harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{c}\pi_x= p_x + \frac{\lambda}{2}\left( x \left( x p_x + y p_y\right)+ \left( x p_x + yp_y\right) x \right), \pi_y = p_y + \frac{\lambda}{2}\left( y \left( x p_x + y p_y\right)+ \left( x p_x + yp_y\right) y \right).\end{array}
\lim_{\ell \rightarrow \infty}\frac{1}{t_\ell}\left ( \int_X F(x+t_\ell e_j,y) d\mu(y) - \int_X F(x,y) d\mu(y)\right) = \lim_{\ell \rightarrow \infty}\int_X G_j(t_\ell,y)d\mu(y) = \int_X \partial_{x_j}F(x,y)d\mu(y).
\frac{a_{n}x^{n+1}}{n+1}+\frac{a_{n-1}x^{n}}{n}+...+\frac{a_{2}x^{3}}{3}+\frac{a_{1}x^{2}}{2}+a_{0}x+c=c+\sum_{i=0}^{n}\frac{a_{i}x^{i+1}}{i+1}
f_Y(y) = \frac{a}{r}\, \left( \frac{e^{-y}}{r}\right)^{-1-a}\, e^{-\left( \frac{e^{-y}}{r}\right)^{-a}}\, \left\lvert -e^{-y}\right \rvert & = \frac{a}{r}\, r^{a+1}\, e^{-y(-a-1) -y}\, e^{-r^a \, e^{ya}}& = a \, r^a \, e^{ya}\, e^{-r^a \, e^{ya}}& = b \, q \, e^{b y}\, e^{-q \, e^{b y}}
i \overline{{\psi}}_{R}\gamma_{\mu}\psi_{R}= i \overline{{\psi}}\gamma_{\mu}\frac{1 - \gamma_{3}}{2}\psi \longleftrightarrow \frac{1}{\sqrt{2 \pi}}( \partial_{\mu}\Lambda + \epsilon_{\mu \nu}\partial^{\nu}\Lambda ) .
\left(\|Q^{-1}\|_{l_1}\sum_{j=1}^N|x_j|\right)^2 \le \|Q^{-1}\|_{l_1}^2\left(\sum_{l=1}^N(1+l)^{-2s}\right)\|x\|^2_{\mathfrak{l}^s}&\le N\|Q^{-1}\|^2_{l_1}\|x\|^2_{\mathfrak{l}^s}.
V_{0}+ V_{v^{2}}\rightarrow - \frac{4 \alpha_{S}}{3 r}\left[1 - \frac{1}{2}\; \left( \bf v_{\rm 1}\cdot v_{\rm 2}+v_{\rm 1}\cdot \hat{r}\; v_{\rm 2}\cdot \hat{r}\right) \right].
P^{K}f^{\pi}_{u,v}= f^{\pi}_{u, E_{\pi}^{K}v}, P_{K}f^{\pi}_{u,v}= f^{\pi}_{E_{\pi}^{K}u, v}, Q_{K}f^{\pi}_{u,v}= f^{\pi}_{E_{\pi}^{K}u, E_{\pi}^{K}v}.
\begin{cases}\partial_t y(x,t)-\Delta y(x,t)=\chi_\omega\chi_{(\tau,T)}u(x,t), \mbox{in}\Omega\times (0,T), y(x,t)=0, \mbox{on}\partial \Omega\times (0,T), y(x,0)=y_0(x), \mbox{in}\Omega.\end{cases}
\oint_{C}\frac{w^{- 1}d w}{2 \pi \sqrt{- 1}}\frac{( 1 - r \zeta^{2}) ( w^{2}- q^{6}) ( w - r q^{4})}{( w - q^{2}\zeta^{2}) ( w - q^{4}\zeta^{2}) ( w - q^{4}\zeta^{- 1})}e^{R ( w ) + R ( q^{6}w^{- 1})}| 0 \rangle^{( 0 )}= ( \zeta \leftrightarrow \zeta^{- 1}) | 0 \rangle^{( 0 )}.
\frac{u_i'(\tilde{x}_t^i)}{\delta^i u_i'(\tilde{x}_{t+1}^i)}=\frac{u_j'(\tilde{x}_t^j)}{\delta^j u_j'(\tilde{x}_{t+1}^j)}, 1 \leq j < i; t=0,1,\ldots
\overline{\gamma_{i,j}}= \begin{cases}\pm [(q_i^1)_{\pm 1}] \; j = j_{\pm}(i) \; \; j_{-}(i) \not= j_{+}(i); [(q_i^1)_{+1}] - [(q_i^1)_{-1}] \; j = j_{-}(i) = j_{+}(i); 0 \; j \notin \{j_{-}(i), j_{+}(i)\}. \end{cases}
L^{\frac{2}{r}-\frac{2}{r_0}}\lambda_p^{\frac{2}{r}-\frac{2}{r_0}}< N(r)^{\frac{2}{r}-\frac{2}{r_0}}\to 1 r \to r_0^-, \mbox{with}r_0=\frac{4}{\alpha-1},
\dim H \geq x-\lfloor\tilde\varepsilon x\rfloor+\ell_{i+1}''-\lfloor\tilde\varepsilon x\rfloor+\sum_{j=1}^i \big(\ell_{j}-\lfloor 2^{j-i-1}\tilde \varepsilon x\rfloor\big) &\geq r+x-\ell_{i+1}'-3\tilde\varepsilon x &\geq r+x-(1+\tilde\varepsilon)^{-i-1}(r+x).
