harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
B_R =C_{\alpha,\beta}R^{-2(\alpha+\beta)}\int^R_0(R^2-t^2)^{\beta-1}t^{2\alpha+1}(1-R^{-2}t^2)B^{\alpha+1}_t dt &=C_{\alpha,\beta}R^{-2(\alpha+\beta)-2}\int^R_0(R^2-t^2)^{\beta}t^{2\alpha+1}B^{\alpha+1}_t dt
\left|{\frac{d x}{d \lambda}}\right| = H^{-{\frac{2}{( D - 2 ) \Delta}}}H_{p}^{-{\frac{2 ( p + 1 )}{( D - 2 ) \Delta_{p}}}}\sqrt{H^{-{\frac{4}{( D - 2 ) \Delta}}}\left( E^{2}H_{p}^{\frac{4 ( D - p - 3 )}{( D - 2 ) \Delta_{p}}}-{\frac{{\cal J}^{2}}{x^{2}H_{p}^{\frac{4 ( p + 1 )}{( D - 2 ) \Delta_{p}}}}}\right) - \epsilon},
{}_{2}\phi_{1}\left( \begin{matrix}{a \phantom{c}b}\\{\phantom{a}c \phantom{b}}\\ \end{matrix}; q , z \right) = \sum_{n = 0}^{\infty}\frac{( a ; q )_{n}( b ; q )_{n}}{( c ; q )_{n}( q ; q )_{n}}z^{n}, \qquad ( a ; q )_{n}= \prod_{j = 0}^{n - 1}( 1 - a q^{j}) .
\mu^2_n \phi_n(z)=\left(m^2-\frac{g^2N}{2\pi}\right)\left(\frac{1}{z}+\frac{1}{1-z}\right)\phi_n(z)-\frac{g^2N}{2\pi}\int_0^1 dz' \frac{\phi_n(z')}{(z-z')^2},
A_{\mu}d x^{\mu}= \frac{1}{2 g r}\left\{\begin{array}{l}{\displaystyle{\tau_{\phi}^{n}\left[ H_{1}d r + ( 1 - H_{2}) r d \theta \right]}}\\{\displaystyle{- n \left[ \tau_{r}^{n}H_{3}+ \tau_{\theta}^{n}\left( 1 - H_{4}\right) \right] r \sin \theta d \phi}}\\ \end{array}\right\}
\left[\sum_{\lambda\geq T}|\langle W, f_{\lambda}\rangle|^2+ \sum_{\mathfrak{a}\in \mathcal{E}_k}\int_{|1/4+t^2|>T}|\langle W, E_{\mathfrak{a}}(.,1/2+it) \rangle|^2 dt\right]^{1/2}= O_{A}\left( \frac{X^{1+\epsilon}}{\sqrt{md_1}T^{A}}\right),
g: =e_{\theta}\cdot \textrm{curl}\f &=-e_{\theta}\cdot \textrm{curl}\ (h-\nabla \Phi) &=-\partial_zh^{r}+\partial_rh^{z}&=-v\cdot \nabla \omega^{\theta}+\frac{u^{r}}{r}\omega^{\theta}+\frac{\partial_z|u^{\theta}|^{2}}{r}.
C_{p}= [ A \Gamma ( 1 + p ) \zeta_{R}( 1 + p ) ]^{\frac{1 - 2{\cal D}( 0 )}{2 ( 1 + p )}}\frac{\exp \left( \frac{d}{d s}{\cal D}( s ) |_{s = 0}\right)}{[ 2 \pi ( 1 + p ) ]^{\frac 1 2}}\, \mathrm{,}
u^{L}=\lim_{\substack{(r,x)\to(t,\overline{y}(t)) x<\overline{y}(s)}}u(r,x)\ ,u^{R}=\lim_{\substack{(r,x)\to(t,\overline{y}(t)) x>\overline{y}(s)}}u(r,x) \ ,\dot{\overline{y}}(t)=\sigma_{i}(u^{L},u^{R})\ .
\gamma \cdot (v,g_f) = (\phi_B (\gamma) \cdot v, \gamma g_f) \mapsto (\phi_B(b) \phi_B (\gamma) \cdot v, \gamma g_f b_f ) = (\phi'_B (\gamma) \phi_B (b)\cdot v, \gamma g_f b_f ) = \gamma \cdot (\phi_B (b)\cdot v, g_f b_f),
\sharp(b) = \sharp_0(b) + \sharp_1(\cdot)b + \sharp_2(\cdot\,,\cdot)b + \sharp^b(\cdot)b\in\mathfrak{X}(M)\oplus\Omega^1(Q,Q)\oplus\Omega^2(Q,B^*)\oplus(\Gamma(B)\otimes\Gamma(B^*))
{{\mathcal{E}}_{\mu \nu}}= -{{\left({\frac{\tilde{\kappa}}{\kappa}}\right)}^{4}}\left[{\mathcal{U}}\left({u_{\mu}}{u_{\nu}}+{\frac{1}{3}}{h_{\mu \nu}}\right) +{{\mathcal{P}}_{\mu \nu}}+{{\mathcal{Q}}_{\mu}}{u_{\nu}}+{{\mathcal{Q}}_{\nu}}{u_{\mu}}\right] ,
(4\pi n^2|D|y)^{-1/4}e^{-\frac{\pi i}{4}}\mathcal{M}_{-\frac{1}{4}, \frac{1}{4}}^+(4\pi n^2|D|y) e^{2\pi in^2 Dx}= (4\pi n^2|D|)^{-1/4}e^{-\frac{\pi i}{4}}(4\pi n^2|D|e^{\pi i})^{1/4}q^{n^2D}&= q^{n^2D}.
0 = \lim_{t\to\infty}\frac 1t \log (X_0/X_t) - \frac 1t \int_0^t \log(1/p) - p(1-X_s)ds = - \log(1/p) + p(1-\mathbf E[X_\infty]), \intertext{i.e.}\mathbf E[X_\infty] = 1 - \tfrac 1p \log(1/p).
