harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\small\begin{array}{ll}\frac{1}{(m-n-1)}Ric(U_i, U_i) = \frac{1}{(m-n-1)}\hat{R}ic(U_i, U_i) - \lambda^2 (m-n-1) \|\nabla (f + \log \lambda)\|^2 &+ \frac{\lambda^4}{4}(m-n-1) \| grad_{\mathcal{H}}\frac{1}{\lambda^2}\|^2 -\lambda^2 (m-n-1) g\left(\nabla f, \nabla_{\mathcal{H}}\frac{1}{\lambda^2}\right).\end{array}
+\int_{ST_2}d\tau_2\hat\mu(a_2)\{-{3\over2}\partial_2\hat\zeta(z_{12})-\hat\zeta(z_{12})\partial_2+{1\over2}D_2\hat\zeta(z_{12}) D_2\}{\delta \Gamma\over \delta\hat\mu(a_2)}
rarctan(\frac{y}{x})=\frac{1}{1}\cdot\frac{ry}{x}-\frac{1}{3}\cdot\frac{ry^{3}}{x^{3}}+\frac{1}{5}\cdot\frac{ry^{5}}{x^{5}}-\cdot\cdot\cdot,where\frac{y}{x}\le1.
u=\begin{pmatrix}u_{11}u_{12}u_{12}^* u_{22}\end{pmatrix}\coloneqq \rho_{m,j}^{(2)}(d_j^*d_j)-\rho_{m,n}^{(2)}\big(\rho_{n,j}^{(2)}(d_j)^*\rho_{n,j}^{(2)}(d_j)\big),
\mathcal E[u]=\|\sqrt{\mathcal K_\lambda}u\|^2 - \frac1p \mathcal P[u] \geq\|\sqrt{\mathcal K_\lambda}u\|^2-\frac{C_{N,\tau,\alpha,\lambda}}{p}\|\sqrt{\mathcal K_\lambda}u\|^{2p}&=f\big(\|\sqrt{\mathcal K_\lambda}u\|^2\big).
\omega^{12}= -i \{\frac{1}{2}d\ln (\frac{d\overline{g}_{1}}{d\overline{z}_{1}}\frac{dz_{1}}{dg_{1}}) + \frac{(g_{1}d\overline{g}_{1}- \overline{g}_{1}dg_{1})}{(1-g_{1}\overline{g}_{1})}\},
\tau_{w}=\mu(\frac{\partial u}{\partial y})_{y=0},v_{w}=v(x,0,t),\delta_{1}=\int_{0}^{\infty}(1-\frac{u}{U})dy,\delta_{2}=\int_{0}^{\infty}\frac{u}{U}(1-\frac{u}{U})dy
\begin{array}{r l}{\operatorname{t a n}\left( \frac{\pi}{4}+ \frac{\Theta}{2}\right) =}{( - 1 )^{n}\left( \frac{E + m}{E - m}\right)^{1 / 2}\left( \frac{k}{2 m}\right)^{2 \nu + 1}}\end{array}
\sum\limits_{d=n}^{+\infty}\left( \sum\limits_{p=0}^{n-1}\frac{(-1)^{d-n+1}\gamma_{p}^{(\nu ,n)}B_{2(d-n+p+1)}\left(\nu+\frac{1}{2}\right)}{(n-1)!(d-n+p+1)(d-n)!}\right) t^{d}.
L^{\mathrm{BEP}}f(x) = \sum_{1 \leq i<j \leq N}x_i x_j \left( \frac{\partial}{\partial x_i}- \frac{\partial}{\partial x_j}\right)^2f(x) - 2(k_i x_i - k_j x_j) \left( \frac{\partial}{\partial x_i}- \frac{\partial}{\partial x_j}\right) f(x),
{\partial}_{x_l}\left(g_{kt}g^{ts}\left({\partial_{x_j}e_{is}}{}-{\partial_{x_i}e_{js}}\right)\right)={\partial}_{x_k}\left(g_{lt}g^{ts}\left({{\partial_{x_j}}e_{is}}-{{\partial_{x_i}}e_{js}}\right)\right).
d=\frac12\big(Q^{(1)}+Q^{(2)}+Q^{(3)}+Q^{(123)}\big), e_{1}=-\frac{1}4\big(Q^{(3)}-Q^{(2)})(Q^{(1)}-Q^{(123)}\big), e_{2}=\frac{1}4\big(Q^{(2)}-Q^{(1)}\big)\big(Q^{(3)}-Q^{(123)}\big).
\sum_{\substack{|\Im(\rho)| \leq T \Re(\rho) > \sigma}}\frac{x^{\Re(\rho) - 1}}{|\Im(\rho)|}\leq 2\sum_{k = 0}^{K -1}\frac{x^{-\nu_1\left(t_k\right)}}{t_k}N_0(\sigma, t_{k+1}) =s_2(x,\sigma,K),\ .
\lambda = 6 \frac{\kappa^{2}}{\tilde{\kappa}^{4}}, ~ \Lambda = \frac{4 \pi}{\tilde{M}_{p}^{3}}\left[ \tilde{\Lambda}+ \left( \frac{4 \pi}{3 \tilde{M}_{p}^{3}}\right) \lambda^{2}\right]
\| \partial_v f^n_t - \partial_v f^{n-1}_t \|_{L^1}\le \int_0^t\!\mathrm{d}s\, \|(G(M^{n-1}(s) - G(M^{n-2}(s)) \partial_{vv}f^{n-1}_s \|_{L^1}+ \int_0^t\!\mathrm{d}s\, \| \partial_x f^n_s - \partial_x f^{n-1}_s \|_{L^1}\,.