J(g)= -\mu_1+\big((w+\bar{w})\partial_{\eta}^2 \bar{w}\big)(e^{-\beta_0\tau}M(1-\eta)^{\alpha_0}+\mu_1 \tau) &+e^{-\beta_0\tau}M(1-\eta)^{\alpha_0}[-w^2\alpha_0(1-\alpha_0) (1-\eta)^{-2}+\beta_0] \leq -\mu_1+e^{-\beta_0\tau}M(1-\eta)^{\alpha_0}[-b^2\alpha_0(1-\alpha_0) +\beta_0]<0,
\gamma = \frac{a_i^{4k^2}}{2^{12k^4+5k^2}}\ge\frac{a_i^{4k^2}}{2^{13k^4}}\,, \alpha = \frac{a_i^{36k^2}}{3^6\cdot 2^{108k^4+53k^2+2}}\ge \frac{a_i^{36k^2}}{2^{109k^4}}\,, \rho = \frac{a_i^{4k}}{2^{12k^3+4k}}\ge \frac{a_i^{4k}}{2^{13k^3}}\,.
\ell_{i,j}^{2}=(s_{j}\hdots s_{i})\ell_{i+1,j}(s_{i}\hdots s_{j})\ell_{i,j-1}&=(s_{j}\hdots s_{i})\ell_{i,j}\ell_{i,j-1}&=(s_{j}\hdots s_{i})(s_i\hdots s_j)\ell_{i,j-1}^{2}
C = \left\{\lambda \in \mathbb{E}\,\, \middle| \,\, \begin{aligned}n_{\alpha}p< \langle \lambda + \rho, \alpha^{\vee}\rangle < (n_{\alpha}+1)p, \,\, (n_{\alpha}-1)p < \langle \lambda + \rho, \alpha^{\vee}\rangle < n_{\alpha}p, \quad\quad\,\,\,\,\, \end{aligned}\right\}.
\allowdisplaybreaks \int_{\phi(|x|^{-2})^{-1}}^\infty p(t,x)\,dt \leq c_2\int_{\phi(|x|^{-2})^{-1}}^\infty \phi^{-1}(t^{-1})^{d/2}\,dt=c_2\int_0^{|x|^{-2}}y^{d/2}\left(-\frac{1}{\phi(y)}\right)'\,dy =c_2\int_0^{|x|^{-2}}y^{d/2-1}\frac{y\phi'(y)}{\phi(y)^2}\,dy\leq c_2\int_0^{|x|^{-2}}\frac{y^{d/2-1}}{\phi(y)}\,dy\,,
\frac{d}{dt}h^{\alpha}_{ij}=-\sum_{\beta}(R_{\alpha\beta})_{ij}V_{\beta}+((\nabla^{\bot})^2_{e_i,e_j}V^{\bot})_{\alpha}+(A^{V^{\bot}}A_{\alpha})_{ij}+\left(\left(\nabla^{\top}_{V^{\top}}A\right)_{\alpha}\right)_{ij},
h_{l , p , q}( \zeta ) = \sum_{n = 1}^{\infty}\left[ \int_{\mathrm{a r c c o s h}( y )}^{\infty}\frac{u d u}{\sqrt{\cosh u - y}}\frac{l^{2}+ p e^{- u}}{\sinh^{q}( u / 2 )}\right]_{y = 1 + n^{2}\zeta^{2}/ 2},
W_{2p}=\frac{2p-1}{2p}\cdot\frac{2p-3}{2p-2}\cdot\cdot\cdot\frac{1}{2}W_{0}=\frac{(2p-1)!!}{(2p)!!}\cdot\frac{\pi}{2}=\frac{(2p)!}{2^{2p}(p!)^{2}}\cdot\frac{\pi}{2},
\|e^{itH_0}J_a^*\left[v_t-E_+(t-s)v_s\right]\| &\leq \|e^{itH_0}J_a^*\| \|v_t-E_+(t-s)v_s\| &=\|J_a^*\| \|e^{i(t-s)H}(v_t-E_+(t-s)v_s)\| &=\|J_a^*\| \|v_s-e^{i(t-s)H}E_+(t-s)v_s\| .
\omega^2 = \omega_0^2 + \omega_\mathrm{p}^2\dfrac{\varepsilon_\mathrm{m}+\lambda_\mathrm{n_0}^{(3)}}{\lambda_\mathrm{n_0}^{(3)}(1-\varepsilon_\infty^{-1}\varepsilon_\mathrm{m})+\varepsilon_\mathrm{m}}+ \mathcal{O}(\delta^\mathrm{h}) \mbox{and}\gamma\omega \sim \delta^h.
\tilde y_1 ={{P}_1^{-1}}y_1 +{q P_2^{-1}}\tilde y_2 + \varphi ,\ \ \tilde y_2 ={{Q}_2^{-1}}y_2, \ \ \tilde t = Q_2 t , \ \p={P}_1{P}_2, \ \p'={Q}_1{Q}_2 - q^2{P_1 P_2^{-1}}.
\alpha_{\{( 2 , 3 ) \}}^{3 3}= \frac{1}{6 t}+ \frac{t}{2}\quad ; \qquad \alpha_{\{( 2 , 2 ) , ( 2 , 1 ) \}}^{3 3}= t \quad ; \qquad \alpha_{\{( 2 , 1 ) , ( 2 , 1 ) , ( 2 , 1 ) \}}^{3 3}= - t^{3}
F(\vec{x};v,q,t) :=\frac{\sum\limits_\mu\frac{C^-_\mu\big(t;q,t^2\big)}{C^-_\mu\big(q;q,t^2\big)}\prod\limits_{1\le i\le n}\frac{1-q^{\mu_i}t^{-2i}v}{1-t^{-2i}v}P_{\mu^2}(\vec{x};q,t)}{\sum\limits_\mu\frac{C^-_\mu\big(t;q,t^2\big)}{C^-_\mu\big(q;q,t^2\big)}P_{\mu^2}(\vec{x};q,t)}
\Delta{C}_{n+1}= 0, \Delta{C}_{t}= \big(\mu_1(t)(\gamma_t-1)-\mu_0(t)(\alpha_t-1)\big)+H(\alpha_t)-H(\gamma_t) &\qquad\qquad+\log\Big(\frac{1+2^{\mu_1(t)+\Delta{C}_{t+1}}}{1+2^{\mu_0(t)+\Delta{C}_{t+1}}}\Big),~~t\in\{n,\ldots,0\}.