S_n(i)-S_{n-1}(i) =\sum_{j=0}^{n-1}\frac{q^{ij}(q;q)_{n-1+j}(1-q^{n+j}-1)}{(q^2;q^2)_j}+\frac{q^{in}(q;q)_{2n}}{(q^2;q^2)_n}&=-q^n S_{n-1}(i+1)+q^{in}(q;q^2)_n.
\left(u\alpha_{1}\right)\left(v\alpha_{2}\right) =\rho^{v_{2}}uv\alpha_{1}\alpha_{2}\ \left(v\alpha_{2}\right)\left(u\alpha_{1}\right) =\rho^{\frac{1}{2}u_{2}\left(u_{2}-1\right)+v_{3}u_{2}-u_{3}v_{2}-u_{2}^{2}-u_{2}v_{2}}\tau^{u_{2}}uv\alpha_{1}\alpha_{2},
\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big(\varepsilon(h,x+y)[[\alpha(x),\alpha(y)],\alpha(h\cdot z)]+ \varepsilon(x+y,z)[[h\cdot z,\alpha(x)],\alpha^2(y)] &+\varepsilon(x,y+z)\big(2[\alpha(h)\cdot[y,z],\alpha^2(x)]-\varepsilon(x,y+z)[[h\cdot y,\alpha(z)],\alpha^2(x)]\big)\Big)=0.
S_{n}^{j}= \sum_{q}\left( X_{q}^{1}X_{n - q}^{j}- X_{q}^{j}X_{n - q}^{1}\right) - 2 \pi R n X_{n}^{j}= \sum_{q}\left( [ X_{q}^{1}, X_{n - q}^{j}] \right) - 2 \pi R n X_{n}^{j}
\mathcal{D}_{\mu}\phi \equiv \partial_{\mu}\phi - \vartheta_{\mu}{}^{i}\Gamma_{r}( M_{i}) \phi \, , \hspace{1 c m}\mathcal{D}_{\mu}\psi \equiv \partial_{\mu}\psi + \psi \vartheta_{\mu}{}^{i}\Gamma_{r}( M_{i}) \,
K_\mathfrak{i}^{\pm}=\frac{1}{\sqrt2}(M_{0i}\pm M_{1i}), \mathfrak{i}=2,3 F_{i}^{\pm}=\frac{1}{\sqrt2}(\frac{M_{0i}}{l_1}\pm J_{i}),i=1,2,3, L_{i}=\frac{1}{2}\epsilon_{ijk}M_{jk}, i,j,k=1,2,3, P^{\pm}= \frac{1}{\sqrt2}(J_0\pm J_1), R=M_{01}, T=M_{23}.
L_{\alpha, s}(z) = L_{Y, \alpha, s}(z)= 0 \; \; \; \mbox{if $\alpha > 1, \;$ and}\; L_{\alpha, s}(z) = L_{Y, \alpha, s}(z) = 1 \; \; \mbox{if $\frac{1}{1+\beta}< \alpha \le 1$}.
|0 \rangle = \Psi_{\beta \over 2}^{\dagger}|-\beta \rangle=\Psi_{\beta \over 2}^{\dagger}\Psi_{3 \beta \over 2}^{\dagger}\cdot \cdot \cdot | - \infty \rangle = \prod_{i=0}^{\infty}\Psi^{+}_{-m_i^{0}}| - \infty \rangle
{\rm O}(\epsilon^1):\rho c_v\partial_{t_1}g_i^{(0)}+d_{1i}g_i^{(0)}=-\frac{1}{\Delta t}\left({{{\bf{M}}}}^{-1}{\Lambda}{{{\bf{M}}}}\right)_{ij}g_j^{(1)}+{{w}_i \bar F^{(1)}},
\partial_t m_t = \div \left( \nabla a(m_t,\cdot)m_t\right)\,,\,\,\,\,\, a(m,x):=\frac{\delta F}{\delta m}(m,x) + \frac{\sigma^2}{2}\log \left( \frac{m(x)}{\pi(x)}\right) - \frac{\sigma^2}{2}\operatorname{KL}(m|\pi).
\begin{array}{rcl}x^{+}= f(x,F_{u}\circ\Psi(x,z,v_{[0,R-A]})) z^{+}= \psi_{c,[1]}(x,z,v_{[0,R-A]}) v_{[0,R-A-1]}^{+}= v_{[1,R-A]}\end{array}
i\int_0^1 \left( \Im (\overline{\xi(x)}\xi^\prime(x)) + \Im (\overline{\eta(x)}\eta^\prime(x))\right)dx+\int_{0}^{1}{\xi^\prime(x)}\overline{\eta(x)}dx-\int_{0}^{1}\overline{\xi^\prime(x)}{\eta(x)}dx-\int_{0}^{1}|\eta^{\prime}(x)|^2 dx =\lambda\int_{0}^{1}|\xi(x)|^2dx+\lambda\int_{0}^{1}|\eta(x)|^2dx,
\int_0^\infty\eta(t)\int_\Omega \big(e_\epsilon(d_\epsilon) \nabla\cdot X-\nabla d_\epsilon\odot\nabla d_\epsilon:\nabla X\big)=\int_0^\infty\eta(t)\int_\Omega \big\langle \partial_t d_\epsilon+u_\epsilon\cdot\nabla d_\epsilon, X\cdot\nabla d_\epsilon\big\rangle,
C(T) \int_0^T \big(||\mathcal{\mathbf{B}}_{n+q+1}^{\ast}W_{n+q+1}||_{G_{n+q+1}}^2 +\sum_{l=0}^{q}|| \mathcal{\mathbf{B}}_{n+l}^{\ast}U_{n+l}||_{G_{n+l}}^2 \big)\, dt &\ge \int_0^T e_1(W_{n+q+1})\,dt\,.
