harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}w(k,A,b)(z) =w_0(k,b)+A\cos(z)+\frac{1}{2}A^2\left(\frac{1}{m(k)-1)}+\frac{\cos(2z)}{m(k)-m(2k)}\right) &\qquad+\mathcal{O}(A(A^2+b^2)) c(k,A,b) =c_0(k,b)+A^2\left(\frac{1}{m(k)-1}+\frac{1}{2(m(k)-m(2k))}\right) &\qquad+\mathcal{O}(A(A^2+b^2))\end{aligned}
(fv_{i})(y,m_{1},m_{2})=e^{2\pi i(yz_{4i}+m_{1}z_{5i}+m_{2}z_{6i}-z_{1i}z_{4i}-z_{2i}z_{5i}-z_{3i}z_{6i})}f(y-z_{1i},m_{1}-z_{2i},m_{2}-z_{3i}).
L_{18}= -\int_{\Omega}(H\times H)_{tt}\cdot u_{t}x=\int_{\Omega}\big(H\otimes H-\frac{1}{2}|H|^2I_3\big)_{tt}: \nabla u_{t}x \leq C|\nabla u_t|_2|H_t|^2_4+C|\nabla u_t|_2|H_{tt}|_2| H|_{\infty}.
I_1 = (e^{g_n}(z_a)-e^{g_n}(z_a'))e^{ik\cdot H_n(z_a)}e^{i\omega\varphi_n(a)}v(z_a), I_2 = e^{g_n}(z_a')(e^{ik\cdot H_n(z_a)}-e^{ik\cdot H_n(z_a')})e^{i\omega\varphi_n(a)}v(z_a), I_3 = e^{g_n}(z_a')e^{ik\cdot H_n(z_a')}e^{i\omega\varphi_n(a)}(v(z_a)-v(z_a')).
\int_{\mathbb{Z}_p}(1+t)^{\tfrac{x}{2}}d\mu_{-1}(x) = \sum_{m=0}^\infty \int_{\mathbb{Z}_p}x^m d\mu_{-1}(x) \frac{1}{2^m}\frac{1}{m!}\Big(\log(1+t)\Big)^m &=\sum_{m=0}^\infty E_m 2^{-m}\sum_{n=m}^\infty S_1(n,m) \frac{t^n}{n!}&= \sum_{n=0}^\infty \left( \sum_{m=0}^n 2^{-m}E_m S_1(n,m) \right) \frac{t^n}{n!}.
\nabla U^p(x) \cdot b(x) = 2p\frac{1}{\delta'}\sum_{i = 1}^d (\delta' |x|_1 - d\lambda')^{2p-1}(\kappa'(x_{i - 1}- x_{i+1})x_i - \delta'x_i +\lambda') = \frac{-2p}{\delta'}\left(\delta'|x|_1 - d \lambda'\right)^{2p}\le 0.
\langle B^{(n)}_{st}, e_{i_1}\otimes \dots \otimes e_{i_n}\rangle = \sum_{j=1}^n (-1)^{j-1}\int_{A^n_j}\prod_{l=1}^{j-1}K(s,r_l) [ K(t,r_j) - K(s,r_j)] \prod_{l=j+1}^n K(t,r_l) \circ dW^{i_1}_{r_1}\dots \circ dW^{i_n}_{r_n}
S = - \frac{1}{4 \pi \alpha^{\prime}}\int \! d^{2}\xi [ \sqrt{- h}h^{\alpha \beta}G_{A B}{\partial_{\alpha}}X^{A}{\partial_{\beta}}X^{B}+ \epsilon^{\alpha \beta}B_{A B}{\partial_{\alpha}}{X^{A}}{\partial_{\beta}}{X^{B}}] ,
d(z_{1},z_{2})\ge S^{-}(z_{2})-S^{-}(z_{1})\ge S^{-}(z_{2})-S^{+}(z_{1})= &=S^{-}(z_{2})-S^{-}(x)+S^{+}(x)-S^{+}(z_{1})=d(x,z_{2})+d(z_{1},x)\ge d(z_{1},z_{2}),
W_i(\hat{x}) = \frac{(x_i - \hat{x}_i) (\sigma_i(x) - \hat{\sigma}_i(x) + \sigma_i(x) \, \hat{\sigma}_i(x))}{1 -\sigma_i(x)}\prod_{j \ne i}{\frac{\hat{x}_i - x_j}{\hat{x}_i - \hat{x}_j}}\, ,