-F =19H-6\sum_{j=1}^{10}E_j,-2F=38H-12\sum_{j=1}^{10}E_j,-D_i=6H-2\sum_{j=1}^{10}E_j +E_i, D_i-F = 13H-4\sum_{j=1}^{10}E_j -E_i,D_i-2F = 32H-10\sum_{j=1}^{10}E_j -E_i.
\phi([g][h]) = \phi([r(g)][g][h]) = \phi([gg^{-1}][g][h]) = \phi([g][g^{-1}][g][h]) = \phi([g][g^{-1}][gh]) = \pi(g)\pi(g^{-1})\pi(gh) = \pi(g)\pi(g^{-1})\pi(g)\pi(h) = \pi(g)\pi(h) = \phi([g])\phi([h]),
|D_{\Phi(y)_1}\log|{\bf k}-\Phi(x)-\overline{\Phi(y)}|||D_{y_1}\Phi_1(y)|=\frac{|\Phi(y)_1+\Phi(x)_1-k|}{|{\bf k}-\Phi(x)-\overline{\Phi(y)}|^2}|D_{y_1}\Phi_1(y)| \le \frac{1}{{|\bf{k}}-\Phi(x)-\overline{\Phi(y)}|}|D_{y_1}\Phi_1(y)|.
\tilde{P}_N(\alpha_{1},\alpha_{2};\beta_{1};q;z)=P_{N}(\alpha_{1},\alpha_{2};\beta_{1},\alpha_{2};q;z)=\sum_{m=0}^{N}\frac{(q^{-N};q)_{m}(\alpha_{1};q)_{m}}{(q;q)_{m}(\beta_{1};q)_{m}}z^{m}.
(\nabla^\varphi v)^t\mathbf{N}=(\partial_1^\varphi v\cdot\mathbf{N},\partial_2^\varphi v\cdot\mathbf{N},\partial_z^\varphi v\cdot\mathbf{N})^t=(\partial_1v\cdot\mathbf{N},\partial_2v\cdot\mathbf{N},0)^t+\frac{1}{\partial_z\varphi}(\partial_zv\cdot\mathbf{N})\mathbf{N}.
\begin{array}{ll}(|\partial_t^{\ell}\mathcal{D}^{\alpha}\phi|^2)_t begin{align*}8pt]= -2\sum\limits_{\ell_1+|\alpha_1|>0}\partial_t^{\ell}\mathcal{D}^{\alpha}\phi\partial_t^{\ell_1}\mathcal{D}^{\alpha_1}v\cdot\nabla(\partial_t^{\ell_2}\mathcal{D}^{\alpha_2}\phi)-v\cdot\nabla|\partial_t^{\ell}\mathcal{D}^{\alpha}\phi|^2. begin{align*}6pt]\end{array}
C^\infty(\Sigma) =\bigoplus_{\delta< c}\hat{E}_\delta \oplus \bigoplus_{\delta\geq c}\hat{E}_\delta \oplus \overline{D_\eta \mathcal{J}_0^0}; \overline{D_\eta \mathcal{J}_0^0}= \mathcal{J}_0^-\oplus (\mathcal{J}_0^-)^\perp.
\psi_{v}( \vec{x^{\prime}}, t ) \equiv \exp \left[ i m \vec{v}\vec{x^{\prime}}- i \left( \frac{m{\vec{v}}^{\ 2}}{2}\right) \tau ( t ) \right] \ \psi_{0}( \vec{x^{\prime}}- \tau ( t ) \vec{v}, t ) ,
f\left( t\right) =\sum\limits_{k=-\infty}^{\infty}f\left( t_{k}\right) \frac{\int\limits_{I}K\left( x,t\right) \overline{K\left( x,t_{k}\right)}dx}{\left\Vert K\left( x,t_{k}\right) \right\Vert_{L^{2}\left( I\right)}^{2}}.
\nu_{h}(t_n,\zeta)= (1-\theta)\int_{t_n}^{t_n+\zeta h}(f(t_n,x(t_n),x(qt_n))-f(t_n,y(t_n),y(qt_n))){\rm d}t +\theta\int_{t_{n}}^{t_n+\zeta h}(f(t_{n+1},x(t_{n+1}),x(qt_{n+1}))-f(t_{n+1},y(t_{n+1}),y(qt_{n+1}))){\rm d}t +\int_{t_n}^{t_n+\zeta h}(g(t_n,x(t_n),x(qt_n))-g(t_n,y(t_n),y(qt_n))){\rm d}B(t).
-[R][R]=-[\begin{matrix}0&-z&y\\z&0&-x\\ -y&x&0\end{matrix}]^{2}=[\begin{matrix}y^{2}+z^{2}&-xy&-xz\\ -yx&x^{2}+z^{2}&-yz\\ -zx&-zy&x^{2}+y^{2}\end{matrix}].
{}_{p}\Psi_q^k(z)={}_{p}\Psi_q^k\Bigg[\left.\begin{matrix}(a_i,\alpha_i)_{1,p}(b_j,\beta_j)_{1,q}\end{matrix}\right|z\Bigg]:=\sum_{n=0}^{\infty}\frac{\prod_{i=1}^{p}\Gamma_k(a_i+n\alpha_i)}{\prod_{j=1}^{q}\Gamma_k(b_j+n\beta_j)}\frac{z^n}{n!},
\mu_{1 , 2}^{2}= m_{3 / 2}^{2}+ V_{o}+{\tilde{\mu}}^{2}, \; \; \mu_{3}^{2}= - B m_{3 / 2}{\tilde{\mu}}, \; \;{\tilde{\mu}}= e^{\frac{1}{2}K_{o}}\mu \frac{W_{o}^{*}}{| W_{o}|},
B_n(q^{-1}) =\prod_{k=0}^{n}(1+q^{-(3k+1)})(1+q^{-(3k+2)}) begin{align*}5pt] =q^{-d_n}\prod_{k=0}^{n}(1+q^{(3k+1)})(1+q^{(3k+2)}) begin{align*}5pt] =q^{-d_n}B_n(q).