C_{i k ~ p r}^{~ j n ~ m s}\theta_{m}^{~ p}t_{s}^{~ r}\otimes t_{n}^{~ l}+ D_{i r ~ k}^{~ p j ~ n}\otimes \theta_{p}^{~ r}t_{n}^{~ l}+ C_{p n ~ b d}^{~ r l ~ m s}t_{i}^{~ p}S ( t_{r}^{~ j}) t_{k}^{~ n}\otimes \theta_{m}^{~ b}t_{s}^{~ d}=
(4\kappa_{max}+ \frac{2}{n}) \sum\limits_{u_{ii}> 0}{F^{ii}u_{ii}}= (4\kappa_{max}+ \frac{2}{n}) \sum\limits_{i =1}^n{F^{ii}u_{ii}}= (4\kappa_{max}+ \frac{2}{n})(k-l).
{\rm mult}_\lambda(\mathbf{m}) =\sum_{\sigma\in S_n}\varepsilon(\sigma)Q\Big(\sigma\Big((\ell_1+1+m)\sum_{j=1}^ne_j+\frac12\sum_{j=1}^n(n+1-2j)e_j\Big)-\lambda-\frac{n+1}2\sum_{j=1}^ne_j\Big) =\sum_{\sigma\in S_n}\varepsilon(\sigma)Q\Big(\sum_{j=1}^n(\ell_1+1+m-\ell_j)e_j-\sigma\Big(\sum_{j=1}^nje_j\Big)\Big).
\int_{S_{\infty}}\lambda_{\xi\eta\xi^{-1},z}^2\d\nu_{\xi\eta^{-1}\xi^{-1}}=1-\sum\nolimits_{\psi\in\{\xi^2,\xi\eta^{-1}\xi^{-1},\xi\eta^{-1}\xi,\xi\eta^{-2},\xi\eta^2,\xi\eta\xi^{-1},\xi\eta\xi\}}\nu_{\psi}(S_{\infty}).
\begin{aligned}L^{\rm resc,h}_n(u)-L^{\rm resc,h}_n(v) \geq B^{\rho_-}_n(u)-B^{\rho_-}_n(v) + (u^2-v^2) &+ \left(\frac{\beta_1}{1-\rho_-}-2(1+\gamma)^{5/3}\gamma^{1/3}\right)\frac{(u-v)}{\beta_2}n^{1/3},\end{aligned}
\frac{3\pi^2}{2(4n+ \frac{1}{2})^2}\sum_{i=1}^{n}\Big(2i-\frac{3}{4}\Big)y_{2i-1,4n}^p\leq \frac{3\pi^2}{(4n+ \frac{1}{2})^2}\sum_{i=1}^{n}iy_{2i-1,4n}^p.
\psi_{t}\left( z\right) :=\mathbf{E}\left( z^{2N_{d}\left( t\right)-N_{b}\left( t\right)}\right) =\Phi_{t}\left( z^{-1},z^{2}\right) =\phi_{0}\left( \frac{A_{t}\left( z\right) z^{-1}+B_{t}\left( z\right) z}{C_{t}\left( z\right) z^{-1}+D_{t}\left( z\right)}\right) .
M_1 =x_0x_1^{a_1}\cdots x_n^{a_n}= \sum_{i=1}^{r_1}L^\prime_i,\ \ \ \M_2 = y_0^{b_0}y_1^{b_1}\cdots y_m^{b_m}=\sum_{i=1}^{r_2}L^{\prime\prime}_j;
\sum_{i = 0}^{N-1}x_{s,t'_i}\otimes y_{t_i,t_{t+1}}= \sum_{i = 0}^{N-1}(\vartheta_h x)_{s-h,\hat{t}'_{i}}\otimes (\vartheta_h y)_{\hat{t}_i, \hat{t}_{i+1}}
S = \int_{\tau^{\prime}}^{\tau^{\prime \prime}}d \tau ( \dot{x^{\mu}}p_{\mu}+ \frac{i}{2}( \dot{\theta^{\mu}}\theta_{\mu}+ \dot{\theta_{5}}\theta_{5}) - N{\cal H}- M{\cal Q}_{0}) - \frac{i}{2}\theta ( \tau^{\prime \prime}) \cdot \theta ( \tau^{\prime}) - [ B ]_{\tau^{\prime}}^{\tau^{\prime \prime}},
E [ f^{(0)}+ f^{(1)}] = E [ f^{(0)}] + \frac{\delta E}{\delta f}[ f^{(0)}] \cdot f^{(1)}+ \frac{1}{2}\frac{\delta^2 E}{\delta f^2}[ f^{(0)}] \cdot (f^{(1)})^2 + \cdots.
\begin{array}{c}{u_{i}^{\mu}v_{\mu}^{j}= G_{i}^{~ j}\equiv [ \operatorname{e x p}( - W ) ]_{i}^{~ j},}{{G^{- 1}}_{i}^{~ j}( u_{j}^{\mu}v_{\nu}^{i}+ u_{\nu j}v^{\mu i}) = \delta_{\nu}^{\mu}.}\end{array}
c_{\Omega_4}^{\Omega_0}= q^{-1/6}\varphi(q)^{-4}\dfrac{1}{2}q^{1/20}(\varphi(-q^{1/2})^2\varphi(q)^{-2}H(-q^{1/2})-\varphi(q^{1/2})^2\varphi(q)^{-2}H(q^{1/2})) = q^{-7/60}\dfrac{1}{2}\varphi(q)^{-6}(\varphi(-q^{1/2})^2H(-q^{1/2})-\varphi(q^{1/2})^2H(q^{1/2})).
