harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Lambda^{(2:2)}_{n,2,\lambda}=\frac{16\cdot \left(1+o(1) \right)}{90^2n^4\delta^4_n(k)}\cdot\sum\limits_{\mathbf{P}_{k}\in \mathcal{W}_{k}(\lambda)}\sum\limits_{\mathbf{i}\in \mathcal{I}_2 \times \mathcal I_2}\varphi_{k-1}\left( \mathbf{P}_{k},\mathbf{i}\right) .
S_t((Y,S, \widetilde W)_0, dW_\cdot)=e^{-\mu t}S_0+ e^{-\mu t}\int_0^t e^{\mu s}F^2\left(Y_s\left((Y,S,\widetilde W )_0,dW_\cdot\right)\right)dW_s, t\ge0.
f_r\left(s\right)=-1+e^{-\frac{i\beta\eta}{2}\omega^2_r s}, f_{r1}\left(s\right)= \frac{\omega^3_r}{4}\left(\frac{\beta^2\eta^2}{2}+\gamma-\eta^2\right) s e^{-\frac{i\beta\eta}{2}\omega^2_r s}
(\alpha'_{n}(b), \beta'_{n}(b), \tau'_{n}(b),\mu'_{n}(b)) :=(z^{(1)}_{n}(b),z^{(2)}_{n}(b),z^{(3)}_{n}(b),z^{(4)}_{n}(b)) \ , (\alpha'_{n}(b), \beta'_{n}(b), \tau'_{n}(b),\mu'_{n}(b)) :=(z^{(3)}_{n}(b),z^{(4)}_{n}(b),z^{(1)}_{n}(b),z^{(2)}_{n}(b))
\frac{1}{2}\frac{d}{dt}\|\bar u_\varepsilon\|_2^2+\|\nabla\bar u_\varepsilon\|_2^2=\int_\Omega[(\nabla_H\cdot v_\varepsilon)\bar u_\varepsilon-(\partial_z v_\varepsilon\cdot\nabla_H)\bar v_\varepsilon]\cdot\bar u_\varepsilon d\textbf{x}.
N_{R}( p_{n}, t ) = \int_{- V / 2}^{V / 2}\frac{d x d y}{V^{2}}e^{i p_{n}( x - y )}\frac{\exp [ 2 \pi i / V ( x - y ) ] \exp [ 2 \pi i Q_{t o p}( x - y ) / V}{1 - \exp [ 2 i ( x - y ) / V ]}.
B_{4}=-\frac{b_{4}^{\prime \prime}}{2b_{1}}+\frac{1}{2b_{1}^{2}}(4b_{1}^{\prime \prime}b_{4}-4b_{2}^{\prime}b_{4}+6b_{3}b_{4}+5b_{1}^{\prime}b_{4}^{\prime}-3b_{4}^{\prime}b_{2})+\frac{1}{b_{1}^{3}}(3b_{1}^{\prime}b_{2}b_{4}-4(b_{1}^{\prime})^{2}b_{4})
\widetilde{C_{ont,l}}(\beta_a)(X\theta,Y\xi) &:=\frac{1}{2}\big(1_{(\theta,\xi)=(1,-1)}C_{ont,l}(\beta_a)(X,Y)-1_{(\xi,\theta)=(1,-1)}C_{ont,l}(\beta_a)(Y,X)\big), &(\forall X,Y\in \hat{I}_0,\theta,\xi\in \{1,-1\}).
J^{( 4 )}( 4 ; 1 , 1 , 1 , 1 ) ={\textstyle{\frac{1}{1 2}}}\; \mathrm{i}\pi^{2}\; \frac{\Omega^{( 4 )}}{V^{( 4 )}}= \frac{2 \; \mathrm{i}\pi^{2}}{m_{1}m_{2}m_{3}m_{4}}\; \frac{\Omega^{( 4 )}}{\sqrt{D^{( 4 )}}}.
\int_0^rt^{a+n-1}[\log(2+t)]^b\,dt &\quad=\int_0^Nt^{a+n-1}[\log(2+t)]^b\,dt+\int_N^r\cdots &\quad<\int_0^Nt^{a+n-1}[\log(2+t)]^b\,dt+\frac{2}{a+n}r^{a+n}[\log(2+r)]^b,
\langle J\psi,J\varphi\rangle_\Delta = \langle J\psi,J\varphi\rangle+\langle\Delta^{1/2}J\psi,\Delta^{1/2}J\varphi\rangle = \overline{\langle \psi,\varphi\rangle}+\overline{\langle\Delta^{-1/2}\psi,\Delta^{-1/2}\varphi\rangle}= \overline{\langle J\psi,J\varphi\rangle_{\Delta^{-1}}}\,.
\langle \Omega | \Phi_{1}( z_{1}) \Phi_{2}( z_{2}) \Phi_{3}( z_{3}) | \Omega \rangle \sim ( z_{1}- z_{2})^{h_{3}- h_{2}- h_{1}}( z_{1}- z_{3})^{h_{2}- h_{3}- h_{1}}( z_{2}- z_{3})^{h_{1}- h_{2}- h_{3}}.
2l\sum\limits_{P\in Br}\left(1-\frac{1}{m_P}\right) \geq 2l(\frac{s+t}{2}\cdot \frac{1}{2}+\frac{r}{2}(1-\frac{1}{2l})+\frac{k}{4}(1-\frac{1}{l})) &=\frac{l}{2}(s+t)+r\frac{2l-1}{2}+k\frac{l-1}{2}.
