_{i=1}^n \epsilon_i a_i x_i \Big\| \Big)\}, \mathrm{RUD}((x_n)_n)):= \inf\{C>0: \|\sum_{i=1}^n a_i x_i\|\leq C\mathbb{E}\Big(\Big\|\sum_{i=1}^n \epsilon_i a_i x_i \Big\| \Big)\},
\widehat{b}^\dagger_N =1-2 \widehat{\alpha}^\dagger_N, \widehat{a}^\dagger_N =\frac{\widehat{C}_N(q)}{\sqrt{N}}\bigg|_{q=e^{-\pi/\sqrt{3N}}}, \intertext{and}\widehat{X}^\dagger_N =4\widehat{\alpha}^\dagger_N\left(1-\widehat{\alpha}^\dagger_N\right).
(1\; 2\; 3)\cdot \frac{\xi_3}{\xi_4}=\frac{\xi_1}{\xi_4}= \frac{\xi_1}{\xi_2}\frac{\xi_2}{\xi_3}\frac{\xi_3}{\xi_4}=\omega^{-2}\frac{\xi_3}{\xi_4}= \omega f_3
G_p(\sigma)>\frac{1}{\sigma\alpha_p-G_{p+1}(\sigma)}>\frac{1}{\sigma\alpha_p-\frac{1}{\sigma\alpha_{p+1}-G_{p+2}(\sigma)}}&>\dots>\frac{1}{\sigma\alpha_p-\frac{1}{\sigma_{p+1}\alpha_{p+1}-\frac{1}{\sigma_{p+2}\alpha_{p+2}-\dots}}}=F_p(\sigma),
S = \int d^{4}x \sqrt{- \tilde{g}}\left\{\tilde{R}- 2 \tilde{g}^{a b}\nabla_{a}\Phi \nabla_{b}\Phi -{\frac{1}{2}}e^{- 2 \Phi}\tilde{g}^{a e}\tilde{g}^{b f}F_{a b}F_{e f}\right\}.
\int_0^\infty\frac{(e^{-\lambda_{1}s}-e^{-\lambda_{2}s})_+}{\sqrt{s}}\Big[\int_0^\infty \tau e^{-\tau^2}d\tau\Big]ds =\frac{\sqrt{\pi}(\sqrt{\lambda_2}-\sqrt{\lambda_1})_{+}}{2\sqrt{\lambda_2\lambda_1}}
{\operatorname{Rest}}_{x_n=0}\circ ({\operatorname{id}}\oplus \iota_{\frac{\partial}{\partial x_n}}) \colon{\mathcal{E}}^i({\mathbb{R}}^n) \to{\mathcal{E}}^i({\mathbb{R}}^{n-1}) \oplus{\mathcal{E}}^{i-1}({\mathbb{R}}^{n-1})
*R*^{\mu \nu}{}{}_{\rho \sigma}=-R^{\mu \nu}{}{}_{\rho \sigma}+\delta^\mu_\sigma R^\nu_\rho +\delta^\mu_\rho R^\nu_\sigma-\delta^\mu_\sigma R^\nu_\rho-\delta^\nu_\rho R^\mu_\sigma +\frac{1}{2}R(\delta^\nu_\rho \delta^\mu_\sigma-\delta^\mu_\rho \delta^\nu_\sigma ).
S O ( 1 , 1 ) \times \frac{S O ( 1 , p )}{S O ( p )}\times \frac{S O ( 4 , 5 - p )}{S O ( 4 ) \times S O ( 5 - p )}\qquad ( n_{V}= p + 1 \, , \, n_{H}= 5 - p ) , \quad 0 \leq p \leq 5 .