\Omega_1 = \left(\begin{array}{cc}I \gamma 0 I \end{array}\right) \quad\Omega_2 = \left(\begin{array}{cc}I 0 \alpha I \end{array}\right) \quad\Omega_3 = \left(\begin{array}{cc}{A^{-1}}^T 0 0 A \end{array}\right)
m_\omega(\xi) = \frac1{\alpha}+ \sum_{k=1}^\infty \frac{2 \alpha (\xi^2 - \omega)}{\alpha^2 (\xi^2 - \omega) +k^2 \pi^2}= \frac1{\alpha}+ \sum_{k=1}^\infty \frac{2 \alpha}{\alpha^2 + \frac{k^2 \pi^2}{\xi^2 - \omega}}.
I(u_{\theta}) =A(u_{\theta})+B(u_{\theta})+C(u_{\theta}) &\le\theta^{2-2\beta}A(u)+\theta^{4-\beta}H(u)+\theta^{(1-\frac{3}{2}\beta)p+3\beta}C(u) &=\theta^{2}\left(I(u)+(\theta^{-2\beta}-1)A(u)+(\theta^{(1-\frac{3}{2}\beta)p+3\beta-2}-1)C(u)+\theta^{2-\beta}H(u)-B(u)\right) &=\theta^{2}(I(u)+f(\theta,u)),
\mathcal{S}_{\mathcal{M}}^{2Du}\left( \tau\right)\overset{\tau\rightarrow\infty}{\approx}\left[ \left( \frac{\lambda_{+}}{\lambda_{+}^{\prime}}\right) \cdot\mathcal{S}_{\mathcal{M}}^{3Du}\left( \tau\right) \right] \frac{\lambda_{+}}{\lambda_{+}^{\prime}}=\frac{1}{\sqrt{2}}<1
\mu_{R,1}^{(1)}(k,\ell,m,n) :=\int\overline{\mathcal{K}}_{R}(x,k) \mathcal{K}_{M_{1}}(x,\ell)\overline{\mathcal{K}_{M_{2}}(x,n)}\mathcal{K}_{M_{3}}(x,m)\,dx, \mu_{R,1}^{(2)}(k,\ell,m,n) :=\int\overline{\mathcal{K}}_{S}(x,k)\mathcal{K}_{M_{1}}(x,\ell) \overline{\mathcal{K}_{M_{2}}(x,n)}\mathcal{K}_{R}(x,m)\,dx, M_{i}\in\{S,R\}.
\sum_{k \geq 1}(2k-1)^2 q^{2k-1}= \sum_{k \geq 1}k^2 q^k - \sum_{k \geq 1}(2k)^2 q^{2k}& = q \frac{1+q}{(1-q)^3}- 4 q^2 \frac{1+q^2}{(1-q^2)^3}& = q \frac{1 + 6 q^2 + q^4}{(1-q^2)^3}.
\{x^{\mu}, \theta_{\alpha}, \bar{\theta}_{\dot{\alpha}}\}: \qquad \delta \theta_{\alpha}= \varepsilon_{\alpha}, \quad \delta \bar{\theta}_{\dot{\alpha}}= \bar{\varepsilon}_{\dot{\alpha}}, \quad \delta x_{\alpha \dot{\alpha}}= - 2 i \theta_{\alpha}\bar{\varepsilon}_{\dot{\alpha}}- 2 i \bar{\theta}_{\dot{\alpha}}\varepsilon_{\alpha}
B=(\begin{matrix}B_{1,1}&B_{1,2}&\cdot\cdot\cdot&B_{1,N}\\B_{2,1}&B_{2,2}&\cdot\cdot\cdot&B_{2,N}\\ \vdots&\vdots&&\vdots\\B_{N,1}&B_{N,2}&\cdot\cdot\cdot&B_{N,N}\\ \end{matrix}).
\begin{aligned}\Vert \langle y\rangle^\lambda \zeta_i \Vert_{L^\infty_y}\lesssim \vartheta_0^{-3}\bigg(C(u_e,h_e,u_b) &+\sum_{|\alpha|\leq 5}\Vert(u+u_e\phi(y)+u_b(1-\phi(y))^{\frac{1}{2}}\langle y\rangle^{\lambda-1}D^\alpha (u,h)\Vert_{L^2_y}\bigg)^3.\end{aligned}
q_{*}\lambda_{Z}^{\eta\cdot z}(b) =\int_{G}b(p(\eta\cdot z,\gamma))\,d\lambda_{Z}^{\eta\cdot z}(\gamma) =\int_{G}b(\gamma^{-1}\eta\cdot z) \,d\lambda^{r_{Z}(\eta\cdot z)}(\gamma) = \int_{G}b(\gamma^{-1}\cdot z)\,d(\eta^{-1}\cdot \lambda^{r(\eta)})(\gamma) = q_{*}\lambda_{Z}^{z}(b).
U_{\lambda}( \lambda_{1}, \dots , \lambda_{N}) = - \sum_{a = 1}^{N}\frac{1}{2}\frac{\lambda_{a}^{N + 1}- ( \alpha - 1 ) \lambda_{a}^{N}+ ( 1 - \alpha + \beta ) \lambda_{a}^{N - 1}}{f_{a}( \vec{\lambda})}
J^*(u,v)=\frac{\xi(w)}{\zeta(2)}\sum_{c\mid w}\chi(c)\sum_{(d,D)=1}d^{-1}\int_0^\infty \phi\left(\frac{dT}{x}\right)h\left(\frac{cx}{u}\right)h\left(\frac{cx}{v}\right)\frac{dx}{x}.
S_{\mathrm{d i l 1}}={\frac{1}{2 \kappa_{4}^{2}}}\int d^{4}x \sqrt{- g}\left[ R - ( \partial \tilde{d})^{2}- \frac{3 2 k^{2}( \Delta + 4 )}{\Delta^{2}}\left( 2 e^{- a \sqrt{4 / 3}\tilde{d}}- e^{- 2 a \sqrt{4 / 3}\tilde{d}}- 1 \right) \right] .