\left| \left| w_{\dot{\alpha}}^{~ u}\right| \right| = \left| \left| \bar{w}_{~ u}^{\dot{\alpha}}\right| \right| = \rho \, T \, \left( \begin{matrix}{{0_{[ N - 2 ] \times [ 2 ]}}}\\{{1_{[ 2 ] \times [ 2 ]}}}\\ \end{matrix}\right)
\sum_{n=0}^{\infty}\mathbf{E}Y_{n}\, \leq\,\frac{c_\beta}{1+\beta}\,c_{(\ref{65})}c^{-1-\beta}\,t^{\gamma(1+\beta)}\sum_{n=0}^{\infty}2^{-(n+1)\gamma(1+\beta)}\, =\,\frac{c_\beta}{1+\beta}\,c_{(\ref{65})}c^{-1-\beta}\,t^{\gamma(1+\beta)}\,\frac{2^{-\gamma(1+\beta)}}{1-2^{-\gamma(1+\beta)}}\,.
E(\mu) \dot{\tilde{x}}(t,\mu) = \tilde{f}(\tilde{x}(t,\mu),\mu) = f(\tilde{x}(t,\mu)+x^*(\mu),\mu) begin{align*}1ex] y(t,\mu) = \tilde{g}(\tilde{x}(t,\mu),\mu) = g(\tilde{x}(t,\mu)+x^*(\mu),\mu) ,
\phi_3(g_1+g_3+g_5+g_7)=\phi_3(g_1+g_4+g_6+g_7)=\phi_3(g_2+g_3+g_6+g_7)=\phi_3(g_2+g_4+g_5+g_7) = \phi_3(g_1+g_2+g_3+g_4)=\phi_3(g_1+g_2+g_5+g_6)=\phi_3(g_3+g_4+g_5+g_6)=0,
P^\omega(C_t^c \mid S) = \frac{1}{P^\omega(S)}\left[P^\omega\left(C_t^c\cap S\cap A_{m(t)}\right)+P^\omega\left(C_t^c\cap S\cap A_{m(t)}^c\right)\right] &\leq c(\omega)\left[P^\omega\left(C_t^c \mid A_{m(t)}\right)+P^\omega\left(A_{m(t)}^c \mid S\right)\right] .
\Delta_{1}=-\left(\left(\gamma\left(\frac{\lambda}{c_{1}}\right)-1\right)-\frac{\lambda^{2}}{c_{1}^{2}}\gamma\left(\frac{\lambda}{c_{1}}\right)\right)=-\left(\boldsymbol{g}\left(\frac{\lambda}{c_{1}}\right)-\boldsymbol{g}\left(0\right)\right).
\begin{aligned}\frac{1}{2}\frac{{\rm d}}{{\rm d}t}\|u_h\|^2_{0,I_h}+ \|w_h\|^2_{0,I_h}+ a_h(u_h,u_h) + \sqrt{\epsilon}\,b_h(w_h,u_h) &+ \sqrt{\epsilon}\,b_h(u_h,w_h) + \left(\left(\rho_h u_h - \rho_h V_h\right), u_h\right) = 0.\end{aligned}
{\cal L}= \frac{1}{2g}[ \sum_{i=1}^{9}\hbox{tr}\dot{X}^{i}_n \dot{X}^{i}_{-n}- \sum_{j=2}^9 \hbox{tr}S^{j}_n(S^j_n)^+- \frac{1}{2}\sum_{j,k=2}^9 \hbox{tr}T^{jk}_n(T^{jk}_n)^+],
\begin{aligned}\Vert D^{\tilde{\gamma}}(u+u_b)\cdot D^{\gamma-\tilde{\gamma}+(1,0)}u \Vert_{L^2_y}&\leq C\sum_{|\alpha |\leq m}\Vert (u+u_e\phi(y)+u_b(1-\phi(y)))^{\frac{1}{2}}\langle y\rangle^{l+k}D^\alpha u\Vert_{L^2_y}^2+|u_b|\Vert u\Vert_{H^{m}_0}.\end{aligned}
r(0,2,2,2;n) = \frac1{(q)_n}\frac{\left(-q^{1+n}; q^2\right)_{\frac{n}{2}}}{\left(-q^2; q^2\right)_{\frac{n}{2}}}=\frac{\left(-q; q^2\right)_n}{(q)_n \left(-q^2; q^2\right)_{\frac{n}{2}}\left(-q; q^2\right)_{\frac{n}{2}}}=\frac{\left(-q; q^2\right)_n}{\left(q^2; q^2\right)_n}.
\sum_{k+1\leq i_1<\ldots<i_{r-1}\leq n \atop i_1,\ldots,i_{r-1}\not=l,j}\ddot{f}_{i_1}\ldots\ddot{f}_{i_{r-1}}=0, j=k+1,\ldots ,n\,\, \mbox{and $j\neq l$}
c_j={{j_{j+1/2}(i S_n \bar{\Omega}_n) h_{j+1/2}^{(1)}(i S_n \bar{\Omega}_n)- j_{j-1/2}(i S_n \bar{\Omega}_n) h_{j-1/2}^{(1)}(i S_n \bar{\Omega}_n)}\over{j_{j+1/2}^2(i S_n \bar{\Omega}_n) -j_{j-1/2}^2(i S_n \bar{\Omega}_n)}},
\frac{1}{2}\left\|u^{n+1}\sqrt{\varphi}\right\|_{L^{2}(\mathbb{R})}^{2}+ \Delta t \int_{\mathbb{R}}\mathcal{H}w_{x}(\varphi w)_{x}\,dx -\frac{\Delta t}{3}\int_{\mathbb{R}}w^{3}\varphi_{x}\,dx = \frac{1}{2}\left\|u^{n}\sqrt{\varphi}\right\|_{L^{2}(\mathbb{R})}^{2}.