\Big{(}\mathcal{D}_tE_u^{n-\theta},v\Big{)}-\gamma(\nabla E_\sigma^{n-\theta},\nabla v)+(\nabla E_u^{n-\theta},\nabla v)+(f(u(t_{n-\theta}))-f^{n-\theta}(u),v)= (R_1^{n-\theta},v),~\forall v\in H_0^1,
\hat{H}=\frac{1}{2}\hat{p}^2_z+\hbar\omega\left[a^\dagger_+a_++a^\dagger_-a_-\right]\ \ \ ,\ \ \ \hat{\phi}=\hat{p}_z+\hbar g\left[a^\dagger_+a_+-a^\dagger_-a_-\right]\ \ \ .
g^{[2]}_0(x_0,0,x_1) =\frac{g^{[1]}_0(x_0,0)-g_0^{[1]}(0,x_1)}{x_0-x_1}=\frac{\frac{x_0f_0(x_0)-0}{x_0-0}-\frac{0-x_1f_0(x_1)}{0-x_1}}{x_0-x_1}& =\frac{f_0(x_0)-f_0(x_1)}{x_0-x_1}=f^{[1]}_0(x_0,x_1).
\sin(\theta) =(1-(\bar{\mathbf{n}}\cdot\mathbf{n}(Y))^2)^{1/2}&\leqslant (1+(1-2\epsilon^2)^{-3/2}) \kappa h ( 4 - (1+(1-2\epsilon^2)^{-3/2})^2 \kappa^2 h^2 )^{1/2}&\leqslant (1+(1-2\epsilon^2)^{-3/2}) ( 4 - (1+(1-2\epsilon^2)^{-3/2})^2 \epsilon^2 )^{1/2}\kappa h
\smash{\Bigl|\Bigl|T^{\,n}\Bigl(\sum_{s=1}^{j_{m}+j}z_{s}\Bigr)-x_{l}\Bigr|\Bigr|}\smash[t]{<\sum_{u=0}^{j-1}2^{-(j_{m}+u)}+\|T^{\,i}z_{j_{m}+j}\|}
A_{{\mathbf x}}{\mathcal H}(w,w')=(a_{11}^1{\mathbf x}_1(\alpha\bar w_1w_1'+\beta \bar w_2w_2'+\beta \bar w_3w_3'),a_{22}^2{\mathbf x}_2(\delta \bar w_2w_2'+\delta \bar w_3w_3')).
\eta =\sum\limits_{i\in I^g(\bar x)}\lambda^1_i\nabla g_i(\bar x)+\sum\limits_{j\in\mathcal P}\rho^1_j\nabla h_j(\bar x) &+\sum\limits_{l\in I^{0+}(\bar x)}\mu^1_l\nabla G_l(\bar x)+\sum\limits_{l\in I^{+0}(\bar x)}\nu^1_l\nabla H_l(\bar x)
h^{-1}((T^{n}- I) f)(S(0)) =b(S(0))\frac{f(S(0)+\Delta)-f(S(0))}{\Delta^2}\to \frac{1}{2}b(S(0))f''(S(0)), h^{-1}((T^{n}- I) f)(S(1)) =b(S(1))\frac{f(S(1)-\Delta)-f(S(1))}{\Delta^2}\to \frac{1}{2}b(S(1))f''(S(1)).
H_{2}^{- 1}\Big ( H_{6}\delta^{i j}\partial_{i}\partial_{j}b + 2 \delta^{i j}\partial_{i}H_{6}\partial_{j}b \Big ) - \frac{\kappa T_{2}}{4 \pi^{2}\alpha^{\prime}}( 4 N - M ) \delta ( x^{3}) \cdots \delta ( x^{5}) = 0 ,
\lim_{n\rightarrow \infty}\mathbf{E}\left[ \exp \left\{-\lambda \frac{Z_{N}(n)}{b_{N}n}\right\}\Big|\mathbf{Z}( n) \neq \mathbf{0};\mathbf{Z}( 0)=\mathbf{e}_{i}\right] =1-\Big( \frac{\lambda}{1+\lambda}\Big)^{1/2^{N-i}}.
\lim_{x\to 1}\dfrac{\widehat\psi(x,z)}{\widehat\varphi(x,z)}=\dfrac{\lim\limits_{x\to 1}\dfrac{\psi(x,z)}{\varphi(x,z)}}{\lim\limits_{x\to 1}\dfrac{\psi(x,z)}{\varphi(x,z)}+1}=\dfrac 1{z^2}-\dfrac 1{z\tan z}.
\bigtriangleup k= \frac{(p^{e'}+n-1-\deg(h(x)))(\sqrt{q}+1)-(\sqrt{q}+n-1)(p^{e'}+1)}{(p^{e'}+1)(\sqrt{q}+1)}= \frac{(n-2)(\sqrt{q}-p^{e'})-(\sqrt{q}+1)\deg(h(x))}{(p^{e'}+1)(\sqrt{q}+1)}\geq 1.
g^{- 1}\left( \begin{array}{c}{T_{1}}\\{\vdots}\\{T_{6}}\\ \end{array}\right) g = \left( \begin{tabular}{c | c}{a ( g )}&{0}\\ \hline{b ( g )}&{d ( g )}\\ \end{tabular}\right) \left( \begin{array}{c}{T_{1}}\\{\vdots}\\{T_{6}}\\ \end{array}\right) .
\left\langle \psi^n T_a, T_j, \tau^{(k)}\right\rangle^\textnormal{coh}_{0,k+2,d}= \sum_{u+v=k}\frac{k!}{u!v!}\left\langle \psi^n T_a, T_j, \tau_2^{(u)}, \tau'^{(v)}\right\rangle^\textnormal{coh}_{0,k+2,d}
( n + 1 ) S_{n , l}S_{n , l}^{*}= t_{- 1}( S_{n , l}S_{n + 1 , l}^{*}+ S_{n , l}^{*}S_{n - 1 , l}) ( 1 - S_{n , l}S_{n , l}^{*}) - l ( 1 - S_{n , l}S_{n , l}^{*}) ,
\phi^{(\mathfrak{r})}:M\setminus\Sigma \rightarrow M\setminus\Sigma'; \phi^{(\mathfrak{r})}(p)=p\mbox{for all}p\in M\setminus N_R, &(t,z)\mapsto (t,z+\chi^{(\mathfrak{r})}(z)\eta(t))\mbox{on}N_{R}
\limsup_{k\to +\infty}\int_{Q_1}f(a_1, \xi_k^{(1)}, \overline{v}_k^{(1)},\nabla \xi_k^{(1)})dx \leq \liminf_{k \to +\infty}\int_{Q_1}f(a_1, u_k^{(1)}, v_k^{(1)},\nabla u_k^{(1)})dx &=\eta^{N-1}K_p(a_1,0,c,d,e_N)+\int_{Q_1}f(a_1, u(x), v(x),0)dx.