\frac{\partial \Sigma^{\check{r}}( \tau , \vec{\sigma}_{o})}{\partial \tau}= - [ I^{- 1}( \tau , \vec{\Sigma}( \tau , \vec{\sigma}_{o}) ) ]_{\; i}^{\check{r}}\, \frac{\partial \alpha^{i}( \tau , \vec{\Sigma}( \tau , \vec{\sigma}_{o}) )}{\partial \tau}= + \frac{J^{\check{r}}( \tau , \vec{\Sigma}( \tau , \vec{\sigma}_{o}) )}{J^{\tau}( \tau , \vec{\Sigma}( \tau , \vec{\sigma}_{o}) )}.
R_n^{(\alpha,\beta)}(\cos 2\theta)R_n^{(\alpha,\beta)}(\cos 2\theta^\prime) = &=\frac{c_{\alpha,\beta}}{a^{2\alpha}}\cdot \int_0^1 R_n^{(\alpha,\beta)}(2t^2-1) \Bigl(\int_{0}^\pi (a^2-b^2-t^2+2bt\cos\psi)_+^{\alpha-\beta-1}\sin^{2\beta}\!\psi d\psi\Bigr)t^{2\beta+1}dt.
\int_{M}G(p,y)\,|f(y)|\,dy = \int_{\mathcal{L}_{p}\left(0, \,A \exp \left(B\, \omega(a)\right)\right)}G(p,y)\,|f(y)|\,dy \,\,\,+ \int_{\mathcal{L}_{p}\left(A \exp \left(B\, \omega(a)\right),\infty\right)}G(p,y)\,|f(y)|\,dy \,.
\begin{array}{c c c}{{K_{a b}^{- 1}=}}&{{p \left( \begin{array}{c c}{{\sqrt{g}}}&{{{\frac{1}{\sqrt{g}}}}}\\{{{\frac{1}{\sqrt{g}}}}}&{{\sqrt{g}}}\\ \end{array}\right)}}&{{+ ( 1 - p ) \left( \begin{array}{c c}{{g}}&{{{\frac{1}{g}}}}\\{{{\frac{1}{g}}}}&{{g}}\\ \end{array}\right)}}\\ \end{array}
h := | f_{11}(1)f_{11}(2)\ldots f_{11}(i) f_{11}(i) f_{11}(i+1)\ldots f_{11}(m),\ f_{21}(1)f_{21}(2)\ldots f_{21}(i) f_{21}(i+1) f_{21}(i+1)\ldots f_{21}(m)| h' := | f_{11}(1)f_{11}(2)\ldots f_{11}(i) f_{11}(i+1) f_{11}(i+1)\ldots f_{11}(m),\ f_{21}(1)f_{21}(2)\ldots f_{21}(i) f_{21}(i) f_{21}(i+1)\ldots f_{21}(m)|.
\int_{\Omega_\rho}\left[(x^\perp\cdot\nabla u-u^\perp)\cdot v+u\cdot(x^\perp\cdot\nabla v-v^\perp)\right]\,dx =\int_{\Omega_\rho}\mbox{div $[x^\perp (u\cdot v)]$}\,dx &=\int_{\partial\Omega}(\nu\cdot x^\perp)(u\cdot v)\,d\sigma
\left\langle u,\mathsf{L}_{j}\left( s\right) \overline{v}\right\rangle_{\mathbb{R}^{3}}=\left\langle \psi,\gamma_{\operatorname*{N};j}^{\operatorname*{ext}}\left( s\right) \overline{v}\right\rangle_{\Gamma j}\qquad\forall v\in H^{1}\left( \mathbb{R}^{3},\mathbb{A}_{j}^{\operatorname*{ext}}\right) .
\lim_{{\cal L}\to\infty}\frac{\langle H\rangle^{{\cal L}}_{min}}{{\cal E}^{{\cal L}}_{0}}=\lim_{{\cal L}\to\infty}\frac{2\sqrt{\frac{{\cal L}+3}{2}}}{({\cal L}+2)}\Big(\frac{\Gamma(\frac{{\cal L}+3}{2})}{\Gamma(\frac{\cal L}{2}+1)}\Big)=1.
\mathcal{P}_{n+1}(x) =\rho_n (x-c_{n})\mathcal{P}_n(x)-d_{n}(x^2+\omega^2)\mathcal{P}_{n-1}(x), n\geq 1, & \mathcal{P}_{0}(x) = 1, \mathcal{P}_1(x) = x-c_0,
r_{1\mu \nu}[F(z)]=\frac{\pi}{2}\sum_{k}{\mathrm{Res}}_{{\mathrm{Im}}z_k=0}\left[ F(z)\sum_{l=1}^{2}\frac{H^{(l)}_{\mu}(\lambda z)}{H^{(l)}_{\nu}(z)}\right] +\pi \sum_{k}\sum_{l=1}^{2}{\mathrm{Res}}_{(-1)^l{\mathrm{Im}}z_k<0}\left[ F(z)\frac{H^{(l)}_{\mu}(\lambda z)}{H^{(l)}_{\nu}(z)}\right] .
|\Phi(s)| \le \frac{1}{|s(s+1) \cdots (s+m-1)|}\int_{\eta}^{\eta+\epsilon}|\phi^{(m)}(t)| t^{\sigma+m-1}dt \ll_{m}\frac{\epsilon^{1-m}(\eta+\epsilon)^{\sigma+m-1}}{|s(s+1) \cdots (s+m-1)|}.