\psi_{\bullet n}=e^{-in(ka+\pi)}\bigg[\frac{t_r}{t_l}\bigg]^{\frac12n}U_{n-1}\bigg(\frac{t_ue^{-ika}+t_de^{+ika}}{2\sqrt{t_rt_l}}\bigg), \psi_{\circ n}=e^{-in(ka+\pi)}\bigg[\frac{t_r}{t_l}\bigg]^{\frac12(N-n)}U_{N-n}\bigg(\frac{t_ue^{+ika}+t_de^{-ika}}{2\sqrt{t_rt_l}}\bigg),
J_\kappa(1_k;\theta)=\prod_{i=1}^l \,\prod_{j=1}^{\kappa_i}\,(k+1-i+(j-1)/\theta)=\theta^{-|\kappa|}\,\prod_{i=1}^l \frac{\Gamma(\!(k+1-i)\theta+\kappa_i)}{\Gamma(\!(k+1-i)\theta)}.
I_n(t,s;v) = \int_{\Sigma_n(t,s)}v(t_n)\otimes \dots\otimes v(t_1) d\L^n, I_n^\flat (t,s;v) = \int_{\Sigma_n^\flat (t,s)}v(t_n)\otimes \dots\otimes v(t_1) d\L^n.
\{\theta_{1}, \theta_{2}\}_{\mathrm{c . b .}}= \beta_{1}I + \beta_{2}P_{1 2}+ \left\{\theta_{1}[ \theta_{2}, r ] \right\}+ \beta_{3}( \theta_{1}\theta_{1}+ \theta_{2}\theta_{2}) + \beta_{4}( \theta_{1}P \theta_{2}+ \theta_{2}P \theta_{1}) .
\mbox{Im}\left[ \mbox{Li}_2\left(r e^{{\mbox{\scriptsize{i}}}\theta}\right) \right]= \omega \ln r +{\textstyle{1\over2}}\left[\mbox{Cl}_2\left(2\theta\right)+\mbox{Cl}_2\left(2\omega\right)+\mbox{Cl}_2\left(2\chi\right) \right],
\delta_{c_1}\circ \delta_{c_2}= q \delta_{c_1}+ q \delta_{c_2}+ v \delta_{c_3}+ w \delta_{c_4}, &\delta_{c_2}\circ \delta_{c_1}= q \delta_{c_1}+ q \delta_{c_2}+ w \delta_{c_3}+ v \delta_{c_4},
g_k(y) = y^{\nu}\int_y^{\infty}x^{-m+n}f_k(x) (x-y)^{m-n-\nu-1}dx & = y^{\nu}\int_0^1 \left(\frac{y}{x}\right)^{-m+n}\left(\frac{y}{x}- y\right)^{m-n-\nu-1}f_k\left(\frac{y}{x}\right) \frac{y dx}{x^2},
W = \{(k_x, k_y, 0) |\a(k_x, k_y) = -1, b(k_x, k_y) = 0 \}&\cup \{(k_x, k_y, \pi) |\a(k_x, k_y) = 1, b(k_x, k_y) = 0 \}, \pi(W) = \{(k_x, k_y) |\a(k_x, k_y) = \pm 1, b(k_x, k_y) = 0 \}= a^{-1}(\{\pm 1 \}) \cap b^{-1}(0), F = \{(k_x, k_y) |\ \lvert a(k_x, k_y) \rvert \le 1, b(k_x, k_y) = 0 \}= a^{-1}([-1, 1]) \cap b^{-1}(0).
\sum\limits_{k=1}^{n_1+1}u_k+\sum\limits_{k=n_1+2}^{n_2}u_k\leq \frac{t}{Tr_2}<\sum\limits_{k=1}^{n_1+1}u_k+\sum\limits_{k=n_1+2}^{n_2}u_k+u_{n_2+1}\,.
|T|= \begin{pmatrix}0 |\lambda_{1}| |\lambda_{2}| \ddots |\lambda_{n-1}| \end{pmatrix}\ \ \{\text and}\ \ |T^*|= \begin{pmatrix}|\lambda_{1}| |\lambda_{2}| \ddots |\lambda_{n-1}| 0 \end{pmatrix}
S_{\textrm{IPLA}}(R) = \sum_{i=1}^{N_{max}}\frac{R}{i\cdot\pi}\cdot & \int_{0}^{\infty}\bigg[ \mathcal{I}m \Big\{\Psi_{\textrm{IPLA}}(t;i, R) - \Psi_{\textrm{IPLA}}(t;i-1,R) \Big\}\bigg] dt,
\mathcal{E}(t):= \begin{cases}|\frac{(1+t^2)^{1/4}}{\sqrt{\xi/k^2}}Z_{NR}|^2 + |Z_R|^2 t< 0, |\frac{1}{\sqrt{\xi/k^2}}Z_{NR}|^2 + |(1+t^2)^{-1/4}Z_{R}|^2 t>0. \end{cases}
\left ( \delta F_p \right ) \left ( \phi_0, \cdots, \phi_{p+1}\right )= \sum_{i = 0}^{p+1}(-1)^i \, F_p \left ( \phi_0, \cdots,\phi_{i-1}, \widehat{\phi}_i, \phi_{i+1}, \cdots, \phi_{p+1}\right)
-{\frac{1}{1 6 \pi G}}\int_{r = r_{+}}\! \! d^{n - 2}x \, \left\{\sqrt{\sigma}\left( \sigma^{a c}n^{b}- \sigma^{a b}n^{c}\right) \left({\hat{\xi}}^{t}\nabla_{c}\delta g_{a b}- \nabla_{c}{\hat{\xi}}^{t}\delta g_{a b}\right) + 2{\hat{\xi}}^{a}\delta \pi_{a}{}^{r}-{\hat{\xi}}^{r}\pi^{a b}\delta g_{a b}\right\},
\widetilde{K}_n(x,y) = \left( \frac{2}{k A_k}\right)^{\frac{\alpha}{k}}y^{\alpha}e^{- \frac{2n}{k A_k}y^k}\frac{1}{2\pi i (x-y)}\begin{pmatrix}0 1\end{pmatrix}U_n^{-1}(y)U_n(x) \begin{pmatrix}1 0 \end{pmatrix},
\displaystyle{T ( H \gg H_{0}) = T_{1}\frac{2 4 m^{4}}{\omega^{3}}\bigg [ \frac{\omega_{1}}{\omega_{2}\sqrt{4 m^{2}- \omega_{2}^{2}}}\mathrm{a r c t a n}\left( \frac{\omega_{2}}{\sqrt{4 m^{2}- \omega_{2}^{2}}}\right) + \frac{\omega_{2}}{\omega_{1}\sqrt{4 m^{2}- \omega_{1}^{2}}}\mathrm{a r c t a n}\left( \frac{\omega_{1}}{\sqrt{4 m^{2}- \omega_{1}^{2}}}\right) - \frac{\omega}{4 m^{2}}\bigg ]}
sl(u)=\frac{1}{2}\sqrt{2}sd(\sqrt{2}u;\frac{1}{2}\sqrt{2})=\frac{1}{2}\sqrt{2}\frac{sn(\sqrt{2}u;\frac{1}{2}\sqrt{2})}{dn(\sqrt{2}u;\frac{1}{2}\sqrt{2})},cl(u)=cn(\sqrt{2}u;\frac{1}{2}\sqrt{2}).