E_{\pm}= \pm \frac{3}{2}m \, \mathrm{s e c h}^{2}\left( \frac{m}{2}\Delta x \right) \left\{\frac{1}{\alpha^{4}}\left[ ( - 2 \alpha^{2}+ 3 \alpha ) + \left( \frac{n m}{4}\Delta x \right) ( \alpha^{2}+ 2 \alpha - 3 ) \right] \right\}.
S :v\mapsto u=\arg\min_{w}I_{\varepsilon,v}(w), I_{\varepsilon,v}(w) =\int_{0}^{T}\exp(-t/\varepsilon)\left(\frac{\varepsilon}{2}|w^{\prime}|^{2}+\phi(w)-(f(v),w)\right) \mathrm{d}t,
\| s(v_t^\delta) \|_{H^1}= \| v_t^\delta -m_{\eta(v_t^\delta)}\|_{H^1}&\leq \| v_t^\delta -m_{\eta(v_t )}^\delta \|_{H^1}+ \| m_{\eta(v_t )}^\delta - m_{\eta(v_t )}\|_{H^1}+ \| m_{\eta(v_t )}-m_{\eta(v_t^\delta )}\|_{H^1}&\leq \|s^\delta (v_t)\|_{H^1}+ C\delta + C' \| v_t - v_t^\delta \|_{L^2},
\sum_{t=0}^{k-1}\binom{k-1}{t}(-p)^{-t}U_{mk+t}^{(k)}=(-p)^{1-k}U_{m}\sum_{t=0}^{k-1}\binom{k-1}{t}(-p)^{k-1-t}U_{m}^{k-1-t}U_{m+1}^{t}&=(-p)^{1-k}U_{m}\sum_{t=0}^{k-1}\binom{k-1}{t}(-pU_{m})^{k-1-t}U_{m+1}^{t}.
[\theta_i,\theta_j] = [z_i,z_j] + \beta [z_i, y_{ij}] + p_{ij}[y_{ij},z_i+z_j] \hphantom{[\theta_i,\theta_j] =}{}+ \sum_{k \not= i,j}\big(p_{ik}p_{jk}\big[y_{ij}+y_{ik}, y_{jk}\big]+p_{ik}p_{ji}\big[ y_{ij},y_{ik}+y_{jk}\big] \big) =0.
\overline{r}_1=\{R_{1213}(R_{2323}-c) - R_{1223}R_{1323}\}S_{12121}- \{R_{1213}R_{1323}- R_{1223}(R_{1313}-c)\}S_{12122}& - \{(R_{1212}-c)(R_{2323}-c)- R_{1223}{}^2\}S_{12131}+ \{(R_{1212}-c)R_{1323}- R_{1213}R_{1223}\}S_{12132}& +\{(R_{1212}-c)R_{1323}- R_{1213}R_{1223}\}S_{12231}- \{(R_{1212}-c)(R_{1313}-c) - R_{1213}{}^2\}S_{12232}.
F_{S}= \left( F_{S}^{( 0 )}, F_{S}^{( 1 )}\left( k_{1}\right) , F_{S}^{( 2 )}\left( k_{1}, k_{2}\right) , \ldots , F_{S}^{( n )}\left( k_{1}, k_{2}, \ldots , k_{n}\right) , \ldots \right) \; ,
\lambda({\mbox{\boldmath$\rho$}},{\mbox{\boldmath$\dot\rho$}}) =({\mbox{\boldmath$\rho$}}{\mbox{\boldmath$\dot\rho$}})\left(\frac{\mu{\mbox{\boldmath$\dot\rho$}}^2}{8}\left(1-\frac{3\mu}{m}\right)+ \frac{1}{4}\left(1-\frac{\mu}{m}-4B\right)U(\rho)\right)\,,
H ( x , y ) = 1 +{\frac{A}{( x + y )^{2}}}+{\frac{B}{( x + y )^{2}}}\left( \mathrm{l n}\left({\frac{( x + y )^{2}}{\rho_{0}^{2}y}}\right) - 1 +{\frac{x}{y}}\right) ,
\epsilon\int_{0}^{1}\rho^{\epsilon}_x\bar{\sigma}_x+\int_{0}^{1}u^{\epsilon}_x\bar{\sigma}+\int_{0}^{1}\rho^{\epsilon}_x\bar{\sigma}+\lambda\int_{0}^{1}\rho^{\epsilon}\bar{\sigma}=\int_{0}^{1}f\bar{\sigma}.