\begin{array}{ll}C_0:=\max \{3^{p+q}ACM^{p+q-1},3^rBCM^{r-1}, &\quad2^{p+q+3}3^{p+q-1}(p+q)ACM^{p+q-1}, 2^{r+3}3^{r-1}rBCM^{r-1}, &3^{p+q}(p+q)ACM^{p+q-1},3^rrBCM^{r-1}, &\quad2\cdot3^{p+q-1}pACM^{p+q-1}\}.\end{array}
\Lambda_2 = z_2 ( v_2 \xi + u_2 ) - y_2 = z_2 \left( s \epsilon_k + (-1)^{k - 1}( s \ell + s' ) \epsilon_{k - 1}\right) - y_2 = (-1)^{k - 1}z_2 s \epsilon_{k - 1}( \ell - \rho ),
\int_0^1x^{\mu-1}(1-x^r)^{\nu-1}{\rm log}x{\rm d}x = \frac{1}{r^2}B\left(\frac{\mu}{r},\nu \right) \left(\psi\left(\frac{\mu}{r}\right) - \psi\left(\frac{\mu}{r}+\nu\right)\right)
{\cal O}_{1 / 2}^{[ m , 0 , 0 ]}= [ W^{1}( \theta_{2 , 3 , 4}, \bar{\theta}^{1}) ]^{m}\quad \mathrm{a n d}\quad{\cal O}_{1 / 2}^{[ 0 , 0 , m ]}= [ \bar{W}_{4}( \theta_{4}, \bar{\theta}^{1 , 2 , 3}) ]^{m}\, .
W^1_n = \widetilde{W}^1_n \cup Z^1_n \cup Z^2_n, \W^2_n = Z^3_n \cup Z^4_n, \widetilde{W}^2_n = Z^1_n \cup Z^3_n, \ \widetilde{W}^3_n = Z^2_n \cup Z^4_n.
\epsilon^{\lambda \mu \nu \rho}\mathbf{F}_{\lambda \mu}\mathbf{F}_{\nu \rho}\longrightarrow \epsilon^{\lambda \mu \nu \rho}\left\{X_{\lambda}( \sigma ) \ , X_{\mu}( \sigma ) \right\}_{\mathrm{P B}}\left\{X_{\nu}( \sigma ) \ , X_{\rho}( \sigma ) \right\}_{\mathrm{P B}}\equiv 0
G_D [x_2, x_1] = \sum_{n_1}\sum_{n_2}...\sum_{n_{D-1}}\,e^{i(n_1 \theta_1 + n_2\theta_2 ...+n_{D-1}\theta_{D-1})}\, G_{n_1 n_2...n_{D-1}}[x_2, x_1],
\beta_m\leq\frac{2}{B_m}\sum_{j=1}^m v_j\mathbf{P}(T_g>j-1)\le \frac{2C_2\mathbf{E}Z^*_m}{B_m^{1-1/3}}\sum_{j=1}^m \frac{v_j}{B_j^{4/3}}.\le \frac{2 C_2\mathbf{E}Z^*_m}{B_m^{2/3}}\sum_{j=1}^m \frac{v_j}{V_j^{2/3}}.
\bar{D}D \Phi_{2}= i \mu_{f}\nu_{f}e^{\Phi_{2}- \Phi_{1}}, \qquad \bar{D}D \Phi_{1}= e^{2 \Phi_{1}- \Phi_{2}}\bar{\Psi}^{+ b}\Psi^{- b}, \qquad \bar{D}\Psi^{- b}= 0 = D \bar{\Psi}^{+ b}.
V_{4m+4n}Q_{4m+4n}W_{4m+4n}B\begin{pmatrix}a 0 0 b \end{pmatrix}W_{4m+4n}^* Q_{4m+4n}^* V_{4m+4n}^* &= \begin{pmatrix}Q_{4m}W_{4m}B(a) W_{4m}^* Q_{4m}^* 0 0 Q_{4n}W_{4n}B(b) W_{4n}^* Q_{4n}^* \end{pmatrix}\; ,
S_{e f f}= \frac{1}{2}( E^{a}E^{b}- B^{a}B^{b}) \left( \delta^{a b}+ \frac{2 2}{3}\frac{g^{2}}{( 4 \pi )^{2}}\ln \left( \frac{1}{\mu^{2}}\sqrt{2 g^{2}( B^{a}B^{b}- E^{a}E^{b})}\right) \right)
( \mathrm{M a s s})^{2}\geq{\frac{1}{1 6}}( M^{( 0 )}+ L )_{a b}( \alpha_{0}^{a}\beta_{0}^{a}){\cal M}^{( 0 )}\left( \begin{matrix}{\alpha_{0}^{b}}\\{\beta_{0}^{b}}\\ \end{matrix}\right) ,
\iiint \delta(x-y)\delta(y-z) \varphi(x,y,z)\d x \d y \d z= \int \varphi(x,x,x) \d x &\frac{\mathcal{P}}{x-\omega}\frac{\mathcal{P}}{y-\omega}=\frac{1}{y-x}\left(\frac{\mathcal{P}}{x-\omega}-\frac{\mathcal{P}}{y-\omega}\right) + \pi^2 \delta(x-\omega)\delta(y-\omega)
-\nabla_x \log \rho_{0|t}(x\,|\,y) = -\nabla_x \log \rho_0(x) - \nabla_x \log \rho_{t|0}(y\,|\,x) &~~~~~~ = -\nabla_x \log \rho_0(x) + \frac{\alpha e^{-\alpha t}}{1-e^{-2\alpha t}}(e^{-\alpha t}x-y).