p_k^{ij}= \mathbb P(\hat X_{k+1}= x_{k+1}^j \vert \hat X_{k}= x_k^i) \textrm{and}\Lambda_{k}^{ij}= \mathbb{E}\big(Z_{k+1}\mathbb{1}_{\{\hat X_{k+1}=x_{k+1}^j \}}\big \vert \hat X_k=x_k^i \big).
\left<A_{N}f,A_{N}f\right> = \left<\frac{1}{N}\sum_{i=1}^{N}T^{n_{i}}.f,\frac{1}{N}\sum_{j=1}^{N}T^{n_{j}}.f\right> &= \frac{1}{N^2}\left<\sum_{1\leq i,j\leq N}T^{n_{i}-n_{j}}.f,f\right> &= \frac{1}{N^2}\sum_{k=-n_{N}+1}^{n_{N}-1}d_{N}(k)\left<T^{k}.f,f\right>,
\widehat{h}=\displaystyle\frac{\widehat{H}}{{\cal M}}=\displaystyle\frac{2\pi}{l}\left( \widehat{L_{0}}+\widehat{\overline{L}_{0}}-\displaystyle\frac{c}{12}\widehat{I}+\kappa \displaystyle\frac{l^{2-h}}{\left( 2\pi \right)^{1-h}}\widehat{B}\right) \: .
\zeta_{p,p^{\prime}}(s,\,c)=\frac{\pi}{\sqrt{c}}\,\frac{1}{s-1}+\frac{\pi}{\sqrt{c}}\left(2C_{p^{\prime}}^{(1)}-\log\left(4c\right)\right)+\frac{\pi^{2}}{3}\,\frac{1+\frac{3}{\pi p}(1+\frac{1}{\pi p})}{\left(1+\frac{1}{\pi p^{\prime}}\right)\left(1+\frac{1}{\pi p}\right)^{2}}+H_{p,p^{\prime}}(1,c)+O\left(s-1\right)
2 D ( c_{1}+ 6 c_{2}+ 4 2 c_{3}) H_{0}^{4}- \{4 D - [ 3 c_{2}+ 4 ( 2 D - 1 ) c_{3}] \Lambda \}H_{0}^{2}-{\frac{1}{1 2}}\left[ ( D - 2 ) \Lambda - ( D - 4 ) \left({\frac{2 c_{1}}{D ( D - 1 )}}+{\frac{c_{2}}{D}}+ c_{3}\right) \Lambda^{2}\right] = 0 .
S^{N KK}_{WZ}=\mu_6\int_{R^{6+1}}{\rm Tr}\left( i_{\hat k}{\hat N}^{(8)}+l_p^2 i_{\hat k}{\hat{\tilde C}}\wedge{\hat{\cal F}}+l_p^4{\hat C}_{{\hat \mu}{\hat \nu}{\hat \rho}}D{\hat X}^{\hat \mu}D{\hat X}^{\hat \nu}D{\hat X}^{\hat \rho}\wedge{\hat{\cal F}}\wedge{\hat{\cal F}}+\dots\right)\, .
\lambda_1(t q_1+(1-t) q_2)= \int_{\Omega}|\nabla \phi_1^t|^2\, dx+\int_{\Omega}(t q_1+(1-t) q_2)|\phi_1^t|^2\, dx = & t( \int_{\Omega}|\nabla \phi_1^t|^2\, dx+\int_{\Omega}q_1|\phi_1^t|^2\, dx) +(1-t)( \int_{\Omega}|\nabla \phi_1^t|^2\, dx+\int_{\Omega}q_2|\phi_1^t|^2\, dx) > &t\lambda_1( q_1) +(1-t)\lambda(q_2),~~\forall t \in (0,1),
H_{B}^{\mathrm{b a r e}}= \frac{1}{2}\sum_{k = 1}^{\infty}\Bigl ( - \sigma_{k + 1}^{x}\sigma_{k}^{x}- \sigma_{k + 1}^{y}\sigma_{k}^{y}+ \sigma_{k + 1}^{z}\sigma_{k}^{z}\Bigr ) + h \sigma_{1}^{z}~ ,
a_1:=w_{1,1}, \a_2:=w_{2,1}, \ldots, a_h:=w_{h,1}, \a_{h+1}:=w_{1,2}, \ldots, a_{2h}:=w_{h,2}, \ldots, a_{kh}:=w_{h,k}, b_1:=z_{1,k}, \b_2:=z_{2,k}, \ldots, b_h:=z_{h,k}, \b_{h+1}:=z_{1,k-1}, \ldots, b_{2h}:=z_{h,k-1}, \ldots, b_{kh}:=z_{h,1}.
\begin{aligned}\max_{\{B_{k,\ell}\}_{k\in \mathcal{K}}}~~{\sum_{k =1}^K(1-2^{-\frac{B_{k,\ell}}{N_k-1}}) \alpha_{k,\ell}N_k}~~\mathrm{s.t.}~~~~ \sum_{k=1}^K B_{k,\ell}\leq B_{\ell},~\forall \ell, B_{k,\ell}\in \mathbb{N}_+. \end{aligned}
\sigma_{1/w}=\begin{bmatrix}-1 1 -4(\alpha+1) 4\alpha+3 \end{bmatrix}\tau_0 \begin{bmatrix}1 -1/4 0 1 \end{bmatrix}\begin{bmatrix}\sqrt{N^{\prime\prime}/4}0 0 1/\sqrt{N^{\prime\prime}/4}\end{bmatrix}.