A = N^{-7/2}\mathbb{E}\sum_{\substack{j_1,j_2,p_1 p_2,q_1,q_2}}W_{p_1j_1}\overline{W_{p_2j_2}}\mathbb{E}_{k}(G^{(1)}_{kp_1j_1})_{j_1q_1}\mathbb{E}_{k}(G^{(2)}_{kp_1j_1})_{q_1p_1}& \times\overline{\mathbb{E}_{k}(G^{(1),p_1j_1}_{kp_2j_2})_{j_2q_2}}\overline{\mathbb{E}_{k}(W_{p_1 j_1}(G^{(2),p_1j_1}_{kp_2j_2})_{q_2j_1}(G^{(2)}_{kp_2j_2})_{p_1p_2})}
\begin{aligned}\hat{f}_{\mathcal{A}}({\bf x})\geq \sum_{i\in [n]}\left(\sum_{(i,i_2,\cdots,i_m)\notin I}|a_{ii_2\cdots i_m}|y_i^{1-m}y_{i_2}y_{i_3}\cdots y_{i_m}x_i^m\right)-\sum_{(i_1,i_2,\cdots,i_m)\in \Delta_{\mathcal{A}}}|a_{i_1i_2\cdots i_m}|x_{i_1}x_{i_2}\cdots x_{i_m}.\end{aligned}
\begin{array}[c]{ll}\bar{Y}^{t,x;\bar{u}}(s)=W(s,\bar{X}^{t,x;\bar{u}}(s)), \bar{Z}^{t,x;\bar{u}}(s)=V(s,\bar{X}^{t,x;\bar{u}}(s),W(s,\bar{X}^{t,x;\bar{u}}(s)),W_x(s,\bar{X}^{t,x;\bar{u}}(s)),\bar{u}(s)),s\in\lbrack t,T]\end{array}
\frac{a_{p,q}(x)-a_{p,q}(x_n)}{x-x_n}=\frac{\pi_q(0^m1\overline{a_{n,m+2}a_{n,m+3}\cdots})}{\pi_p(0^m1\overline{a_{n,m+2}a_{n,m+3}\cdots})}&\ge \frac{\pi_q(0^m10^{\infty})}{\pi_p(0^m1^{\infty})}=\frac{p-1}{p}\left(\frac{p}{q}\right)^{m+1}
\begin{array}{c}{{\lbrack e_{i}, f_{j}\rbrack = \delta_{i j}\frac{k_{i}- k_{i}^{- 1}}{Q_{i}- Q_{i}^{- 1}}}}\\{{k_{i}e_{j}k_{i}^{- 1}= Q_{i}^{a_{i j}}e_{j}, ~ ~ k_{i}f_{j}k_{i}^{- 1}= Q_{i}^{- a_{i j}}f_{j}}}\\{{k_{i}k_{i}^{- 1}= k_{i}^{- 1}k_{i}= 1 , ~ ~ k_{i}k_{j}= k_{j}k_{i}}}\\{{\lbrack e_{i}, e_{i}\rbrack = \lbrack f_{i}, f_{i}\rbrack = 0 , \mathrm{i f}a_{i i}= 0}}\\ \end{array}
\int_{B_1}\varphi(x)\,dx =\int_0^1\int_{\partial B_t}\varphi(y)\,d\mathcal{H}^{n-1}(y)\,dt=\int_0^1 t^{n+1}\int_{\partial B_1}\varphi(y)\,d\mathcal{H}^{n-1}(y) &=\frac{1}{n+2}\int_{\partial B_1}\varphi(y)\,d\mathcal{H}^{n-1}(y).
D_{\vec{m}\vec{n}}= D_{m_{1}n_{1}}^{\theta_{( 1 )}}( z_{1}, \bar{z}_{1}) D_{m_{2}n_{2}}^{\theta_{( 2 )}}( z_{2}, \bar{z}_{2}) \dots D_{m_{p / 2}n_{p / 2}}^{\theta_{( p / 2 )}}( z_{p / 2}, \bar{z}_{p / 2}) ,
{\hat{v}}( \eta ,{\bf x}) = \frac{1}{( 2 \pi )^{3 / 2}}\int \mathrm{d}^{3}{\bf k}[{\hat{c}}_{\bf k}( \eta ) e^{i{\bf k}\cdot{\bf x}}+{\hat{c}}_{\bf k}^{\dagger}( \eta ) e^{- i{\bf k}\cdot{\bf x}}] ,
\lim_{\eta\to0}\int_{- \infty}^{+\infty}e^{i \gamma x}\, \frac{\partial^{\alpha, \eta}f}{\partial|x|^{\alpha}}\, \mathrm{d}x= 2 C_{\alpha, 0}\, |\gamma|^{\alpha}\cos\biggl( \frac{\pi\alpha}{2}\biggr)\, \hat{F}(\gamma)
\mu_{Mas}(E^u_{(0,\cdot)}(0),E^s_{(0,\cdot)}(0)) =\mu^{(-,0)}_{Mas}(E^u_{(0,\cdot)}(0),E^s_{(0,\cdot)}(0))+\mu^{(+,0)}_{Mas}(E^u_{(0,\cdot)}(0),E^s_{(0,\cdot)}(0)), \mu_{Mas}(E^u_{(1,\cdot)}(0),E^s_{(1,\cdot)}(0)) =\mu^{(-,0)}_{Mas}(E^u_{(1,\cdot)}(0),E^s_{(1,\cdot)}(0))+\mu^{(+,0)}_{Mas}(E^u_{(1,\cdot)}(0),E^s_{(1,\cdot)}(0)).
Z_{O(2)NLS}= \sum_{n=0}^{\infty}{\zeta^n \over (n!)^2}\int \prod_{j=1}^{n}d^2z_j \exp \left[ (2\pi)^2 \beta \sum_{i,j}Q_i Q_j{1 \over 2\pi}\ln{R_0 \over |z_i-z_j|}\right] ,
\sum_{i = 1}^{r}\, \Lambda ( H_{i}) \Lambda^{\prime}( H^{i}) = ( \lambda , \lambda^{\prime}) \, , ~ \forall \Lambda = ( \lambda , \kappa , \tau ) \, , \, \Lambda^{\prime}= ( \lambda^{\prime}, \kappa^{\prime}, \tau^{\prime}) \, \in \,{\cal H}^{*}
{\alpha}_{1}= \displaystyle \frac{g^{2}C_{V}}{4 \pi}, \quad{\alpha}_{2}= - \displaystyle \frac{g^{2}C_{V}}{3 2 \pi}, \quad{\beta}_{1}= - \displaystyle \frac{g^{2}C_{V}}{1 6 \pi}, \quad{\beta}_{2}= 0 .