W = - i c ( \omega - \omega^{- 1}) \,{\frac{\eta_{1}+ \eta_{2}^{1 / N_{1}}+ \eta_{3}^{1 / N_{1}}+ \eta_{1}^{- 1}+ \eta_{2}^{- 1 / N_{1}}+ \eta_{3}^{- 1 / N_{1}}}{( 1 - 2 \xi ) \left( \eta_{1}- \eta_{1}^{- 1}\right) + ( 1 + \xi ) \left( \eta_{2}^{1 / N_{1}}+ \eta_{3}^{1 / N_{1}}- \eta_{2}^{- 1 / N_{1}}- \eta_{3}^{- 1 / N_{1}}\right)}}\ .
\mathbf{T}=\left[\begin{array}{cccc|cccc}-1 0 -1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 -1 0 0 0 0 \hline 0 1 0 0 -1 0 0 0 -1 0 -1 0 0 1 0 -1 0 1 0 1 -1 0 1 0 0 0 -1 0 0 0 0 -1\end{array}\right].
\begin{aligned}\min_{\{\lambda_n\}}~ -\sum_{n=1}^{KN_S}\tilde{w}_n \ln \left( 1 + \gamma_n \lambda_n \right) s.t.~ \sum_{n=1}^{KN_S}\lambda_n = KN_S, &\lambda_n \ge 0,~\mathrm{for}~n = 1, \cdots, KN_S,\end{aligned}
(1-f(x_{\lfloor|\Gamma|/2\rfloor+1}))\ll C_p(f)^{\frac{1}{p-1}}\left(\frac{1}{\deg(v)^{\frac{1}{p-1}}}+\sum_{n=1}^{\lfloor\log_2(|\Gamma|/\deg(v))\rfloor}\max_{\substack{A\subset X v\in A A |\Gamma|/2^{n+1}<|A|\le|\Gamma|/2^n}}j_{A,p}\right).
{\prod_{\bf k}}^{1 / 2}\delta \left({\cal E}_{a , \bf k}^{M \mathrm{R}}-{\cal E}_{a , - \bf k}^{M \mathrm{R}}\right) \delta \left({\cal E}_{a , \bf k}^{M \mathrm{I}}+{\cal E}_{a , - \bf k}^{M \mathrm{I}}\right) .
\begin{aligned}\sup_{0\leq\tau\leq t}\|(u,v,w)(\tau)\|_{s}^{2}+\int_{0}^{t}\|v(\tau)\|_{s+1}^{2}d\tau+ &+\int_{0}^{t}\|\left(u_{t}(\tau),v_{t}(\tau),w_{t}(\tau)\right)\|_{s-1}^{2}d\tau\leq M^{2}\end{aligned}
\frac{\partial^{2}{Z_{0}}_{l}( r )}{\partial r_{*}^{2}}- V ( r ){Z_{0}}_{l}( r ) = - \frac{4 \pi q_{s}m_{0}f ( r ) Y_{l}( 0 )}{r^{( d - 2 ) / 2}}( \frac{d t}{d \tau})_{r_{0}}^{- 1}\delta ( r - r_{0}) \, ,
\left|\nabla_{z}K_{N}^{1}(z,\zeta)\right|\lesssim\frac{1}{\prod_{j=1}^{n-1}\tau_{j}^{2}\left(z,\delta_{\Omega}(z)\right)}\frac{1}{\left|z-\zeta\right|}\frac{1}{2^{-i}\delta_{\Omega}(z)},
P ( l ) = \frac{2 \pi \lambda}{l !^{2}( l + 1 ) !^{2}}\left( \frac{\omega r_{0}}{2}\right)^{2 l + 2}\left| \frac{\Gamma^{2}\left( \frac{l + 2}{2}- \frac{i \lambda}{2}\right)}{\Gamma \left( 1 - i \lambda \right)}\right|^{2}\, .
\left\lbrace \begin{array}{lcr}(dd^c v_\lambda)^n=(\Vert u_{\lambda}\Vert_{C^0(\bar \Omega)}^{-1}-\lambda v_\lambda)^nf^n\omega^n \textnormal{in}\Omega v_\lambda =0 \textnormal{in}\partial\Omega \Vert v_\lambda \Vert_{C^0(\bar \Omega)}= 1.\end{array}\right.
y_{k}\left [ l \right ] = \mathbf{h}_{k}^{H}\mathbf{x}\left [ l \right ]+z_{k}\left [ l \right ] = \mathbf{h}_{k}^{H}\mathbf{p}_{\mathrm{c}}s_{\mathrm{c}}\left [ l \right ]+ \mathbf{h}_{k}^{H}\sum_{i \in \mathcal{K}}\mathbf{p}_{\mathrm{p},i}s_{\mathrm{p},i}\left [ l \right ] +z_{k}\left [ l \right ],
y^{(m)}_{r_1,\ldots,r_k}=\Bigl(\prod_{i=1}^k\mu(r_i)g(r_i)\Bigr)\sum_{\substack{a_1,\ldots,a_k r_i\mid a_i\forall i}}\frac{y_{a_1,\ldots,a_k}}{\prod_{i=1}^k\varphi(a_i)}\sum_{\substack{d_1,\ldots,d_k d_i\mid a_i,r_i\mid d_i\forall i d_m=1}}\prod_{i=1}^k\frac{\mu(d_i)d_i}{\varphi(d_i)}.