S^{(d)}_{z}(x) =\frac{C_z}{x^{|z|+1}}F^{(d)}_C\left(\begin{array}{c}\frac{|z|}{2}+\frac{1}{2},\frac{|z|}{2}+1 begin{align*}3pt] z_1+1,\ldots,z_d+1\end{array};\,\frac{4}{x^2},\ldots,\frac{4}{x^2}\right) &=\frac{1}{x^{|z|+1}}\sum^{\infty}_{n=0}A^{(d)}_z(n)\frac{(\frac{|z|}{2}+\frac{1}{2})_n(\frac{|z|}{2}+1)_n}{(h+1)_n}\frac{(\frac{4}{x^2})^n}{n!},
\frac{\mathcal{H}_{d-1}(W\cap\partial O)}{|O|^\frac{d-\zeta}{d}}\geq \frac{c_H}{|O|^\frac{d-\zeta}{d}}\geq\frac{c_H}{R^{\theta \cdot \frac{d-\zeta}{d}}}= \frac{c_H}{R^{1-\zeta}}.
\mathbf{P}_{\alpha}(g) (T^{-1}(g_0) s_j)=J_{\alpha}QgPT^{-1}(g_0)s_j & \phantom{abc}=J_{\alpha}Q\boldsymbol\sigma_{\!\alpha}(g)(g_+^{-1})_\alpha\chi^{-n}g_+P g_+^{-1}\chi^j=\boldsymbol\sigma_{\!\alpha}(g)g_+^{-1}\chi^{n-j-1},
\Lambda_{\omega}(F,G) = \sum_{j=1}^m \int_{\mathbb{R}^2}\Big( \int_{\mathbb{R}}F(x+u,y)G(x,y+u)(\phi\ast\omega)_{2^{k_{j}}}(u) du \Big) & \Big( \int_{\mathbb{R}}F(x+v,y)G(x,y+v) (\phi_{2^{k_{j}}}-\phi_{2^{k_{j-1}}})(v) dv \Big) \,dx dy.
X \ = \ \S_1 \d_1 \S_2 \ \ldots d_{i-1}\!-\! 1 \S_i \d_{i}-1 \ldots \d_{j-1}\ \ \sigma_j \ \ \d_j \ \ \ldots \S_k Y \ = \ \S_1 \d_1 \S_2 \ \ldots \ \d_{i-1}\ \ \ \sigma_i \ \ \d_i \ldots \d_{j-1}\!-\! 1 \S_j \d_{j}-1 \ \ \ldots\S_k
d X^{(\epsilon)}(t) =f(t, X^{(\epsilon)}_t) d t+g(t,X^{(\epsilon)}_t)d W(t)+h_0(t,X^{(\epsilon)}_t)\int_{|z|<\epsilon}|\lambda(z)|^2 \nu(dz) dB(t) &+ \int_{|z|\geq \epsilon}h_0(t,X^{(\epsilon)}_t)\lambda(z)\tilde N(d t, d z) X^{(\epsilon)}_0 =\eta \, .
\frac{144}{\pi^2}\log^3(T/5)\log^3(T)\left(\frac{\pi^2}{144\log^2(T/5)}+\frac{0.1052}{\log^3(T/5)}-E(T)\right) &=\log(T/5)\log^3(T)-\log(T)\log^3(T/5)+\frac{144}{\pi^2}0.1052\log^3(T)-6.59125\log^3(T/5) &=2\log(5)\log(T/5)\log(T)\log(T/\sqrt{5})+\frac{144}{\pi^2}0.1052\log^3(T)-6.59125\log^3(T/5).