\begin{cases}t_{i,en}(x_0,t_0)=\max\left(0,t_0-\dfrac{x_0+R}{\lambda_i}\right) \forall\: i\in [1,p], t_{i,en}(x_0,t_0)=\max\left(0,t_0-\dfrac{x_0-R}{\lambda_i}\right) \forall\: i\in [p+1,n].\end{cases}
\begin{aligned}L_4(q) = -1 + \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{8n^2 + 3n - 4j^2 - 3j}+ q^{8n^2 + 13n + 5- 4j^2-3j}& + \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{8n^2 + 11n + 4 - 4j^2 -j}+ q^{8n^2 + 21n + 14 - 4j^2 -j}\end{aligned}
V_{C}( p , 0 ) = V_{C}( 0 , p ) = V_{0}e^{\mu \alpha_{C}}\int_{\mu}^{\infty}d z \, \frac{\exp ( - \alpha_{C}z )}{z^{2}+ p^{2}}= V_{0}\mu e^{\mu \alpha_{C}}\int_{0}^{1}d y \, \frac{\exp ( - \alpha_{C}\mu / y )}{\mu^{2}+ ( p y )^{2}}.
= \exp\left\{\frac{1}{(1-x)^3(1-y)^5}\sum_{n=1}^{\infty}(n^3x^{n+1}y^{n+1}-2n^3x^{n+1}y^{n+2}+n^3x^{n+1}y^{n+3}) \frac{z^n}{n^4}\right\}
\langle z_-,j | n_{12},j \rangle = Q_{n_{12}-1}^{\mu_{12}}(z) \iint d\lambda_{x_\pm}d\lambda_{y_\pm}\langle z_-,j | x_\pm y_\pm \rangle \displaystyle\sum\limits_{n=0}^{j+1}\langle x_\pm y_\pm |n,1+j-n\rangle \langle n,1+j-n | 1,j \rangle.
\begin{pmatrix}v_1 v_2\end{pmatrix}=-\frac{1}{k^2+x_1^2+x_2^2}\begin{pmatrix}x_1x_2 -k^2-x_1^2 k^2+x_2^2 -x_1x_2\end{pmatrix}\begin{pmatrix}\partial_{x_1}\varphi \partial_{x_2}\varphi\end{pmatrix}.
\gamma_{\textrm{NDN}}^{\textrm{OFDMA}}=\frac{\left(1-\beta_{\textrm{N}}\right)P_\textrm{N}{|h_{\textrm{BN}}|^2}}{{d_{\textrm{BN}}^{\alpha}\sigma_{\textrm{N}}^2}\left( 1-\theta\right)},
\left \langle A_{{i_1}{j_1}}\left({{t_1}}\right)...A_{{i_n}{j_n}}\left({{t_n}}\right)\right \rangle_c = W_{{i_1}{j_1}...{i_n}{j_n}}^{\left( n \right)}\left({{t_1}-{t_2}, \ldots ,{t_1}-{t_n}}\right)
L= \int_{\mathbb R}\zeta_k \nu_{n_k}\mathcal T(\varphi_k \mathcal H \nu'_{n_k}) +\zeta_k \nu_{n_k}\mathcal T[\mathcal H, \varphi_k] \nu'_{n_k}+ \zeta_k \nu_{n_k}\mathcal T \mathcal H(\varphi_k' \nu_{n_k}) dx\equiv L_1+L_2+L_3.
\int \Big[ \hat{f}_n(t)-f_n(t) \Big]^2 dt = \textnormal{O}_\textnormal{p}\Big( \frac{(\textnormal{log(log}n))^2}{n^2h_n^4}+\frac{\textnormal{log(log}n)}{n^{3-2\alpha}h_n^2}\Big) = \textnormal{o}_\textnormal{p}\Big(\frac{1}{n \sqrt{h_n}}\Big)
\begin{array}{l c l}{{\{p^{n}, f \}_{\theta}}}&{{=}}&{{\sum_{k = 0}{\theta}^{2 k}f^{( 2 k + 1 )}p^{n - 2 k - 1}}}\\{{\{p^{- n}, f \}_{\theta}}}&{{=}}&{{- \sum_{k = 0}{\theta}^{2 k}c_{2 k + n}^{2 k + 1}f^{( 2 k + 1 )}p^{- 2 k - n - 1}}}\\ \end{array}
\mathbf{v}=-\frac{1}{4\pi}\mathbf{\nabla}\int_{E_{3}^{\prime}}\frac{\mathbf{\nabla}^{\prime}\cdot \mathbf{v}(\mathbf{r}^{\prime})}{r_{12}}dV^{\prime}+\frac{1}{4\pi}\mathbf{\nabla}\times \int_{E_{3}^{\prime}}\frac{\mathbf{\nabla}^{\prime}\times \mathbf{v}(\mathbf{r}^{\prime})}{r_{12}}dV^{\prime}.
\begin{array}{r l}{{A_{\lambda}^{( n n )}( x | a_{1}, \cdots , a_{n}| b_{1}, \cdots b_{n}) =}}&{{\displaystyle \prod_{j = 1}^{n}( x - a_{j}\tau ) \, g_{\lambda}^{( n )}( x | b_{1}, \cdots , b_{n})}}\\{{}}&{{+ \tau^{2 ( \lambda - 1 )}\displaystyle \prod_{i = 1}^{n}( x - b_{i}\tau^{- 1}) \, g_{\lambda}^{( n )}( x | a_{1}, \cdots , a_{n}) .}}\\ \end{array}
\Sigma^{+}=\{e_{1}\pm e_{2}, e_{1}, e_{2}, 2e_{1}, 2e_{2}\},\W^{+}=\{e_{1}\pm e_{2}, e_{1}, e_{2}\}, \Pi =\{\alpha_{1}=e_{1}-e_{2}, \alpha_{2}= e_{2}\}, tilde{\alpha}=\alpha_{1}+2\alpha_{2}=e_{1}+e_{2}
S=[\begin{matrix}S_{HH}&S_{HV}&S_{Hh}&S_{Hv}\\S_{HV}&S_{VV}&S_{Vh}&S_{Vv}\\S_{Hh}&S_{Vh}&S_{hh}&S_{hv}\\S_{Hv}&S_{Vv}&S_{hv}&S_{vv}\end{matrix}]
\frac{1}{t}\big( G^t_{\epsilon}(z) - G_{\epsilon}(z) \big) = G_z (\epsilon) \frac{1}{t}\big( L_0 - L_t \big) G_{\epsilon}(z) + \big( G_z^t (\epsilon) - G_z (\epsilon) \big) \frac{1}{t}\big( L_0 - L_t \big) G_{\epsilon}(z)
\|\left(\Phi(X^*AY)\Phi(Y^*AY)^{-1}\Phi(Y^*AX)\right)^{\frac{p}{2}}\Phi(X^*AX)^{-\frac{p}{2}}\| &\le \frac{(M-m)^{p}(M^\alpha+m^\alpha)^{\frac{p}{\alpha}}}{2^{2+\frac{p}{2}}M^{\frac{3p}{4}}m^{\frac{3p}{4}}(M+m)^{\frac{p}{2}}}.