I_{1,\ast}(t)+\Vert\varphi(t)\Vert^2_{L^2(\mathbb R)}\le \frac{C_3}{\varepsilon}\int_0^t(I_{1,\ast}(s)+\Vert\varphi(s)\Vert^2_{L^2(\mathbb R)})ds+C_1\varepsilon I_{1,\ast}(t)+\frac{C_3}{\varepsilon}\Vert{\mathbf F}\Vert^2_{H^1_\ast(\Omega_t)}\,,
\mathbf{R}^{(g)}_{code}(l) =\sum_{\left\{m \; \vert \beta_m^l > 0\right\}}\beta_m^l \boldsymbol{\phi}_m^l \left[\boldsymbol{\phi}_m^l\right]^H \mathbf{SNR}^{(g)}_{mimo}(l) =\boldsymbol{\Gamma}_l\operatorname{diag}\left[\left\{\lambda_n^l\right\}_{n=1}^{D}\right]\left(\boldsymbol{\Gamma}_l\right)^{-1}
\|\Phi_{\#}^y\|^2_{\#}= 4 \alpha \|(h_{\alpha}- e_{\alpha}+H_f^+)^{-1/2}Pu_{\alpha}\otimes A_{y}^+\Omega\|^2 &=4 \alpha \int_{\mathbb{R}^3_+\times \mathbb{R}^3}\mathrm{d}k\, \mathrm{d}x \; \left| \frac{Pu_{\alpha}}{(h_{\alpha}-e_{\alpha}+|k|)^{1/2}}\lambda_y(k) \right|^2+O(\alpha^3 L^2 e^{-L}).
\begin{aligned}\max_{\mathbf{W}_k}~ \sum_{m=1}^{M_{MS}}||(\mathbf{w}_k^{(m)})^H\mathbf{H}_k||_1^2 s.t.~ |\mathbf{W}_k^{(i,j)}| = \frac{1}{\sqrt{N_{MS}}},~\forall i,j.\end{aligned}
\begin{array}{c}{\psi = \psi_{0}\{t h [ \frac \gamma 2 ( r_{+}- R ) ] - \epsilon / \lambda \}, z < 0 ,}\\{\psi = \psi_{0}\{t h [ \frac \gamma 2 ( r_{-}- R ) ] - \epsilon / \lambda \}, z > 0 ,}\\{r_{\pm}= \sqrt{x^{2}+ y^{2}+ \left( z \pm b \right)^{2}}, b > R .}\\ \end{array}
\begin{array}{l l}{D_{\mu}\phi = ( \nabla_{\mu}- i \frac{q}{2 \sqrt{\beta}}H_{\mu}) \phi ,}{D_{\mu}\sigma = ( \nabla_{\mu}+ i \frac{e}{2}A_{\mu}- i \frac{\alpha e}{2 \sqrt{\beta}}H_{\mu}) \sigma ,}\end{array}
\lim_{\ell \rightarrow 0}\frac{1}{\sqrt{2\ell+1}}{\mathbf D}^{-}(E, \ell) =Q^{-}(E) = \sqrt{\frac{2\pi}{i}}{\rm Ai}(-E), \lim_{\ell \rightarrow 0}\frac{1}{\sqrt{2\ell+1}}{\mathbf D}^{+}(E, \ell) =Q^{+}(E)=- \sqrt{\frac{2\pi}{i}}\frac{d}{dE}{\rm Ai}(-E) .