\left( 1 - i \varepsilon \int d^{4}z \frac{\partial F^{\beta}}{\partial \alpha}\mid_{_{\alpha_{0}}}\delta \alpha ( \partial . A - F )^{\alpha}( x ) i \int d^{4}x M^{' - 1 \alpha \beta}( x , z ; A ; \alpha_{0}) + O [ \left( \delta \alpha \right)^{2}] + O [ g ] \right)
\ln\frac{(x+1)^2}{(x+\eta)(x+1-\eta)}+\ln \frac{x+c_0+\mathrm{MC}_1(x)}{x+1+c_0+\mathrm{MC}_1(x+1)}= \frac{2 \kappa_0 - \eta^2 + 2 \eta^3 - \eta^4}{x^3}+\frac{-3 \kappa_0 - 3 \kappa_0 \lambda_0 - 3 \kappa_0 \eta + 2 \eta^2 + 3 \kappa_0 \eta^2 - 3 \eta^3 - \eta^4 + 3 \eta^5 - \eta^6}{x^4}&+\frac{g(x)}{x^5}+O\left(\frac{1}{x^6}\right),
\# \big\lbrace z \in \sigma_{\textup{disc}}\big( H(b_0,e V) \big) : -r_0 e^2 \leq z < -r e^2 \big\rbrace = C_{m}\left( \frac{1}{2b_0}\right)^{1/m}r^{-1/m}\big( 1 + o(1) \big),
\begin{array}{ccc}\partial^\mu C_{n\mu}^F=0, \quad \partial_\mu b_n^{F\mu}=0, \quad \partial_\mu \bar{C}_n^{F\mu}=0, \partial^\mu C_{n\mu}^B=0, \quad \partial_\mu b_n^{B\mu}=0, \quad \partial_\mu \bar{C}_n^{B\mu}=0. \end{array}
kk'^{-1}= ~\textrm{diag}(1,\zeta,1)\kappa(\theta)\textrm{diag}(\xi\bar{\xi}',\eta\bar{\eta}',\gamma\bar{\gamma}')\kappa(-\theta')\textrm{diag}(1,\bar{\zeta}',1) = ~\begin{bmatrix}a \bar{\zeta}'b 0 \zeta c \zeta\bar{\zeta}'d 0 0 0 \gamma\bar{\gamma}'\end{bmatrix},
L^{\xi}= p \cdot \hat{H}+ X^{\xi}, \quad X^{\xi}= i \sum_{\rho \in \Delta_{+}}g_{| \rho |}\, ( \rho \cdot \hat{H}) \left( x ( \rho \cdot q ) - x ( \rho^{\vee}\! \cdot \hat{H}\xi ) \right) \hat{s}_{\rho},
\Lambda_1(\lambda)(f, g) = \sum_{k = 1}^\infty \partial_\nu\phi_k\Big\vert_{\partial B_R}\frac{1}{\lambda_k - \lambda}\Big(\int_{\partial B_R}\partial_\nu\phi_k f{\rm d}s(y) + \int_{\partial B_R}\partial_\nu(\Delta\phi_k)g{\rm d}s(y)\Big)
{\cal H}_{B}={\frac{1}{2}}p^{2}+{\frac{1}{2}}\sum_{\rho \in \Delta_{+}}g_{| \rho |}( g_{| \rho |}- 1 ) | \rho |^{2}\, V ( \rho \cdot q ) + ({\frac{\omega^{2}}{2}}q^{2}) -{\cal E}_{0},
\int_{D_1 \times D_2}\left|h\right|\,\mathrm d \rho^{(2)}\leq c \cdot E\Bigl\|\sum_{n=1}^m X_ng_n\Bigr\|_{k_2}\leq c \biggl( E\Bigl\|\sum_{n=1}^m X_ng_n\Bigr\|_{k_2}^2\biggr)^{1/2}&= c \Bigl(\sum_{n=1}^m\|g_n\|_{k_2}^2\Bigr)^{1/2}=c\left\|h\right\|_{K_s},
f_{\varepsilon}:= \eta\left(|v|^2+\varepsilon\right)^{\frac{p}{2}-2}\left((p-1)|v|^2+\varepsilon\right)\sum_{j=1}^{d}\overline{D_j v}^T A D_j v f := (p-1)\eta |v|^{p-2}\sum_{j=1}^{d}\overline{D_j v}^T A D_j v.
\sum_{\substack{\lambda \vdash n \ell(\lambda) = t}}w(B_{\lambda,\mu}) = \sum_{\substack{b_1,\ldots,b_{r}\ge 1 b_1+\ldots + b_r = t}}\sum_{\substack{\forall 1 \le i \le r: a_{i,1},\ldots, a_{i,b_i}\ge 1 \sum_{j=1}^{b_i}a_{i,j}=\mu_i}}\prod_{i=1}^{r}a_{i,b_i}.
\nabla_{z_{\alpha}}^{0}\left( \begin{array}{c}{\langle v |}{\hat{V}}\end{array}\right) = \frac{1}{\theta_{\alpha}}\left( \begin{array}{c}{\langle v | \widehat{\bar{z}}_{\alpha}}{- \lbrack \widehat{\bar{z}}_{\alpha}, \hat{V}\rbrack}\end{array}\right) \, .
f(1,R) = \, \psi(\exp\{\mu(1) - \mu(R) \})\leq \psi(\exp\{\mu(1) - \mu(\overline R) \}) = \, \psi(\exp\{\mu(d(\overline R)) - \mu(\overline R) \})= \psi \Big( \frac{\sqrt{3}}{2}\Big) = \sqrt{3}.
p_{a}= \frac{1}{\sqrt{k}}\times \frac{1}{l_{s}}\left( \frac{n_{a}}{r_{a}}+ w^{a}r_{a}\right) , \bar{p}_{a}= \frac{1}{\sqrt{k}}\times \frac{1}{l_{s}}\left( \frac{n_{a}}{r_{a}}- w^{a}r_{a}\right) ,
\sum_{k=i_r}^p(-1)^k \binom{k-1}{i_r-1}\binom{p-1}{k-1}=(-1)^{i_r}\binom{p-1}{i_r-1}\sum_{l=0}^{p-i_r}(-1)^l \binom{p-i_r}{l}=(-1)^{i_r}\delta_{p,{i_r}}.
b_0(t,y)-\pi(t,y)\sigma_0^2(t,y)- \int_{\mathbb{R}}\frac{\pi(t,y) \gamma_0^2(t,y,\zeta)}{1+\pi(t,y)\gamma_0(t,y,\zeta)}\nu(d\zeta) &+\sigma_0(t,y)\Phi(t,y)+ \int_{\mathbb{R}}\frac{\gamma_0(t,y,\zeta)}{1+\pi(t,y)\gamma_0(t,y,\zeta)}\Psi(t,y,\zeta)\nu(d\zeta)= 0.
\sum_{a_{i}, b_{i}}H_{a_{0}b_{0}}^{( l_{0})}H_{a_{1}b_{1}}^{( l_{1})}. . . . . H_{a_{n}b_{n}}^{( l_{n})}\sum_{\alpha_{i}, \beta_{i}}( \gamma_{\mu_{0}})_{\alpha_{0}\beta_{0}}( \gamma_{\mu_{1}})_{\alpha_{1}\beta_{1}}. . . . . ( \gamma_{\mu_{n}})_{\alpha_{n}\beta_{n}}
H_I=\int d^{\nu}x \left[ \left(\mu^2(\phi_0) \, \phi_0 -{{\lambda}\over 3}\phi^3_0 \right):\eta(\vec{x}): +{\lambda\over{3!}}\phi_0 : \eta^3(\vec{x}): +{\lambda\over{4!}}:\eta^4(\vec{x}): \right].