\beta=0 \gamma\in \{2m \mid m =0,1,5,6,8,9,10,11,13,\ldots,65,68,71,79\}, \beta=1 \gamma\in \{2m \mid m =13,14,16,\ldots,58,63\}, \beta=2 \gamma\in \{2m \mid m =0,16,\ldots,50,55\}
f_ie_i^{(m)}= e_i^{(m)}f_i + q^2e_i^{(m-1)}\frac{q^{m-1}q^{2k_i}-q^{-m+1}q^{2k_{i+1}}}{q^2-1}+ (q^2-1)\left(q^2e_i^{(m-1)}e_{\overline{i}}f_{\overline{i}}- \frac{1}{2}e_i^{(m-2)}q^{k_{i+1}}e_{\overline{i}}F_{\overline{i+1}}\right)
F_1 (\rho_1, v) = \ (\alpha-\delta_1) \rho_1 - \beta \rho_1 v, F_2 (\rho_1, \rho_2, v) =\ \beta \rho_1 v -\delta_2\rho_2, F_3 (\rho_1, \rho_2, v) =\ b \delta_2 \rho_2 -B \rho_1 v - \delta_v v.
2N^{(1)}_{(f,H_t),\leqslant k_t}(r)\ge 2\sum_{s\ne 1,2,t}N^{(1)}_{(f,H_s),\leqslant k_s}(r)-2\sum_{u=1}^3(N^{(1)}_{(f^u,H_1),>k_1}(r)+N^{(1)}_{(f^u,H_2),>k_2}(r) &+2N^{(1)}_{(f^u,H_t),>k_t}(r)).
\frac{1}{k^2}|E(G)| -\frac{19}{12}(\alpha+\beta)\, \leq\, \frac{1}{6}(\alpha^2+\alpha \sigma+6 \alpha+\sigma^2-3 \sigma+3) + \frac{1}{6}\left((\beta+1)^2+4(\beta+1)-2\right) - \frac{19}{12}(\alpha+\beta) =\, \frac{1}{12}(2 \alpha^2+2 \alpha \sigma-7 \alpha+2 \beta^2-7 \beta+2 \sigma^2-6 \sigma+12).
\{Q_{\epsilon}^{+}, Q_{\epsilon}^{-}\}= a^{+ 2}\alpha \gamma^{*}+ a^{- 2}\alpha^{*}\gamma + \frac{1}{2}\left( \{a^{+}, a^{-}\}- \epsilon K [ a^{-}, a^{+}] \right) ( \alpha \alpha^{*}+ \gamma \gamma^{*}) .
\Sigma\left( \rho_{k},\lambda_{k},\alpha_{\pm}\right) \overset{}{=}-\frac{\Xi_{k}}{4\lambda_{k}}\frac{1+\sqrt{\Delta\left( \rho_{k}\right)}}{1-\sqrt{\Delta\left( \rho_{k}\right)}}\sqrt{\frac{2\alpha_{-}\left(\rho_{k}\right)}{\alpha_{+}\left( \rho_{k}\right)}}>0\forall\rho\in\lbrack0,1)
M_{Y}(t) = 2p(1-p) \bigg[ \frac{(p_{\alpha_{+}}-p_{\alpha_{-}})}{(p_{\alpha_{-}}- \sigma t)(p_{\alpha_{+}}-\sigma t)}\bigg]\exp\bigg\{\mu t +\frac{1}{2}\alpha^{2}\sigma^{2}t^{2}\bigg\}\; \Phi(\alpha \sigma t) .
\bigl\langle G_R(\Lambda_{\psi}(p)\otimes\Lambda(b)),\Lambda_{\psi}(q)\otimes\Lambda(d)\bigr\rangle &=(\psi\otimes\eta)\bigl((q^*\otimes d^*)Q_R(p\otimes b)\bigr)=\eta\bigl(d^*\,(\psi\otimes\operatorname{id})[(q^*\otimes1)Q_R(p\otimes b)]\bigr).