\frac{1}{n^2}\sum_{a=1}^{n}\sum_{B=n+1}^{2n}G_{aB}G_{B \bar a}=\frac{1}{n^2}\sum_{a=1}^{n}\sum_{B\neq \bar{a}}G_{aB}G_{B \bar a}+\frac{1}{n^2}\sum_{a=1}^{n}G_{a \bar{a}}G_{\bar a \bar a},
\begin{aligned}\bar{x}^{k+1}=\bar{x}^k+\gamma_1^k \bar{\zeta}_w^k-\lambda^k \frac{(u\otimes{I_d})^T}{m}\left(y^k-(v\otimes{I_d})\bar{y}^k\right) \qquad-\lambda^k \frac{(u\otimes{I_d})^T}{m}(v\otimes{I_d})\bar{y}^k\end{aligned}
\Gamma_{V D}[ \phi ] = S [ \phi ] - \frac{1}{2}\mathrm{T r \l n}\left[ G^{l i}\left( S_{, i j}- \left\{\begin{array}{c}{k}\\{i \j}\\ \end{array}\right\}S_{, k}- T_{i j}^{k}S_{, k}\right) \right] \ ,
[ D_{\mu}, \Phi ]_{\times}= - i R^{- N}{\theta}_{{\mu}{\nu}}^{- 1}[ L^{\nu}, \Phi ]_{\times}+ i a R^{- N}{\theta}_{{\mu}{\nu}}^{- 1}{\Delta}( R^{N}, R^{- N}[ L^{\nu}, \Phi ] ) .
-h'(s) = \sum_{m=1}^{\infty}-\phi'(ms) & = \sum_{m \leq s^{-1/2}}-\phi'(ms) + \sum_{s^{-1/2}< m \leq s^{-1}}-\phi'(ms) + \sum_{m > s^{-1}}-\phi'(ms) & =: \chi_1(s)+\chi_2(s)+\chi_3(s).
\begin{aligned}\left[c_{p,r}\left(M_{s,t}>\eta\right)\right]^{p}\leq\left[\sum_{l=0}^{r}\left(\frac{\eta}{p(t-s)^{2H}}\right)^{lp}(t-s)^{lHp}\right]e^{-\frac{\eta^{2}}{2(t-s)^{2H}}}=\sum_{l=0}^{r}\left(\frac{\eta}{p(t-s)^{H}}\right)^{lp}e^{-\frac{\eta^{2}}{2(t-s)^{2H}}}.\end{aligned}
R_p(u\leftrightarrow\partial B)^{\frac{1}{p-1}}\ll\frac{1}{\deg(u)^{\frac{1}{p-1}}}+\sum_{n=0}^{\lfloor\log_2(|B|/\deg(u))\rfloor}\max_{\substack{A\subset B u\in A A |B|/2^{n+1}<|A|\le|B|/2^n}}j_{A,p}.
\begin{bmatrix}0 1\end{bmatrix}_{\frac{1}{4},0}\begin{bmatrix}1 0\end{bmatrix}_{\frac{1}{2},0}\equiv\begin{bmatrix}1 0\end{bmatrix}_{\frac{1}{2},0}\begin{bmatrix}1 1\end{bmatrix}_{\frac{1}{4},-\frac{1}{2}}\begin{bmatrix}1 1\end{bmatrix}_{\frac{1}{4},\frac{1}{2}}\begin{bmatrix}0 1\end{bmatrix}_{\frac{1}{4},0}.
{\Phi}(r,z,\omega)=\vert\nabla_x{c}(z,{X}^{M_{{\mathcal{P}}_n}}_{r-})\vert\vert D^Z{X}^{M_{{\ \mathcal{P}}_n}}_{r-}-D^Z{X}^{M_{{\ \mathcal{P}}_m}}_{r-}\vert_{l_2},
\dot s(t)+\frac{s(t)^2}{n-1}-\frac{2s(t)}{n-2}\frac{\left<\nabla g(\Psi_t(v)),\dot\Psi_t(v)\right>}{g(\Psi_t(v))}-\epsilon g(\Psi_t(v))^{\frac{2n-2}{n-2}}\leq 0.