(b\theta)(a_0\otimes\cdots\otimes a_n) =\theta(a_0a_1\otimes a_2\otimes\cdots\otimes a_n) +\sum_{k=1}^{n-1}(-1)^k\theta(a_0\otimes a_1\otimes\cdots \otimes a_{k-1}\otimes a_ka_{k+1}\otimes a_{k+2}\otimes\cdots\otimes a_n) +(-1)^n\theta(a_na_0\otimes a_1\otimes a_2\otimes\cdots\otimes a_{n-1}).
\rho_{\mathrm{ext}}\hat\otimes 1(\omega_1\hat\otimes\omega_2) = (e_1\wedge \omega_1 - \iota(e_1)\omega_1)\hat\otimes \omega_2 1\hat\otimes \rho^j(\omega_1\hat\otimes\omega_2) = (-1)^{|\omega_1|}\omega_1\hat\otimes(e_j\wedge \omega_2 - \iota(e_j)\omega_2),
\sum_{k=1}^m 2 x_-^T(s)\Sigma N_{k}x_-(s) u_k(s) \leq x_-^T(s) \Sigma x_-(s) \left\|u(s)\right\|_{2}^2 +\sum_{k=1}^m x_-^T(s) N_k^T\Sigma N_{k}x_-(s).
\dim_{\mathbb{F}_2}H_n(D_8; \mathbb{F}_2) = n+1 \dim_{\mathbb{F}_2}H_n(C_2\times C_2; \mathbb{F}_2) = n+1 \dim_{\mathbb{F}_2}H_n(C_2; \mathbb{F}_2) = 1 \dim_{\mathbb{F}_2}H_n(C_4; \mathbb{F}_2) = 1 ,
h_{0,j}= ~ (f \cup_\alpha (h \circ_{j - m}g))(a_1, \ldots, a_{m+n+p-1}) j = m, m+1, \ldots, m+p-1 h'_{i, m+i-1}= ~ (-1)^{m +i -1}(h \circ_{i-1}(f \cup_\alpha g))(a_1, \ldots, a_{m+n+p-1}) i = 1, 2, \ldots, p h''_{i, m+p}= ~ (-1)^{m+n+p-1}((h \circ_{i-1}f) \cup_\alpha g)(a_1, \ldots, a_{m+n+p-1}) i = 1, 2, \ldots, p.
\mathbf{Y'(t)}=\mathbf{A}\mathbf{Y(t)}, \mathbf{Y(t_0)}=\begin{bmatrix}\mathbf{w(t_0)}\mathbf{w'(t_0)}\end{bmatrix}, \mathbf{A}=\begin{bmatrix}\mathbf{0}\mathbf{I}-\mathbf{M}^{-1}\mathbf{K}\mathbf{0}\end{bmatrix}.
d_X(x_m,x_n) \leq d_X(x_m,x_{m-1})+d_X(x_{m-1},x_{m-2})+...+d_X(x_{n+1},x_n) &\leq{\delta_1}^{m-1}\d_X(x_1,x_0)+{\delta_1}^{m-2}\d_X(x_1,x_0)+...+{\delta_1}^n\d_X(x_1,x_0) &={\delta_1}^n \Big(1+\delta_1 + ...+{\delta_1}^{m-n-1}\Big)\d_X(x_1,x_0) &= \frac{{\delta_1}^n}{1-\delta_1}\d_X(x_1,x_0) &\rightarrow 0\ \n\rightarrow \infty.
\Gamma_{k_0,\epsilon}:=\Gamma\cap U_{\epsilon}(k_0)=\Gamma_{j,\epsilon}\cup\Gamma_{j,\epsilon}^{*}, j=1, 2, \Gamma_{-k_0,\epsilon}:=\Gamma\cap U_{\epsilon}(-k_0)=\Gamma_{j,\epsilon}\cup\Gamma_{j,\epsilon}^{*}, j=3, 4, \Gamma_\epsilon:=\Gamma_{k_0,\epsilon}\cup\Gamma_{-k_0,\epsilon}.
\partial_t\prod_{j=1}^r\left(\frac{(1+t)^{a_j}\ln(1+t)}{(1+t)^{a_j}-1}\right) =\frac{1}{1+t}\prod_{i=1}^r\left(\frac{(1+t)^{a_i}\ln(1+t)}{(1+t)^{a_i}-1}\right)\frac{\sum_{j=1}^r\left(\frac{t}{\ln(1+t)}-\frac{a_j t(1+t)^{a_j}}{(1+t)^{a_j}-1}\right)}{t}&+\frac{1}{1+t}\prod_{i=1}^r\left(\frac{(1+t)^{a_i}\ln(1+t)}{(1+t)^{a_i}-1}\right)\sum_{j=1}^r a_j\,.