C'_{k_1,j_1}= C_{k_1,j_1}\setminus A_1,C'_{k_2,j_2}= C_{k_2,j_2}\setminus A_2, C'_{k_1,j_2}= C_{k_1,j_2}\cup A_1,C'_{k_2,j_1}= C_{k_2,j_1}\cup A_2.
\eta_1 = \frac{1}{2}\left( -\alpha_3 \pm \sqrt{\alpha_3^2 - \frac{4\alpha_3 \eta_3^2}{y_1+\alpha_0-y_2}}\right), \eta_2 = \frac{-\eta_1(y_1+\alpha_0-y_2)}{\alpha_3}.
ch(J_2(\eta^-)) = ch(\eta^- \otimes (1 \oplus \omega_p \oplus Sym^2 \omega_p))= e^{\frac{1}{2}\psi}\cdot (1 +e^\psi +e^{2\psi})= 3 +\frac{9}{2}\psi +\frac{35}{8}\psi^2 +\frac{51}{16}\psi^3 +\!\cdots
\{J^{L}( x ){}_{,}^{\otimes}J^{L}( y ) \}= u ( x ) T_{\rho}u^{- 1}( x ) \otimes u ( y ) T_{\sigma}u^{- 1}( y ) \; \partial_{x}\partial_{y}{\cal F}_{\rho \sigma}( x , y ) \quad .
\sum_{\ell=0}^{N}R^{\eta^{-\ell}}\leq\sum_{\ell\geq0:R^{\eta^{-\ell}}>2}R^{\eta^{-\ell}}+\sum_{\ell\leq N:R^{\eta^{-\ell}}\leq2}R^{\eta^{-\ell}}& \leq R\sum_{j=0}^{\infty}\left( 2^{1-\eta}\right)^{j}+2\left(N+1\right) .
\theta_z =\big((z-q_c)^{s_c},\ldots,(z-q_{b_z})^{s_{b_z}}\big), \Omega_z =\Big\{1+p^z\delta_{n-r}+{\sum}_{i=1}^rp^{d'_i}\colon (d'_1,\ldots,d'_r)\le\theta_z,\d'_r=0\Big\}.
P_s=\sum\limits_{i=1}^N \frac{w_i e^{-\rho_I \delta_0 (-1 +{}_2F_1(-\frac{1}{\alpha_I}, \hat{m}, 1-\frac{1}{\alpha_I}, -\frac{T R^{2 \alpha_d}\gamma_0}{c_0 t_i \overline{g_p}\hat{m}\delta_0^{\alpha_I}}))}}{e^{\frac{T \sigma^2 R^{2 \alpha_d}}{c_0 t_i \overline{g_p}}}}
\mathcal{D}= \big\{g(\langle \cdot, y^*\rangle) \colon g \in C^2_c(\mathbb{R}), y^* \in D(A^*)\big\}, \mathcal{B}= \big\{g (\langle \cdot, y^*\rangle )\colon g \in C^2_b(\mathbb{R}), y^* \in D(A^*) \big\}.
\left\langle m_{\Lambda}^{*}E^{ij}_0, f \otimes g \right\rangle \equiv \Lambda^{- 2}\left\langle E^{ij}_0, ({m_{\Lambda}}_*f) \otimes ({m_{\Lambda}}_*g) \right\rangle = \left\langle E^{ij}_0, f \otimes g \right\rangle
f_2 =-Jf_4=a^3e_1+a^4e_2-a^1e_3-a^2e_4, f_3 =Rf_4=-a^2e_1+a^1e_2+a^4e_3-a^3e_4, f_1 =-Rf_2=a^4e_1-a^3e_2+a^2e_3-a^1e_4\,.