\phi (\theta ,\varphi )=a\left\{\cos^{2}\frac{\theta}{2}|2\lambda_{\phi}>-\frac{\sqrt{2}}{2}\sin \theta e^{-i\varphi}|2\lambda_{\phi}-\alpha >+\sin^{2}\frac{\theta}{2}e^{-2i\varphi}|2\lambda_{\phi}-2\alpha >\, .\right\}
\mathcal{Z}_{m,n}^{\gamma}(z,\overline{z})= (-1)^{m}(\gamma+m+1)_{n}\left(1-|{z}|^{2}\right)^{-\gamma}\dfrac{\partial^{m}}{\partial z^{m}}\left(z^{n}\left(1-|{z}|^{2}\right)^{\gamma+m}\right)
\gamma_\chi(z^2) V^-(w) = V^+(z)^{-1}H^{\gamma}(z^2)z^{-2h_0}V^-(z)^{-1}V^-(w) = V^+(z)^{-1}V^-(w) H^{\gamma}(z^2)z^{-2h_0}V^-(z)^{-1}&= \left(1-\frac{z^2}{w^2}\right) V^-(w) V^+(z)^{-1}H^{\gamma}(z^2)z^{-2h_0}V^-(z)^{-1}= \frac{w^2 -z^2}{w^2}V^-(w)\gamma_\chi(z^2).
e_{m_{1}...m_{q}}=\frac{1}{N^{[1+\frac 12(m_{1}+\cdots+m_{q})]}}\sum_{k_{0}k_{1}...k_{q}}E_{m_{1}}(a_{k_{0}}-a_{k_{1}})\overline{E_{m_{2}}(a_{k_{1}}-a_{k_{2}})}...\mathbf{C}^{q}E_{m_{q}}(a_{k_{q-1}}-a_{k_{q}}).
h(x,a,b,\theta ) :=\mathbb{E}_x\left(e^{-q\tau_a^-(r)-\theta [a-X_r (\tau_a^- (r))]};\tau_a^-(r) <\tau_b^+ (r)\right) = \hat{J}_a^{(q,r)}(x,\theta ) -\frac{I_a^{(q,r)}(x)}{I_a^{(q,r)}(b)}\hat{J}_a^{(q,r)}(b,\theta ) = J_a^{(q,r)}(x,\theta ) -\frac{I_a^{(q,r)}(x)}{I_a^{(q,r)}(b)}J_a^{(q,r)}(b,\theta ).
G_k^0(x) =G_{k}(x)-l(k)=\frac1{2\pi}\int_0^{\infty}\frac{e^{ix\xi}-\chi(\xi)}{\xi-k}~d\xi, T_k^0(\varphi) =G_k^0*(u\varphi) = T_k(\varphi)-l(k)\langle \varphi, u\rangle.
\psi (X^1,\cdots, X^i+2\pi r_i',\cdots, X^n)=e^{\frac{r_i'}{r}\mathbf{i}(q'-q)^t\mathcal{B}_i}\psi (X^1,\cdots, X^i,\cdots, X^n) \ (i=1,\cdots, s), &\psi (X^1,\cdots, X^i+2\pi ,\cdots, X^n)=e^{\frac{\mathbf{i}}{r}(q'-q)^t\mathcal{B}_i}\psi (X^1,\cdots, X^i,\cdots, X^n) \ (i=s+1,\cdots, n)
\begin{aligned}2\gamma \sum_{i=1}^{n-1}\langle (\bar{z}_i-\bar{x})-(z_i-x_i),x_{i+1}-x_i\rangle & = \sum_{i=1}^{n-1}\left(\|\bar{z}_i-z_i\|^2 + \|z_i^+-z_i\|^2 - \| z_i^+-\bar{z}_i\|^2 \right) & \quad-\frac{1}{\gamma}\sum_{i=1}^{n-1}\|z_i^+-z_i\|^2 + \gamma \left( \|x_{n}-\bar{x}\|^2 - \|x_1-\bar{x}\|^2\right).\end{aligned}
\prescript{}{i_0}{(\hat{s}^*})(i)= \begin{cases}\hat{s}^*(i) i<i_0 \hat{s}^*(i+1) i_0\leq i\end{cases}= \begin{cases}\hat{s}(j-i+1) i<i_0 \hat{s}(j-i) i_0\leq i\end{cases}
k_{2}^{\mp}(v)^{-1}e_{1}^{\pm}(u)k_{2}^{\mp}(v) =\frac{u_{\pm}-v_{\mp}+h}{u_{\pm}-v_{\mp}}e_{1}^{\pm}(u)-\frac{h}{u_{\pm}-v_{\mp}}e_{1}^{\mp}(v), k_{2}^{\mp}(v)f_{1}^{\pm}(u)k_{2}^{\mp}(v)^{-1}=\frac{u_{\mp}-v_{\pm}+h}{u_{\mp}-v_{\pm}}f_{1}^{\pm}(u)-\frac{h}{u_{\mp}-v_{\pm}}f_{1}^{\mp}(v).