\xi(s,\chi) = \Big(\frac{q}{\pi}\Big)^{\frac{s+{\mathfrak a}}{2}}\Gamma\Big(\frac{s+{\mathfrak a}}{2}\Big) L(s, \chi) = e^{A+Bs}\prod_{\rho}\Big(1-\frac{s}{\rho}\Big) e^{s/\rho},
\frac{d\phi_{i}}{dt}=\frac{\partial H^{\prime}}{\partial I_{i}}=\omega_{i}+\frac{K}{N}\sum_{k=1}^{N}[2\sqrt{I_{i}I_{k}}sin(\phi_{k}-\phi_{i})+\sqrt{I_{k}/I_{i}}(I_{k}-I_{i})sin(\phi_{k}-\phi_{i})].
h_e(G) = \frac{|e(S,S^c)|}{|S|}\geq \frac{s(n-s)(d-1)}{s\cdot k\cdot d^{k}}\geq \frac{n}{2}\cdot \frac{(d-1)}{kd^{k}}= \frac{\alpha \cdot \mu_{d,k}(d-1)}{2k\cdot d^k}&= \frac{\alpha \cdot (d^{k+1}-1)}{2k\cdot d^k}.
\begin{array}{r l}{\bar{\phi}_{a b}( z ) : =}&{( \bar{\phi}_{a b}^{( 0 )}( z ) , \cdots , \bar{\phi}_{a b}^{( n - 1 )}( z ) ) ,}\\{\bar{\phi}_{a a + \bar{\epsilon}_{j}}^{( i )}( z ) : =}&{( \tilde{\Phi}( z ) )_{j}^{i}/ C h ( z - ( n - 1 ) / 2 ) ,}\\ \end{array}
\limsup_{n\to+\infty}n(2\ln n)^{\frac{1}{2}}\sup_{y\in I}\Big(\mathbb{P}(\xi^{(n)}([y,y+G_n(x)/S(I)])=0) -D_n(\sqrt{4-y^2}/S(I) \cdot G_n(x)/2)\Big)\leq 0.
L_{-1}^{(kp^t)}V_{-kp^t}\subset \sum_{i=1}^{p^t}L_{-1}^{(kp^t+i)}V_{-kp^t-i}+(L_{1}^{+}V)_0 \subset L_{-1}^{((k+1)p^t)}V_{-(k+1)p^t}+(L_{1}^{+}V)_0
\frac{Lv}{v}\leq \frac{1}{v}(w^2\alpha(\alpha-1)(1-\eta)^{\alpha-2}+4\delta(1-\eta)^{\alpha}) \leq \frac{1}{v}(-b^2(1-\eta)^2\alpha(1-\alpha)(1-\eta)^{\alpha-2}+4\delta(1-\eta)^{\alpha}) < 0in D_{T^*},
D_n^{(1)}=\begin{cases}8(n+2)(2n+1)(2n+3)(4n+7)(4n+9)f_0f_6^3 , 6(n+2)(2n+1)(3n+5)(3n+7)f_0f_5^2 , 3(n+2)(3n+4)(3n+5)f_5^2 .\end{cases}
\sum \limits_{i=1}^{m-1}\sum \limits_{j=1}^{m-1}\sum \limits_{k=1}^{m-1}l_{jk}\tau_{ij}\tau_{ik}= tr( \{\Gamma}^T L{\Gamma}) =tr( L \{\Gamma}{\Gamma}^T ) = \sum \limits_{i=1}^{m-1}\sum \limits_{j=1}^{m-1}l_{ij}h^{ji}= \sum \limits_{i=1}^{m-1}\sum \limits_{j=1}^{m-1}l_{ij}h^{ij}.
\left({}^{c}D_{0+}^{\alpha,\beta,\gamma}f\right)(x)=\left(I_{0+}^{-\alpha+[\operatorname{Re}(\alpha)]+1,-\beta-[\operatorname{Re}(\alpha)]-1,\alpha+\gamma-[\operatorname{Re}(\alpha)]-1}f^{([\operatorname{Re}(\alpha)]+1)}\right)(x)
\bar{u}^{LF}_{\sigma_1}(k_1)=\sum_{\sigma'_1=\pm\frac{1}{2}}\beta^*_{\sigma_1\sigma'_1}(k_1)\bar{u}_{\sigma'_1}(k_1),\quad\bar{u}^{LF}_{\sigma_2}(k_2)=\sum_{\sigma'_2=\pm\frac{1}{2}}\beta^*_{\sigma_2\sigma'_2}(k_2)\bar{u}_{\sigma'_2}(k_2),
\partial_{\mu}\partial_{\nu}\partial_{\rho}\sigma^{2}= \delta_{\mu \nu}( 8 \lambda^{2}x_{\rho}+ 2 c_{\rho}) + \delta_{\mu \rho}( 8 \lambda^{2}x_{\nu}+ 2 c_{\nu}) + \delta_{\rho \nu}( 8 \lambda^{2}x_{\mu}+ 2 c_{\mu}) .
\| ( t_{0}, t_{1}, t_{2}, t_{3}) \|_{c}^{2}= c \sum_{0}^{3}\| t_{i}( + \infty ) \|^{2}+ \int_{0}^{+ \infty}\sum_{0}^{3}\left( \| t_{i}( s ) \|^{2}- \| t_{i}( + \infty ) \|^{2}\right) d s .