\partial_t |{f}|^2+2\nu\gamma(t) |{f}|^2= 2((2\pi ik{f}_{1}+{f}_{2}+{f}_{3})\overline{f})\leq \gamma(t) |f_1|^2+\gamma(t)^{-1}|2\pi ikf|^2 +\nu^{\frac{1}{3}}|f|^2+\nu^{-\frac{1}{3}}|f_2|^2+\nu\gamma(t) |{f}|^2+\nu^{-1}\gamma(t)^{-1}|{f}_3|^2,
T_{n_s}\theta(\lambda_s^n) - \theta(n_s(\lambda_s)^n)T_{n_s}=\sum_{k = 0}^{n - 1}\theta(n_s(\lambda_s)^k\lambda_s^{-k})(\lambda_s^k\cdot c_{n_s})\theta(\lambda_s^n) & = \sum_{k = 0}^{n - 1}(n_s(\lambda_s)^k\cdot c_{n_s})\theta(n_s(\lambda_s)^k\lambda_s^{n-k}).
-\dim H^2(\widehat{G},M)= -\dim H^2(\widehat{G}_1,M)+\dim H^2(\widehat{C},M) -\dim H^1(\widehat{G},M) +\dim H^1(\widehat{G}_1,M) -\dim H^1(\widehat{C},M)+\dim H^0(\widehat{G},M) -\dim H^0(\widehat{G}_1,M)+\dim H^0(\widehat{C},M).
S(z) \begin{pmatrix}1 0 0 0 z^{-\beta}0 0 0 z^{-\beta}\end{pmatrix}D_0^{-n}(z) = \mathcal{O}\begin{pmatrix}1 h_{-\alpha-\frac{1}{2}}(z) h_{-\alpha-\frac{1}{2}}(z) 1 h_{-\alpha-\frac{1}{2}}(z) h_{-\alpha-\frac{1}{2}}(z) 1 h_{-\alpha-\frac{1}{2}}(z) h_{-\alpha-\frac{1}{2}}(z)\end{pmatrix},
\begin{aligned}\eta_m=(KU_1^{k-m-1}e, U_2^{l-m}\bar e)=(C K^\ast C \, U_1^{m-l}e, U_2^{m-k+1}\bar e) = -(K \, U_1^{m-l}e, U_2^{m-k+1}\bar e)=-\eta_{k+l-1-m},\end{aligned}
a_1S(a_2) = a_1(a_2\rightharpoonup T(a_3))=a_1\circ (T(a_2)\rightharpoonup (a_3\rightharpoonup T(a_4)) &=a_1\circ ( (T(a_2)\circ a_3)\rightharpoonup T(a_4))=a_1\circ T(a_2)=\epsilon(a)1
\alpha_1 = \frac{q^{- l_1 + 1/2}}{t_1}, \alpha_2 = \frac{q^{-l_2 +3/2}}{t_2},q^{\lambda}=q^{(h_1+h_2-l_1-l_2-k_1 +k_2 +1)/2}& \beta_1 = \frac{q^{(-h_1+h_2-l_1-l_2-k_1 +k_2)/2 +1}}{t_1}, \beta_2 = \frac{q^{(h_1- h_2-l_1-l_2-k_1 +k_2)/2 +1}}{t_2}.