\begin{array}{c}2\, \Im m\, \displaystyle\int \displaystyle\frac{dx}{i}\varphi '(x+i\epsilon )\log (1+f(x+i\epsilon ))= =-2\, \Re e\, \displaystyle\int_{\Gamma}\displaystyle\frac{du}{u}\log (1+u)-\displaystyle\frac{1}{2}\displaystyle\int dx\, G(x)\left( F^{2}(+\infty )-F^{2}(-\infty )\right) \end{array}
\begin{cases}x_j^{k+1}= \mathrm{prox}_{\frac{\alpha_j}{\beta}f_j}(x_j^k-\alpha_jA_j^\top(\sum_{i=1}^{j-1}A_i(x_i^{k+1}+\theta(x_i^{k+1}-x_i^k))+A_jx_j^{k+1}+\sum_{i=j+1}^sA_i((2+\theta)x_i^k-(\theta+1)x_i^{k-1})-b)- \frac{\alpha_j}{\beta}A_j^\top y^k), ~~j\in\mathbb{N}_s, y^{k+1}= y^k+\beta(\sum_{i=1}^sA_ix_i^{k+1}-b), \end{cases}
\Psi_{1}= \sum_{N = 0}^{\infty}\sum_{l = 0}^{N}\sum_{n = - l}^{l}\left\{_1 a_{N l n}\phi_{N l n}^{( + )}( t , r , \theta , \phi ) +_{1}\! a_{N l n}^{\dagger}\phi_{N l n}^{\ast ( + )}( t , r , \theta , \phi ) \right\}
P:\overset{Def}= \sum_{u=1}^{3}\nu^{(n)}_{(f^u,H_i),\leqslant k_i}+\sum_{u=1}^3\nu^{(n)}_{(f^k,H_j),\leqslant k_j}+2\sum_{\overset{t=1}{t\ne i,j}}^{2n+2}\chi_{T_t}-(2n+1)\chi_{T_i}-(n+1)\chi_{T_j}&-\sum_{u=1}^3\nu^\infty_{F_u^{ij}}+\chi_{\nu_j}-\chi_{S_i}-\chi_{S_j}-2(n-1)\chi_{R_i}-(n-1)\chi_{R_j}\leqslant \nu_{\Phi^\alpha_{ij}}.
V_{x}={\frac{7 \,{x^{6}}\, y}{3}}- 6 \,{x^{5}}\,{y^{3}}+ 5 \,{x^{4}}\,{y^{5}}- 3 \,{x^{2}}\,{y^{6}}+{\frac{{y^{7}}}{3}}= 0
\begin{array}{rcl}\eta_{12}= - \left(P_{12}{}^{t_2}\left( (r_{12}{}^{t_2})^{-1}\right) \right)^{t_2}= - \mbox{tr}_3\left(P_{23}(r_{23}{}^{t_3})^{-1}\right) P_{12}= - D_2 P_{12}.\end{array}
\begin{pmatrix}( g_1, h_{1})_{\Omega}\dots ( g_m, h_{1})_{\Omega}\vdots \vdots \vdots (g_1, h_{m})_{\Omega}\cdots ( g_m, h_{m})_{\Omega}\end{pmatrix}\begin{pmatrix}a_1 \vdots a_m\end{pmatrix}= v_1(\epsilon) + v_{2}(\epsilon),
\sum_{m=\frac{1}{2}}^{\Lambda}\bigl[ \Delta_m(\zeta) \bigl(b^{\dag}_m b_m a_0 + b^{\dag}_m a_0 b_m + a_0 b^{\dag}_m b_m+ (b_m a_0 b^{\dag}_m - a_0 ) \bigr)+ (b \rightarrow d, \zeta \rightarrow - \zeta) \bigr]+{\hat{\rho}}(\zeta) a_0
\frac{d}{dt}\left(\Phi_t^1(x_1^0,x_2)D(t)\right)=x_2\frac{\rho_{0}(D(t)\Phi_t^1(x_1^0,x_2),0)}{D(t)\Phi_t^1(x_1^0,x_2)}\frac{1}{2}\frac{d}{dt}\left(\int_0^tD(s)ds\right)^2 &\Leftrightarrow \frac{d}{dt}U(t,x_1^0,x_2)=\frac{x_2}{2}\frac{\rho_0(U(t,x_1^0,x_2),0)}{U(t,x_1^0,x_2)}\frac{d}{dt}H(t).
2N \left(p\mu-\frac{x_1+x_N}2\right) =2p(x_2+\dots +x_{N-1})-(N-2p)x_1-(N-2p) x_N & \ge 2p(x_k+\dots +x_{N-1})-(N-2p)\delta -(N-2p) a & \ge 2p(N-k)a(1-\Delta)-(N-2p)\delta -(N-2p) a &=a\left[2p(N-k)(1-\Delta)-(N-2p) \right]-(N-2p)\delta
W^0_{k+2}(p_1,\dots,p_{k+1})=(\mathbf{1})\ast(\mathbf{1})\ast\dots\ast(\mathbf{1}), k, &W^0_{l+2}(p_1,\dots,p_{l+1})=(\mathbf{1})\ast(\mathbf{1})\ast\dots\ast(\mathbf{1}), l,
\hat{H_{q}}= \frac{\hbar^{2}( I - I_{3})}{2 I I_{3}}J_{3}^{\prime 2}+ \frac{\hbar^{2}}{2 I}\left( \frac{\gamma}{\sinh \gamma}\hat{J}_{+}^{\prime}\hat{J}_{-}^{\prime}+ [ \hat{J}_{3}^{\prime}] [ \hat{J}_{3}^{\prime}+ 1 ] \right) ~ .
E(u\pi^{-m},f)=Z(0,f,\chi_{triv})+_{t^{m-1}}\frac{(t-q)Z(s,f,\chi_{triv})}{(q-1)(1-t)}+\sum_{\chi \neq \chi_{triv}}g_{\chi^{-1}}\chi(u)_{t^{m-c(\chi)}}Z(s,f,\chi),
h_{Q_1,v}=\min \{h_{L_y,v},h_{L_z,v}\}+O(1), h_{Q_2,v}=\min \{h_{L_x,v},h_{L_z,v}\}+O(1), h_{Q_3,v}=\min \{h_{L_x,v},h_{L_y,v}\}+O(1),
U^{1,i,j}_{n,u,s,t}(h)\, =\, \frac{\lambda^i}{N^i}\partial_\theta\left[ \left( \left( e^{(t-u)L_{\psi_{n-1}}^*}-e^{(s-u)L_{\psi_{n-1}}^*}\right)h\right)^i\right]\big(\varphi^i_j(u)\big)\, ,
[a(D)u] \ \widehat{}\ (\xi) = \lim_{\substack{l \to \infty L^{2}(B(0,j))}}\big( a(D)u_{l}\big) \ \widehat{}\ (\xi)= \lim_{\substack{l \to \infty L^{2}( B(0,j))}}a(\xi) \, \widehat{u_{l}}(\xi)= a(\xi) \, \widehat{[u]}(\xi),
g_{2 0}( q_{1}, q_{2}) ={\frac{g}{4}}({\frac{q_{1}q_{2}}{q_{1}^{2}+ q_{2}^{2}+ q_{1}q_{2}}}) [ \hat{f}^{2}( q_{1}) C ( g , q_{1}) + \hat{f}^{2}( q_{2}) C ( g , q_{2}) + 2 \phi_{0}] .