\left[ Q_{k + 2}^{+}, \, H_{k + 2}\right] = \left[ W^{\prime}, \, Q_{k + 2}^{+}\right] + \left( k + 2 \right) W A^{-}Q_{k + 2}^{+}+ \frac 1 4 \left( k + 1 \right) \left( k + 2 \right) W^{\prime}Q_{k}^{+}.
g([X,Y], Z) =-g(h(X,tY), \varphi Z)+(1-\lambda) g(\widetilde\nabla_{X}Y,Z)+g(\widetilde\nabla_{X}Z, ntY) -g(h(Y,tX), \varphi Z)+(1-\lambda)g(\widetilde\nabla_{Y}X,Z)+g(\widetilde\nabla_{Y}Z, ntX)-2g(\widetilde\nabla_{Y}X, Z).
a^{\mp}= a^{\mp}( \vec{\sigma}\cdot \vec{L}+{\bf 1}) = \frac{1}{\sqrt{2}}\left\{\mp \Sigma_{1}\left( \frac{\partial}{\partial r}+ \frac{1}{r}\right) \pm \frac{1}{r}( \vec{\sigma}\cdot \vec{L}+{\bf 1}) \Sigma_{1}\Sigma_{3}- \Sigma_{1}r \right\}
K_{r}( u ) = \left( \begin{array}{l l}{{2 y_{0}( q t u - q^{- 1}t^{- 1}u^{- 1}) \;}}&{{y_{+}( q u^{2}- q^{- 1}u^{- 2})}}\\{{y_{-}( q u^{2}- q^{- 1}u^{- 2}) \;}}&{{- 2 y_{0}( t^{- 1}u - t u^{- 1})}}\\ \end{array}\right) \, ,
M_j=\max \{p_j(x_j^{*}),u_{j+\frac{1}{2}}^{-},u_{j+\frac{1}{2}}^{+}\}, m_j=\min \{p_j(x_j^{*}),u_{j+\frac{1}{2}}^{-},u_{j+\frac{1}{2}}^{+}\}.
\iint\limits_{X\times Y}\sum_j\frac{1}{\mu(Q_{j})}\int_{Q_{j}}f(x_1,y_2)\,d\mu(y_2)\,\chi_{S_j}(x)\, d\nu(x_1)\,d\mu(x_2) &=\int_X \bigg(\sum_j\int_{S_j(x_1)}\frac{1}{\mu(Q_{j})}\int_{Q_{j}}f(x_1,y_2)\,d\mu(y_2)\,\chi_{S_j}(x)\,d\mu(x_2)\bigg)\,d\nu(x_1) &\leq \int_{X\times Y}f(x_1,y_2)\,d\mu(y_2)\,d\nu(x_1)=\|f\|_{L^1}.
r_t = 0.4 \cdot e^{-j0.9349\pi}, r_r = 0.9 \cdot e^{-j0.9289\pi}, \sigma_{t,1}= 0.35 \cdot e^{-j0.8537\pi}, \sigma_{r,1}= 0.9 \cdot e^{-j0.7464\pi}, \sigma_{t,2}= 0.4 \cdot e^{-j0.4583\pi}, \sigma_{r,2}= 0.9 \cdot e^{-j0.2649\pi}.
2\Phi^i_lh_{00}^{\frac{m+3}{2}}W_0^{m+1}= h_{00}^{m+2}W_0C^i_{(0)l}+ h_{00}^{m+2}C^i_{(11)l}+ h_{00}^{\frac{m+3}{2}}W_0^{m+1}Ci_{(12)l}+ (h_{00})^{m+1}W_0C^i_{(21)l}+ h_{00}W_0^{2m+1}C^i_{(22)l}+ W_0^{2m}h_{00}C^i_{(3)l}+ W_0^{2m+1}C^i_{(4)l},
\mathrm{Re}\Big[\frac{\Gamma'}{\Gamma}(\frac{1}{2}(\frac12+it+a + ib))\Big]=\log N -\sum_{n=0}^N \frac{2(2n+a+\frac12)}{(2n+a+\frac12)^2+(t+b)^2}+ O\big(\frac{1+|t+b|}{N}\big).
\frac{1}{k}\sum_{j=1}^{k}\left (N_j - \binom{\deg h_j+2}{2}\right ) \ge \frac{N - (\sum_{j=1}^k \binom{\deg h_j+2}{2})}{k}\ge \frac{N-\binom{b_N+2}{2}}{k}\geq \frac{\varepsilon_0 N}{b_N}\ge \varepsilon_0\sqrt{N}
\frac{\varphi(4n)}{|\mathcal{H}|}=\frac{|\mathbb{Z}_{4n}^*|}{|\mathcal{H}|}=|\mathcal{O}_S|=\frac{|\mathbb{Z}_{n}^*|}{|\mathcal{H}'|}=\frac{\varphi(n)}{|\mathcal{H}'|}.
\frac{\partial W}{\partial k_{ij}}=\frac{1}{2}\alpha_{ij}+\frac{1}{2}\frac{\partial\alpha_{kl}}{\partial k_{ij}}k_{kl}=\frac{1}{2}\alpha_{ij}+\frac{1}{2}k_{ji}-\frac{1}{2}\delta_{ij}k_{kk}=\alpha_{ij}
S = - 1 6 ( 0 , 6 0 6 )^{9}1 . 2 \frac{2 \pi}{( 4 \pi^{2}\alpha^{\prime})^{1 . 2}g}\int d t \left[ 1 + \frac{( 2 \pi \alpha^{\prime})^{2}}{4}( 2 \partial_{t}X^{i}\partial_{t}X^{i}+ 2 i \overline{{{\theta}}}\Gamma^{0}\partial_{t}\theta ) \right]
I_{Stmod}= -\frac{1}{4}F_{mn[A]}F^{mn}_{[A]}- \frac{1}{2}\mu^2[A_{m}+ \frac{1}{2\mu}l_m][A^{m}+ \frac{1}{2\mu}l_m] + \frac{1}{4}\epsilon^{mnpq}B_{mn}\partial_{p}l_q.