\left( \begin{array}{c c}{{\hat{\alpha}( p )}}\\{{\hat{\beta}( p )}}\\ \end{array}\right) = \left( \begin{array}{c c}{{a ( p )}}\\{{1}}\\ \end{array}\right) \hat{\xi}( p ) + \left( \begin{array}{c c}{{b ( p )}}\\{{1}}\\ \end{array}\right) \hat{\eta}( p )
{\rm Ric}(g)(X,Y)={\rm Tr}\{Z\mapsto R(Z,X)Y\}\,\,{\rm and}\,\,{\rm Ric}(\widehat{g})(\widehat{X},\widehat{Y})={\rm Tr}\{\widehat{Z}\mapsto\widehat{R}(\widehat{Z},\widehat{X})\widehat{Y}\},
M_j = \left[\begin{array}{ccc}H_{uu}(U_{j-1},x_j,t_j) H_{u\mu}(U_{j-1},x_j,t_j) M_{13}H_{\mu u}(U_{j-1},x_j,t_j) H_{\mu\mu}(U_{j-1},x_j,t_j) M_{23}M_{31}M_{32}M_{33}\end{array}\right],
0 = 2^n\sum_{h=0}^{m-1}a_{1,h}b_0(1+2h,1) + \sum_{h=0}^m a_{0,h}b_m(0+2h,0) (q^m) &\cdots &= 2^n\sum_{h=0}^{m-1}a_{1,h}b_{m-1}(1+2h,1) + \sum_{h=0}^m a_{0,h}b_{2m-1}(0+2h,0) (q^{2m-1})
K_{j\bar k}(\zeta,\bar\eta)=\langle\langle \cdot\bar\eta_{\bar k}, \cdot\bar\zeta_{\bar j}\rangle\rangle +\langle\langle [L_j,L_{\bar k}]\cdot\bar\eta,\cdot\bar\zeta\rangle\rangle -i^{n^2}\int_{D_t}\{L_j\cdot\bar\eta,L_k\cdot\bar\zeta\}.
S_{0}= \int d^{2}x \Big ( i \overline{\psi}\gamma^{\mu}\partial_{\mu}\psi - g_{0}\overline{\psi}\gamma^{\mu}\gamma_{5}\psi \partial_{\mu}\phi +{\frac{1}{2}}\partial_{\mu}\phi \partial^{\mu}\phi -{\frac{m^{2}}{2}}\phi^{2}\Big )
\partial_t^j \partial_x^n \partial_y^m J_0(t,x,y;\nu) = \sum\limits_{l=1}^{+\infty}\frac1{2\pi}\int_{\mathbb R}(i\theta)^j \frac{r_1r_2^n e^{r_2x}-r_2r_1^ne^{r_1x}}{r_1-r_2}\widehat\nu(\theta,l)\,d\theta\, \psi_l^{(m)}(y)
\mathcal{A}(x)=\frac{\zeta(3/2)}{\zeta(3)}x^{1/2}+\frac{\zeta(2/3)}{\zeta(2)}x^{1/3}+O(x^{1/6}\cdot \mathrm{e^{-c(\log^{3}x/\log\log x)^{\frac{1}{5}}}})
B = -3 \int \partial_xu(\partial_x^4u)^2 \chi(x+\nu t) \; dx + 3 \int u(\partial_x^4u)^2 \chi'(x+\nu t) \; dx &+ \int (\partial_x^3u)^3\chi(x+\nu t) \; dx- \int u(\partial_x^3u)^2\chi'''(x+\nu t) \; dx.
J_{o r b}= \sum \varepsilon_{i j}R_{p}^{i}\frac{\partial L_{e f f}^{( 1 )}}{\partial \dot{R}_{p}^{j}}= - 4 \pi \kappa \sum_{p}R_{p}^{i}\varepsilon_{i j}a_{j}( \vec{R}_{p}) .
A_\mu=\mp\,{2i\over g}\sum_{I=1}^K\,[\,H_I\,{v_\mu^I\over\rho_I}+g\,\kappa\,E^\pm_{I\,K+1}\,R_\mu^I\,\prod_{I=1}^{K-1}\delta(R^K\cdot R^I)\,].