\frac{F}{V}=-\sum_{\stackrel{n=1}{n ~\rm odd}}^\infty\sum_{k=1}^\infty\frac{2}{k}\left( \frac{1}{4\pi^2\alpha'\tau_2}\right)^5e^{-n^2\beta^2/4\pi\alpha'\tau_2}\sum_Me^{-\pi\alpha'\tau_2 M^2}
\rho_q(\xi) = \lim_{\lambda \rightarrow \infty}\lambda^{-q}\rho(\lambda \xi), \rho_{q-j}(\xi) = \lim_{\lambda \rightarrow \infty}\lambda^{-q+j}\biggl( \rho(\lambda \xi)- \sum_{\ell <j}\lambda^{q-\ell}\rho_{q-\ell}(\xi)\biggl), j\geq 1.
\widetilde{\textbf{c}}_{q,j}^{(n)}=\frac{1}{n!(q-1)!(n-q)!}\sum_{i=0}^{2n-2-j}P_{2n-i}(q)(n+1-q)^{2n-2-j-i}&=\frac{\binom{n}{q-1}}{(n!)^2}\sum_{i=j+2}^{2n}P_i(q)(n+1-q)^{i-2-j}.
T^{\alpha+1}=\{t\in T^{\alpha}: (\exists x\in p[T^{\alpha}_{[t]}])(\forall k)(\forall \alpha_0,\ldots,\alpha_{k-1}<\alpha) &(\forall i_0,\ldots, i_{k-1}\in\omega)\x\not\subseteq^*\bigcup_{j<k}A^{\alpha_j}_{i_j}\};
w_1=\frac{1}{8}{\varphi_2^2 + \frac{1}{4}\sin^2\varphi_1-\frac{c_0}{2}\sin\varphi_1}+\varepsilon h_1(\varphi_1,\varphi_2,\varepsilon),\ \w_2=\varepsilon h_2(\varphi_1,\varphi_2,\varepsilon),\ \ (\varphi_1,\varphi_2)\in\mathcal{K},
\mathrm{trires}\left( \Phi_{A}^{*(2)}\right) =\left( gh_{1}\left( \Phi^{A}\right) ,gh_{2}\left( \Phi^{A}\right) +1,gh_{3}\left( \Phi^{A}\right)\right) =-\mathrm{trigh}\left( \Phi_{A}^{*(2)}\right) ,
D_{ki}= \begin{cases}\frac{(1+\tau_k)\dot{P}_K(\tau_k)+P_K(\tau_k)}{(\tau_k-\tau_i)\big[(1+\tau_i)\dot{P}_K(\tau_i)+P_K(\tau_i) \big]}k\neq i \frac{(1+\tau_i)\ddot{P}_K(\tau_i)+2\dot{P}_K(\tau_i)}{2\big[ (1+\tau_i)\dot{P}_K(\tau_i)+P_K(\tau_i) \big]}k = i\end{cases}
\tilde K_{a,b}(u;m) = \frac{d^{j_m}}{dz^{j_m}}\Big|_{z=0}(1-z)^{-a}(1-u-z)^{-b}, \tilde H_{a,b,c}(u,v;m) = \frac{d^{j_m}}{dz^{j_m}}\Big|_{z=0}(1-z)^{-a}(1-u-z)^{-b}(1-v-z)^{-c}.
= \frac{i \beta}{2 \pi}4 i m \varepsilon^{\mu \nu \alpha}\frac{k_{\alpha}}{\sqrt{k^{2}}}\log \left( \frac{1 - \sqrt{\frac{4 m^{2}}{k^{2}}}}{1 + \sqrt{\frac{4 m^{2}}{k^{2}}}}\right) ,
\!\!\!\!\!\!\! \mathcal{R}_{\rm u}= \int\limits_{0}^{\infty}\frac{B_{\rm u}^{\alpha{\rm D}}}{(g+1) \ln (2)}\exp \left(- \frac{N_o g}{\rho}\right) \mathcal{L}_{I_{\rm u}}\left(\frac{g}{\rho |\mathcal{I}_{\rm u}(\alpha)|^2}\right) dg,
\phi^i_D(u) = \bar p_1 \phi^i_R(u) + \bar p_2 \sum_{j\neq i}u_i u_j (G_{i,i}-G_{j,i}+ G_{i,j}-G_{j,j}) & -(1/\kappa) p(v_1|v_2) \sum_{j \neq i}u_i u_j ( G_{i,j}-G_{j,i}).
S = \int d^{1 0}x \sqrt{- g_{1 0}}\left\{e^{- \varsigma}\left[ R_{1 0}+ \left( \nabla \varsigma \right)^{2}\right] - \frac{1}{4}F_{A B}F^{A B}- \frac{1}{4 8}F_{A B C D}F^{A B C D}- \frac{1}{2}m^{2}\right\},
\begin{array}{llr}F(s,r)-F(s,0)= \int\limits_0^r \mathbb{E}[\{f(\zeta^1(s),\zeta^{2,t}(u))g(u,\zeta^{2,t}(u))+\mathcal{G}^{2,t}(u)f(\zeta^1(s),\zeta^{2,t}(u))\}. \exp\{\int\limits_0^u g(\tilde{u},\zeta^{2,t}(\tilde{u}))d\tilde{u}\}]du.\end{array}
\mathop{\lim}\limits_{N \to \infty}\frac{1}{N}\sum\nolimits_{n=1}^N{{{\left|{{\lambda_{u,n}}}\right|}^2}}= \mathop{\lim}\limits_{L \to \infty}\sum\nolimits_{l=1}^L{{{\left|{{h_{u,l}}}\right|}^2}}.
f_{k,\ell-1}- f_{k-1,\ell}= \beta \sum_{i=0}^{\ell-1}\varrho^{\ell-i-1}\eta_{k,i}+ \frac{\beta^2}{2}\left( \sum_{i=0}^{\ell-1}\varrho^i \right)^2 - \beta \sum_{j=0}^{J}b_j \sum_{i=0}^{j \wedge (\ell - 1)}\varrho^{\ell+j-1-2i}
f(t,\xi,z) = \frac{G_{|\Omega}^{1-s}(x,\xi)}{\Gamma(s)\,\delta(z)}p_\Omega(t,\xi,z)t^{s-1}\le C \vert x-\xi\vert^{2s-N}\left[\frac{\delta(x)\delta(\xi)}{\vert x-\xi\vert^2}\wedge 1\right]t^{s-2-N/2}\delta(\xi)e^{-c_2\vert z-\xi\vert^2/t}.