\left\vert \left\langle y^*,f(x)\right\rangle \right\vert =\left\vert \left\langle y^*,T_{f}(\delta_{U}(x)\right\rangle \right\vert & \leq d_{p}(T_{f})\left\Vert y^*\right\Vert_{L_{p^*(\mu )}}\left\Vert \delta_{U}(x)\right\Vert & =d_{p}(T_{f})\left\Vert y^*\right\Vert_{L_{p^*(\mu )}}
\sum_{n,m \ge 1}B(n,m)x^my^n = \sum_{n,m}\sum_{s_1,\dots,s_n}\frac{(1\cdot t_1)^{s_1}}{s_1!}\cdots \frac{(n\cdot t_n)^{s_n}}{s_n!}x^my^n &=\prod_{j\ge 1}\sum_{s\ge 0}\frac{(jt_jxy^j)^s}{s!}&= \exp \left( x\sum_{j\ge 1}jt_jy^j\right) = e^{xR(y)},
\mathbf E[H_0^\circ(n+1)|\mathcal F_n] = \mathbf E[F_0^\circ(n+1)|\mathcal F_n] = \frac n{n+1}F_0^\circ(n)+\frac1{n+1}\sum_{\ell\geq0}F_\ell^\circ(n)(1-p)^\ell \geq F_0^\circ(n) = H_0^\circ(n),
-\sum_{\substack{i,j=1, \i\not=j}}^n\frac{a_{ij}}{(\alpha_i-\alpha_j)^2}=-C_0\sum_{i=1}^{2k}\sum_{\ell=1}^{2k-i+1}\frac{(-1)^{i-1}}{(\theta_\ell+\cdots+\theta_{\ell+i-1})^2},
\binom{n+\ell-1}{n}+ \frac{\ell-1}{n+1}\binom{n+\ell-1}{n}= \frac{n+\ell}{n+1}\binom{n+\ell-1}{n}= \frac{\ell}{n+1}\binom{n+\ell}{n}begin{align*}0.5em] \binom{n+\ell-1}{n-1}+ \binom{n+\ell-1}{n}= \binom{n+\ell}{n}
{\cal S}^{D 0}= \frac{1}{2 g_{S}}\int d t ~ \mathrm{T r}\left( G_{i j}\dot{X}^{i}\dot{X}^{j}+ \frac{1}{2}\frac{1}{( 2 \pi )^{2}}G_{i j}G_{k l}[ X^{i}, X^{k}] [ X^{j}, X^{l}] + \right.
0\in \partial f(x^{k+1})+\frac{1}{\eta}(x^{k+1}-x^k-\eta A^T\lambda^k) 0 \in \partial g^*(\lambda^{k+1})+\frac{1}{\eta}(\lambda^{k+1}-\lambda^k+\eta A(2x^{k+1}-x^k)) \ .
h_1(\lambda,s)=\begin{cases}(g^\ast_1\tilde{\mathcal{A}})_{(\lambda,s)}\oplus(g^\ast_2\tilde{\mathcal{A}})_{(\lambda,s)},0\leq \lambda\leq\frac{1}{2}(g^\ast_2\tilde{\mathcal{A}})_{(\lambda,s)}\oplus (g^\ast_1\tilde{\mathcal{A}})_{(\lambda,s)},\quad\frac{1}{2}\leq \lambda\leq 1.\end{cases}
\,\big\langle \gamma_1, \gamma_2,\delta_A,\eta_i\big\rangle_{0,\beta_1}\big\langle \eta^i,\gamma_3,\delta_B,\delta \big\rangle_{0,\beta_2}&=\Phi_{0,|A|+3,\beta_1}\otimes\Phi_{0,|B|+3,\beta_2}([\overline{M}_{0,|A|+3}\times\overline{M}_{0,|B|+3}]\boxtimes\delta_A\boxtimes\gamma_1\boxtimes\gamma_2\boxtimes\eta\boxtimes \delta_B\boxtimes\gamma_3\boxtimes\delta ) .
\phi^{\alpha}= \left( \begin{array}{c c}{U_{k + \times k +}}&{0}\\{0}&{U_{k - \times k -}}\\ \end{array}\right) \mathrm{a n d}\phi^{i}= \left( \begin{array}{c c}{0}&{U_{k + \times k -}}\\{U_{k - \times k +}}&{0}\\ \end{array}\right) .
Z_{o n e - l o o p}= \int{\frac{d^{2}\tau}{( \mathrm{I m}\tau )^{{\frac{D}{2}}+ 1}| \eta |^{2 4}}}\sum_{\alpha , \beta}C \Big [ \begin{array}{c}{\alpha}\\{\beta}\\ \end{array}\Big ] \prod_{f \in F^{\prime}}\Theta^{1 / 2}\Big [ \begin{array}{c}{\alpha ( f )}\\{\beta ( f )}\\ \end{array}\Big ] ( \tau ) .