{{dr}\over{d\lambda}}={{dr}\over{ds}}{{ds}\over{d\widetilde{\lambda}}}{{d\widetilde{\lambda}}\over{d\lambda}}= (4^{(1/\beta)}cosh^{\left({2/\beta}\right)}({{\beta r}\over{2L}}))(C^*)(4^{(-2/\beta)}sech^{\left({4/\beta}\right)}({{\beta r}\over{2L}})) = Csech^{\left({2/\beta}\right)}({{\beta r}\over{2L}}),
\sum_{i=0}^{n}\vert \omega_{i,n}\vert \leq \int_{0}^{T}\sum_{i=0}^{n}\bigg\vert \dfrac{\frac{\beta_{i}}{t-t_{i}}}{\sum_{i=0}^{n}\frac{\beta_{i}}{t-t_{i}}}\bigg{\vert}\mathrm{d}t = \int_{0}^{T}\Lambda_{n}(t)\mathrm{d}t,
[\begin{matrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{matrix}]^{T}[\begin{matrix}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{matrix}]=[\begin{matrix}1&0&0\\ 0&1&0\\ 0&0&1\end{matrix}],
\begin{array}{l}{{}^{A}( \lambda \Phi + \mu \Psi )^{( n )}= \lambda{}^{A}\Phi^{( n )}+ \mu{}^{A}\Psi^{( n )}, \quad \forall \lambda , \mu \in C ,}\\{({}^{A}\Phi ,{}^{A}\Psi ) = \S_{n = 0}^{\infty}({}^{A}\Phi^{( n )},{}^{A}\Psi^{( n )}) .}\\ \end{array}
H_{i}^{\pm}(u)=k_{i+1}^{\pm}(u+\frac{1}{2}hi)k_{i}^{\pm}(u+\frac{1}{2}hi)^{-1},K^{\pm}(u)=\prod_{i=1}^{n}k_{i}^{\pm}(u+(i-\frac{n+1}{2})h), &E_{i}(u)=\frac{1}{h}X_{i}^{+}(u+\frac{1}{2}hi),F_{i}(u)=\frac{1}{h}X_{i}^{-}(u+\frac{1}{2}hi).
\frac{d}{ds}e^{isH^{(\Lambda)}}A^{(\Omega)}(t-s)e^{-isH^{(\Lambda)}}=\frac{d}{ds}e^{isH^{(\Lambda)}}e^{i(t-s)H^{(\Omega)}}Ae^{-i(t-s)H^{(\Omega)}}e^{-isH^{(\Lambda)}}&= ie^{isH^{(\Lambda)}}[(H^{(\Lambda)}-H^{(\Omega)}),A^{(\Omega)}(t-s)]e^{-isH^{(\Lambda)}}.
\cos \theta=\frac{\|\mathbf z-\mathbf w_0\|^2+\|\mathbf w_0-\mathbf o\|^2-\|\mathbf z-\mathbf o\|^2}{2\|\mathbf z-\mathbf w_0\|\|\mathbf w_0-\mathbf o\|}=\frac{1+x^2-(k+1)/2k}{2x}=\frac{k-1}{4kx}+\frac{x}{2}=:g(x).
\|\phi\|_{L^{p}(\mathbb C^n)}\le \frac{C_n|\Omega|^{\frac1{2n}}}{p-1}\sum_{j=1}^n \|\partial \phi/\partial\bar{z}_j\|_{L^p(\mathbb C^n)},\ \ \ \forall\,\phi\in C^1_0(\mathbb C^n),
\mathcal{T}_{12}(s, t) = \chi_{\{s \leq t\}}g_z(s) g_z(t), \mathcal{T}_{21}(s, t) = \chi_{\{s \geq t\}}g_z(s) g_z(t), \mathcal{T}_{11}(s, t) = \mathcal{T}_{22}(s, t) = S_{22}(s,t) = 0.
B= \frac{(n-1)^2}{4 \pi^2 a^4}\frac{1}{4}\sum_{l=1}^{\infty}\int_0^{x_0}x^2\frac{3\nu^9-42x^2\nu^7+18x^4\nu^5-2x^6\nu^3}{(x^2+\nu^2)^6},
S^{(m)}_{s}(\theta )=\frac{\sinh \theta +i\cos \frac{\pi mp}{2}}{\sinh \theta -i\cos \frac{\pi mp}{2}}\prod^{m-1}_{l=1}\frac{\sin^{2}\left( \frac{m-2l}{2}\pi p-\frac{p}{4}+i\frac{\theta}{2}\right)}{\sin^{2}\left( \frac{m-2l}{2}\pi p-\frac{p}{4}-i\frac{\theta}{2}\right)}\,.
\eta^2 := \sum_{K\in \mathcal{T}_h}h_K^2 \|\Delta u_h - u_h +f\|^2_K + \sum_{e\in{\cal E}_h^i}h_e \|[\nabla_h u_h]\|^2_e +\sum_{e\in{\cal E}_h^2}h_e \|\nabla_h u_h\cdot n-g\|^2_e.