u_{\underline{{{m}}}}^{i}= G_{{\cal R}}^{i k}H_{L}^{k j}( - l_{\underline{{{m}}}}^{j}+ e^{2 W}r^{( + + ) \underline{{{n}}}}l_{\underline{{{n}}}}^{j}l_{\underline{{{m}}}}^{( -- )}) ~ , \qquad
d_1\triangleq\max\limits_{1\le t'\le t}\left\{\frac{\binom{N_R-1}{r}\binom{N_T}{t'}\binom{N_R-r-1}{t'-1}t'}{\binom{N_R-1}{r}\binom{N_T}{t'}\binom{N_R-r-1}{t'-1}t'+\binom{N_R-1}{r+1}\binom{N_R-r-2}{t'-1}\binom{N_T}{t'-1}}\right\}
U \phi_{\ell}= 2 \alpha \left({\tt \frac{1}{\bar{z}}}\partial_{z}\phi_{\ell}-{\tt \frac{1}{z}}\partial_{\bar{z}}\phi_{\ell}\right) + \frac{\alpha^{2}}{2}\left({\tt \frac{1}{z}{\tt \frac{1}{\bar{z}}}}+{\tt \frac{1}{\bar{z}}}{\tt \frac{1}{z}}\right) \phi_{\ell},
H(t,x,y,u,p,q,r)=H(t,x,y,u,p,q,r,\omega) &=\mathbb{E}[\delta_Y(y)|\mathcal{F}_t] f(t,x,u,y)+b(t,x,u,y)p + \sigma(t,x,u,y)q+\int_{\mathbb{R}}\gamma(t,x,u,y)r(y,\zeta)\nu(d\zeta).
M_{i i}= \left( \begin{array}{c c}{2 ( i - 2 )}&{0}\\{0}&{2 ( i - 1 )}\\ \end{array}\right) + \left( \begin{array}{c c}{\delta M_{\lambda \lambda}}&{\delta M_{\lambda \xi}}\\{\delta M_{\xi \lambda}}&{\delta M_{\xi \xi}}\\ \end{array}\right) ,
\sum_{i=0, 1,2,3,4}\big|{}_0^{1}Err^i_{k,j;n}(t_1, t_2)\big|\lesssim \mathcal{M}(C) 2^{(\alpha^\star + 3\iota + 140\epsilon) M_t- (k+2n)/2}\lesssim 2^{M_t/2}\mathcal{M}(C).
\begin{aligned}\mathbb{E}_n \bigg[ \sup_{0 \le t \le T}\bigg| \frac{1}{\sqrt{n}}\int_0^t \sum_{j \in \mathbb{Z}}\overline{W}_{j-1}(s) \overline{W}_{j}(s) \overline{W}_{j+1}(s) \nabla^n \varphi^n_j (s) ds \bigg|^2 \bigg] & \le C T^2 \| \partial_x \varphi \|^2_{L^2(\mathbb{R})}\le C \frac{T^{3/2}}{n}\| \partial_x \varphi \|^2_{L^2(\mathbb{R})}.\end{aligned}
e^{\textstyle - J \alpha_{-}\Phi ( \sigma , \tau )}= c_{J}\sum_{m = - J}^{J}\,{\frac{( - 1 )^{J - m}\, \lambda_{m}^{( J )}( \varpi )}{\sqrt{\lfloor \varpi \rfloor}\sqrt{\lfloor \varpi + 2 m \rfloor}}}\psi_{m}^{( J )}( x_{+}) \,{\overline{{\psi}}_{m}^{( J )}}( x_{-})
|T_{2}(x)| \leq\int_{|xy^{-1}|\geq R}\frac{\rho(|xy^{-1}|)}{|xy^{-1}|^{Q}}|f(y)|dy = \sum_{k=0}^{\infty}\int_{2^{k}R\leq|xy^{-1}|<2^{k+1}R}\frac{\rho(|xy^{-1}|)}{|xy^{-1}|^{Q}}|f(y)|dy.
\mathcal{P}_s = \begin{bmatrix}I 0 B \hat{A}^{-1}I \end{bmatrix}\begin{bmatrix}\hat{A}_s 0 0 - \hat{S}\end{bmatrix}\begin{bmatrix}I \hat{A}^{-\top}B^\top 0 I \end{bmatrix}\quad \hat{A}_s = \hat{A}(\hat{A}+ \hat{A}^\top - A)^{-1}\hat{A}^\top.
|V(x')u(x)| \leq C_{k}V(x') \int_{B_{0}}{1\over{\big( 1+{{d(x,y)}\over{\rho(x')}}\big)^{k}}}{1\over{d(x,y)^{n}}}\bigg( |Lu(y)| + \qquad\qquad\qquad + \sum_{i,j=1}^{n}|a_{ij}(y)-a_{ij}(x)| |u_{x_{i}x_{j}}(y)| \bigg) dy,
\sum_{n=2^{k-1}}^{2^k-1}1_{[-B+C,B-C]}\left( t + \sum_{i=0}^{n-1}f(T^i x) \right)\ge\beta \tau\sum_{n=0}^{2^k-1}1_{[-B-C,B+C]}\left( t + \sum_{i=0}^{n-1}f(T^i x) \right)