G_b =F_b\big|_{c\leftarrow \frac{F_c |_{b=0}}{d}}=(a+1)^2+\left(\frac{a^3+a^2}{ad}\right)^2 &=(a+1)^2\left\{1+\left(\frac{a}{d}\right)^2\right\}=\frac{(a+1)^2}{d^2}(a^2+d^2),
g_1= t_1 t_3 +{(q - 2) (q - 3)\over 6}t_1^4 + ( - q + 2) t_1^2 t_2+{1\over 2}t_2^2, g_2= t_2 t_3-{(q^2 - 1) (q - 2)\over 5}\,t_1^5+{2 q (q - 1)\over 3}t_1^3 t_2 +( - q + 1) t_1 t_2^2, g_3={1\over 2}t_3^2 +{4 (q^2 - 2 q + 2) (q - 1)\over 15}t_1^6 -(q - 1) (q - 2) t_1^4 t_2 + 2(q - 1) t_1^2 t_2^2 -{1\over 3}t_2^3,
\left( \frac{e^{- s w}}{1 - e^{- w}}+ \frac{e^{- (1-s)\bar{w}}}{1 - e^{- \bar{w}}}\right) =\left( \frac{e^{- s z}}{1 - e^{- z}}+ \frac{e^{- (1-s)\bar{z}}}{1 - e^{- \bar{z}}}\right),
I_{1}( d , a , c ) ={\frac{1}{2 ( a^{2}+ c^{2}- d^{2})}}\ln \left({\frac{c^{2}}{( d + a )^{2}}}\right) +{\frac{a}{( c^{2}- d^{2})^{1 / 2}( a^{2}+ c^{2}- d^{2})}}\operatorname{a r c t a n}\left({\frac{( c^{2}- d^{2})^{1 / 2}}{d}}\right)
\left| \int_{R_l}^{f^{a}}{\bf 1}_E(s) \log^{\frac{p}{p-1}}|B(i s)B(-i s)| ds \right|^{(p-1)/p}<[ f^a e^7 \left|2 \overline M \log f \right|^{\frac{p}{p-1}}\sum_{k=0}^\infty (k+1)^{\frac{p}{p-1}}f^{k}]^{(p-1)/p}\le f^{-1 + 3a/2 + a(p-1)/p}\log f^{-1}
G_{1, \varphi}(x) = I + \varepsilon N_{\varphi}(x) = \left(I + \varepsilon N_{1,\varphi}^{-}(x) + \varepsilon^2 N_{2,\varphi}^{-}(x)\right) \left(I + \varepsilon N_{1,\varphi}^{+}(x) + \varepsilon^2 N_{2,\varphi}^{-}(x)\right),
|U^\omega(x) - U^\omega(y)| = \lim_{n_k\to \infty}n_k^{-\frac{1}{d}}|U^\omega_{n_k}(x) - U^\omega_{n_k}(y)| \leq c_ddM^\frac{d-1}{d}\|f^*\|_{L^\infty([0,M]^d)}^\frac{1}{d}|x-y|^\frac{1}{d}
G ( \vartheta ) = \frac{1}{2 \pi}\int_{- \infty}^{+ \infty}d k e^{i k \vartheta}\frac{\sinh \frac{\pi ( p - 1 ) k}{2}}{2 \sinh \frac{\pi p k}{2}\cosh \frac{\pi k}{2}}, p = \frac{\pi}{\gamma}- 1
\sum^n_{j=0}(-1)^j\mathrm{dim\,}H^j_{b,m}(X,E)=\int_X\Bigr(\mathrm{Tr\,}e^{-t\widetilde{\Box}^+_{b,m}}(x,x)-\mathrm{Tr\,}e^{-t\widetilde{\Box}^-_{b,m}}(x,x)\Bigr)dv_X.
L \overset{\Delta}{=}\underset{\mathbf{v}_1 \ne \mathbf{v}_2}{\sup}\frac{\left\| \mathbf{f}_{\boldsymbol{\psi}}(\mathbf{v}_1) - \mathbf{f}_{\boldsymbol{\psi}}(\mathbf{v}_2) \right\|_2}{\left\| \mathbf{v}_1 - \mathbf{v}_2 \right\|_2}.
|u(y)\psi(y)| \le w^{-s_1}(y)u_1(y)\|\psi\|_{L^{\infty}_{-(s-s_1)}}&= w^{-p_2}(y)w^{-\frac12-p_2}u_1(y)\|\psi\|_{L^{\infty}_{-(s-s_1)}}&=w^{-p_2}(y)u_2(y)\|\psi\|_{L^{\infty}_{-(s-s_1)}},
P(x_1,x_2,x_2) =(e_1 -1)x_1 +\left(\sum_{i=2}^9 (e_i -1)\right)\frac{1-x_1}{2}&=\frac12 \sum_{i=2}^9 (e_i -1) +\frac{x_1}{2}\left[2(e_1 -1) - \sum_{i=2}^9 (e_i -1)\right].
\begin{aligned}\mathrm{Cont}(Q^0)|_{b_l=0}= -(hr+1-b_{I\setminus \{a_l\}}) \mathrm{Cont}(Q) \mathrm{Cont}(Q^j)|_{b_l=0}= (r - b_{I_j}+ |Q_j|) \mathrm{Cont}(Q) .\end{aligned}
\tilde g_{ij}=[\frac{1}{2}(\bar p^*F)^2]_{\tilde y^i\tilde y^j}=\frac{1}{2}\frac{\partial^2(\bar p^*F^2)}{\partial \tilde y^i\partial \tilde y^j}=\bar p^*[\frac{1}{2}F^2]_{y^i y^j}=\bar p^*g_{ij}.
E_{o_-}(\lambda)T_{n_v}\in T_{n_v}E_{o_-}(n_v^{-1}\cdot\lambda) + \sum_{v_2 < v,\mu \in \Lambda(1)}C[q_s^{\pm 1}]T_{n_{v_2}}^*E(\mu) & = T_{n_v}E_{o_-}(n_v^{-1}\cdot\lambda) + \sum_{v_2<v,\mu\in\Lambda(1)}C[q_s^{\pm 1}]E_-(n_{v_2}\mu).
\sum_{\mu \in G\times \widehat{G}}c_\mu\pi(\mu) =\pi(\lambda)^{-1}S_{f, \tau, \Lambda}\pi (\lambda)=\pi(\lambda)^{-1}\left(\sum_{\mu \in G\times \widehat{G}}c_\mu\pi(\mu)\right)\pi (\lambda) &=\sum_{\mu \in G\times \widehat{G}}c_\mu\pi(\lambda)^{-1}\pi(\mu)\pi(\lambda)=\sum_{\mu \in G\times \widehat{G}}c_\mu d_{\mu, \lambda}\pi(\mu).