Ev(k,z) := \left\{\forall j\in [|k, N |] ,\, l_j^{(N)}\leq \sqrt{\frac{1}{2}}W_{\tau_j^{(k)}}- \tau^{(k)}_j +z\leq u_j^{(N)}\right\}, GEv(k,z) := \left\{\forall j\in [|k, N |] ,\, l_j^{(N)}\leq \sqrt{\frac{1}{2}}W_{\tau_j^{(k)}}+z \leq u_j^{(N)}\right\}.
Q=\begin{pmatrix}1 1 0 \cdots 0 -1 0 1 0 \cdots 0 -1 0 0 1 \cdots 0 -1 \vdots \vdots \vdots \ddots \vdots \vdots 0 0 0 \cdots 1 -1 0 0 0 \cdots 0 1\end{pmatrix}.
=\frac{n^{1/2}}{m^2}\cdot\int\limits_{1/R<|y|<1}\frac{\Vert f^{\,\prime}(y)\Vert}{|y|^{2n}}\,dm(y)\leqslant \frac{n^{1/2}R^{\,2n}}{m^2}\cdot\int\limits_{1/R<|y|<1}|\nabla f(y)|\, dm(y)<\infty\,.
\xi_\beta(t) =\left ( 1-\frac{\tau}{t}\right )^\beta :=\exp\left\{\beta \log \left | 1-\frac{\tau}{t}\right | + i \beta \arg \left ( 1-\frac{\tau}{t}\right ) \right \}, \eta_\beta(t)=\left (1-\frac{t}{\tau}\right )^\beta :=\exp\left \{\beta \log \left |1-\frac{t}{\tau}\right | + i \beta \arg \left ( 1-\frac{t}{\tau}\right ) \right \}.
a_r^\epsilon := \begin{cases}a_r, ~ |a_r - a_t| \leq (\mu - \epsilon)|r-t|^{\kappa}a_t + (\mu - \epsilon)(t-r)^\kappa \frac{a_r - a_t}{|a_r - a_t|}, ~ |a_r - a_t| \geq (\mu - \epsilon)|r-t|^{\kappa}. \end{cases}
[ \tilde{\Sigma}( x ) , \tilde{\Sigma}( y ) ] = [{\partial}^{+}\tilde{\Sigma}( x ) ,{\partial}^{+}\tilde{\Sigma}( y ) ] = 0 , [ \tilde{\Sigma}( x ) ,{\partial}^{+}\tilde{\Sigma}( y ) ] = i{\delta}( x^{--}y^{-}) .
\zeta_{2}^{L L T + T R}( 0 ) = \left( \frac{\alpha}{A}\right) ( P r^{2}) - \left( \frac{a_{1}- 2}{A}\right) ( Q r^{2}) + \left[ \frac{2}{3 A}( 3 \alpha - 2 A ) + \frac{a_{1}- 2}{A}\right] \ .
B ( q^{2}) = b_{0}( q^{2}) g^{3}+ g^{5}{\frac{B_{1}( q^{2}, Q^{2})}{\epsilon}}+ g^{5}B_{2}( q^{2}, Q^{2}) + g^{7}\left[{\frac{B_{3}( q^{2}, Q^{2})}{\epsilon^{2}}}+{\frac{B_{4}( q^{2}, Q^{2})}{\epsilon}}+ B_{5}( q^{2}, Q^{2}) \right] + . . .
g_{\alpha}\theta_{\alpha}\psi_{\alpha\pi,\beta\pi}=(h_{\alpha}\theta_{\alpha}^H\psi_{\alpha\pi,\beta\pi}^H)(k_{\alpha}\theta_{\alpha}^K \psi_{\alpha\pi, \beta\pi}^K) & = (h_{\alpha}\psi_{\alpha,\beta}^{H}\theta_{\beta}^{H})(k_{\alpha}\psi_{\alpha,\beta}^{K}\theta_{\beta}^{K}) & = g_{\alpha}\psi_{\alpha,\beta}\theta_{\beta}
I_1 \le C\|u-\bar{u}\|_{L^t(R_0\cup R_1)}^{as}\|u\|_{L^q(R_0)}^{(1-a)s}\le C\big( \|u-\bar{u}\|_{L^1(R_0\cup R_1)}+\|\nabla(u-\bar{u})\|_{L^1(R_0\cup R_1)}\big)^{as}N_0^{(1-a)s/q}\le C(M_0+M_1)^{as}N_0^{(1-a)s/q}.
[|u|^p]_{\alpha;\bar{\Omega}}=\sup_{x\neq y}\frac{||u(x)|^p-|u(y)|^p|}{|x-y|^{\alpha}}=\sup_{x\neq y}p\xi^{p-1}\frac{||u(x)|-|u(y)||}{|x-y|^{\alpha}}&\leq C(\Omega,p)[u]_{\alpha;\bar{\Omega}}\leq C,
\sum_{\sigma \in S_n}x_{\sigma(1)}^{d_1}\ldots x_{\sigma(j)}^{d_j}= \Big(\sum_{\pi \in S_n}x_{\pi(1)}\ldots x_{\pi(j)}\Big)^{d_j}\cdot \sum_{\rho \in S_n}x_{\rho(1)}^{d_1 - d_j}\ldots x_{\rho(j-1)}^{d_{j-1}- d_j},