V(\mathbf{j}'') =3j''_{n-2}-s(j''_{n-2})+2j''_{n-1}-s(j''_{n-1})-c(j''_n,-\alpha_{\mathbf{j}''}(n)-1) & = 3(0)-s(0)+2(16q+7)-s(16q+7)-c(2,8q+2) & = 32q+14-s(q)-s(7)-c(2,8q+2) & = 32q+10-s(q)
R_{T}=\Phi_{\left\Vert \eta_{1}\right\Vert ,\left\Vert y\right\Vert}+\Phi_{\left\Vert \eta_{2}\right\Vert ,\left\Vert y\right\Vert}+\Phi_{\left\Vert \eta_{2}\right\Vert ,\left\Vert z\right\Vert}+\Phi_{\left\Vert \eta_{3}\right\Vert ,\left\Vert x\right\Vert}+\Phi_{\left\Vert \eta_{3}\right\Vert ,\left\Vert z\right\Vert}
a\left(y,(\log n)/2\right)=\frac{y+2\log \left((\log n)/2\right)+(1/2)\log\log\left(( \log n)/2\right)-(1/2)\log \pi}{\sqrt{2\log \left((\log n)/2\right)}}=\frac{y+2\log \log n +(1/2)\log \log \log n-(1/2)\log \pi+o(1)}{\sqrt{(2\log \log n) (1+o(1))}}= \frac{y-\log 2+b_n+o(1)}{a_n(1+o(1))},
\lim_{u\to0^+}u^{k(2k-1)/2}\varOmega_{2k-1}(u)=(-1)^{\frac{(k-1)(k-2)}{2}}\frac{k[\Gamma(k/2)]^{2}}{(2k+1)}\frac{(\det\mathbf N_{k-1})^2}{2^{(k-1)(2k-1)+1}}
e^{\lambda_{\max}^{\mathrm{d}}}=\lim_{n\rightarrow\infty}\rho\left[\left( \mathit{\Phi}(n,\omega)^{{\mathrm{T}}}\mathit{\Phi}(n,\omega)\right)^{1/2n}\right] & = \lim_{n\rightarrow\infty}\lim_{k\rightarrow\infty}\left\Vert \left(\mathit{\Phi}(n,\omega)^{{\mathrm{T}}}\mathit{\Phi}(n,\omega)\right)^{k/2n}\right\Vert^{1/k},
\begin{cases}\widehat{H}(x,P+\nabla \widetilde{u}_0 (x ) )=\ln \widetilde{m}_0 (x ) + \overline{H}, -\div_x \big(\widetilde{m}_0 (x )\int_{\mathcal{Y}^d}\widetilde{m}_1 (x,y ) (P+\nabla\widetilde{u}_0 (x )+\nabla_y\widetilde{u}_1 (x,y ) )dy\big )=0,\end{cases}
Q_Z = Q_1+\frac{121}{12\sqrt{105}}S''-\frac{2}{3}\sqrt{\frac{2}{3}}(J\; S')- \sqrt{\frac{2}{3}}(J'\; S)+ \frac{1}{3}\sqrt{\frac{35}{3}}(J\; J\;S)-\frac{1}{3}\sqrt{\frac{35}{3}}(T \; S)
H_0^-(x) =\frac{(x-b)^2(2x-3a+b)}{(b-a)^3},H_1^-(x)=\frac{(x-a)(x-b)^2}{(b-a)^2}, H_0^+(x) =-\frac{(x-a)^2(2x+a-3b)}{(b-a)^3},H_1^+(x)=\frac{(x-a)^2(x-b)}{(b-a)^2}.
\xi ( t ; \xi_{f}) = \left( \xi_{f}+ \frac{b}{a}\right) e^{a ( t - t_{f})}- \frac{b}{a}\ \ , \ \ \gamma ( \xi_{f}) = \frac{1}{a}\left( \xi_{f}+ \frac{b}{a}\right) \left( 1 - e^{- a ( t_{f}- t_{i})}\right) - \frac{b}{a}( t_{f}- t_{i}) \ \ ,
\xi(\underbrace{1, \ldots , 1}_{k-1}, r ; s+1)= \ (-1)^{r-1}\sum_{\substack{a_{1}+\cdots +a_{r}=k{}^\forall a_p\geq 0}}\binom{s+a_{r}}{a_{r}}\,\zeta(a_{1}+1, \ldots , a_{r-1}+1,a_{r}+1+s) +\sum^{r-2}_{i=0}(-1)^{i}\zeta(\underbrace{1, \ldots , 1}_{k-1}, r-i)\,\zeta(\underbrace{1, \ldots ,1}_{i},1+s),
\begin{aligned}1-{\mathcal M}(L^+_{\gamma}(t_0)) -{\mathcal M}(L^-_{\gamma}(t_0)) \leq \frac{M}{K\underline{R}\cos^{2}\gamma}, \mbox{or}&{\mathcal M}(L^-_{\gamma}(t_0)) \geq 1-{\mathcal M}(L^+_{\gamma}(t_0))-\frac{M}{K\underline{R}\cos^{2}\gamma}. \end{aligned}
l_{\mu}^{*},R_{\mu}^{*},t_{\mu}^{*}=argmin_{l>0,R\in SO(3),t\in\mathbb{R}^{3}}\sum_{i=1}^{N}\rho_{\mu}(\frac{1}{\sigma_{i}}\|s_{i}-lRm_{i}-t\|_{2})
\begin{array}{r c l r c l}{{e^{0}}}&{{=}}&{{\lambda^{1 / 2}/ R \left( d t + \omega_{\aleph}d \varphi \right) \, , \hspace{. 5 c m}}}&{{e^{1}}}&{{=}}&{{\lambda^{- 1 / 2}R d r \, ,}}\\{{}}&{{}}&{{}}&{{}}&{{}}&{{}}\\{{e^{2}}}&{{=}}&{{R \, d \theta \, ,}}&{{e^{3}}}&{{=}}&{{R \, \Omega_{\aleph}\, d \varphi \, ,}}\\ \end{array}
r_1 = \cos( \alpha_1 ) \cos( \phi_2 - \phi_1) - \sin( \alpha_1) \sin( \phi_2 - \phi_1) r_2 = \cos( \alpha_2 ) \cos( \phi_2 - \phi_1) - \sin( \alpha_2) \sin( \phi_2 - \phi_1).
\frac{d \phi_1}{d x}=\frac{\partial W}{\partial \phi_1}= \sqrt{2}(4 \phi_1^2+2 \sigma \phi_2^2-1) \hspace{1cm};\hspace{1cm}\frac{d\phi_2}{d x}=\frac{\partial W}{\partial \phi_2}=4 \sqrt{2}\sigma\phi_1 \phi_2