\Gamma_{\mu}( p^{\prime}, p ; k ) = ( p^{\prime}+ p )_{\mu}L ( p^{2}, p^{2}, k^{2}) + [ k_{\mu}( p^{2}\! - \! p^{2}) - ( p^{\prime}+ p )_{\mu}k^{2}] T ( p^{2}, p^{2}, k^{2}) ,
\hspace{- . 5 c m}\hat{T}_{m}( F ) = - \frac{\delta F}{\delta \phi^{I}}\Big ( \frac{\delta}{\delta \bar{\phi}_{I}}- \mathrm{~ \frac{1}{2}~}m^{2}( P_{-})_{J c}^{I c}\frac{\delta}{\delta \eta_{J}}\Big ) - \mathrm{~ \frac{1}{2}~}( \hbar / i ) \epsilon_{a b}\frac{\delta}{\delta \phi_{I a}^{*}}\frac{\delta^{2}F}{\delta \phi^{I}\delta \phi^{J}}\frac{\delta}{\delta \phi_{J b}^{*}}+ ( i / \hbar ) m^{2}F .
{\cal I}= \int_{{\cal F}}\frac{d^{2}\tau}{I m \tau}\sum_{m_{1}, m_{2}\atop n_{1}, n_{2}}\left( \sum_{b = e v e n}q^{\frac{1}{2}p_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}h_{0}( \tau ) + \sum_{b = o d d}q^{\frac{1}{2}p_{L}^{2}}\bar{q}^{\frac{1}{2}p_{R}^{2}}h_{1}( \tau ) - c ( 0 ) \right)
\begin{array}{rclrclrcl}\mathsf Z(10) = \{(1,0,1),(0,2,0)\}, \mathsf{L}(10) = \{2\}, \Delta(10) = \emptyset, \mathsf Z(12) = \{(4,0,0),(0,1,1)\}, \mathsf{L}(12) = \{2,4\}, \Delta(12) = \{2\}, \mathsf Z(14) = \{(3,1,0),(0,0,2)\}, \mathsf{L}(14) = \{2,4\}, \Delta(14) = \{2\}.\end{array}
\frac{dn_{abs(\lambda,T)}}{dT}=\frac{n_{(\lambda,T_{0})}^{2}-1}{2n_{(\lambda,T_{0})}}(D_{0}+2D_{1}\Delta T+3D_{2}\Delta T^{2}+\frac{E_{0}+2E_{1}\Delta T}{\lambda^{2}-\lambda_{TK}^{2}})
|\mu_g(0)| \geq \frac{1}{4}-\sum_{\gamma \neq e \in \Gamma_g}\frac{|\gamma'(0)|^2}{\rho(0)}& =\frac{1}{4}-\frac{1}{4}\sum_{\gamma \neq e \in \Gamma_g}(1-|\gamma(0)|^2)^2 & > \frac{3}{16}\eqref{cis-5}.
\max_{z_n^{\rm R}\ge 0,z_n^{\rm I}\ge 0}\ln \left(\frac{1}{2}+\frac{1}{2}{\rm{erf}}\big(z_n^{\rm R}+\sqrt{P/2}\big)\right) + \ln\left(\frac{1}{2}+\frac{1}{2}{\rm{erf}}\big(z_n^{\rm I}+\sqrt{P/2}\big)\right) \mathrm{s.t.}~~~{z_n^{\rm R}}^2+{z_n^{\rm I}}^2 = A_n.
R_{\mathcal{S}^{*}_{c}}(F) = \begin{dcases}\sigma_{0}\ 2\alpha+\beta-2\geqslant 0 \sigma_{0}\ 2\alpha+\beta-2<0\ \X(\alpha,\beta)\leqslant 0 \tilde{\sigma_{0}}\ 2\alpha+\beta-2<0\X(\alpha,\beta)> 0,\end{dcases}
\left| B_{N}\right\rangle_{B = 0}= \left( \frac{- \det g_{\mu \nu}}{( 2 \alpha^{\prime})^{p + 1}}\right)^{\frac{1}{4}}\prod_{n = 1}^{\infty}\exp \left( - \frac{1}{n}\alpha_{- n}^{\mu}g_{\mu \nu}\tilde{\alpha}_{- n}^{\nu}\right) \left| \mathbf{0}\right\rangle .
\bar{D}^{\dot{\alpha}}J_{\alpha \dot{\alpha}}^{0}= \frac{1}{3}D_{\alpha}\left\{\left[ 3{\cal W}- \sum_{i}S_{i}\, \frac{\partial{\cal W}}{\partial S_{i}}\right] - \left[ \frac{b}{1 6 \pi^{2}}\mathrm{T r}\, W^{2}+ \frac{1}{8}\sum_{i}\gamma_{i}Z_{i}\bar{D}^{2}\left({S_{i}}^{\dagger}e^{V}S_{i}\right) \right] \right\}
\bar{S}^{\prime}= \int_{0}^{t_{0}}\left[{\textstyle{\frac{1}{2}}}\epsilon_{\alpha \beta}\dot{\phi}_{\alpha}( t ) \dot{\phi}_{\beta}( t ) + \left( j_{\alpha}( t ) + \tilde{L}_{\alpha}( t ) \right) \phi_{\alpha}( t ) \right] d t .
xP_n(u\varphi)(x) = x\int_{\real}\check{\psi_n}(x-y)u(y)\varphi(y)~dy &= \int_{\real}(x-y)\check{\psi_n}(x-y)u(y)\varphi(y)~dy+\int_{\real}\check{\psi_n}(x-y)yu(y)\varphi(y)~dy &= (x\check{\psi_n})*(u\varphi)+ \check{\psi_n}*(xu\varphi).
\|u\|^{p}_{L^{p^{*}}(\mathbb{G})}\leq C\sum_{k\in\mathbb{Z},\,a_{k}\neq0}2^{kp}a^{-sp/Q}_{k}a_{k+1}\leq C\int_{\mathbb{G}}\int_{\mathbb{G}}\frac{|u(x)-u(y)|^{p}}{q^{Q+sp}(y^{-1}\circ x)}dxdy= C[u]^{p}_{s,p,q}.