M^{\dagger}=\begin{pmatrix}\Sigma \Sigma H^{\ast}E_{S_{A}}-\Psi KS_{A}^{\dagger}F_{S_{A}}K^{\ast}\Sigma-S_{A}^{\dagger}H\Phi S_{A}^{\dagger}-S_{A}^{\dagger}H\Phi H^{\ast}E_{S_{A}}-F_{S_{A}}K^{\ast}\Psi KS_{A}^{\dagger}+F_{S_{A}}K^{\ast}\Sigma H^{\ast}E_{S_{A}}\end{pmatrix},
\|\zeta\|_{C(\overline{S_T})}\leq c\|\zeta(\cdot,0)\|_{C(\Gamma(0))}+c\varepsilon\sum_{\xi=\zeta,\zeta_2}\Bigl(\|\xi\|_{C(\overline{S_T})}+\|\xi\|_{C^{2,1}(S_T)}\Bigr)
2 \pi i \omega_{s p}^{( 0 )}( \{q \}) = \delta_{s p}\ln k_{p}+{\sum_{\Gamma}}^{'^{\prime}}\ln \frac{[ u_{s}- g_{\Gamma}( u_{p}) ] [ v_{s}- g_{\Gamma}( v_{p}) ]}{[ u_{s}- g_{\Gamma}( v_{p}) ] [ v_{s}- g_{\Gamma}( u_{p}) ]}
\xi(\theta) = \xi(m_0) + \frac{1}{\sqrt{n}}\sum_{r,s}\delta_r(\hat{\theta}^M-m_0)^s\hat{\xi}^{''}_{rs}(m_0) + \frac{1}{2n}\sum_{r,s}\delta_r\delta_s\hat{\xi}^{''}_{rs}(m_0) + O_p\left( n^{-1}\right) \,.
\overline{\tau}_i(P^{op}_e(i)) =\sum_{j<i}\,\tau_j(P^{op}_e(i)) = \sum_{j<k_1}\,\tau_j(P^{op}_e(i))+\sum_{k_1\leq j<i}\,\tau_j(P^{op}_e(i)) =\sum_{j<k_1}\,\tau_j(P^{op}_e(i))+P^{op}_e(i) =P^{op}_e(i),
(a_xf)(x_1,\cdots ,x_n) = f(x,x_1,\cdots x_n) (a_x^+f)(x_1,\cdots ,x_n) = \delta_{x,x_1}f(x_2,\cdots ,x_n) +\delta_{x,x_2}f(x_1,x_3,\cdots ,x_n)+\cdots + \delta_{x,x_n}f(x_1,\cdots ,x_{n-1})
I = \frac{i s}{4 \pi}\int{\mathrm{T r}}( A \wedge d A + \frac 2 3 A \wedge A \wedge A ) - \frac{i s}{4 \pi}\int{\mathrm{T r}}( \bar{A}\wedge d \bar{A}+ \frac 2 3 \bar{A}\wedge \bar{A}\wedge \bar{A}) \, ,
\det \big(M(\xi)-\lambda I\big) =\det\left(-(\xi_1^2+\xi_2^2+\cdots+\xi_d^2)\left(\begin{smallmatrix}1 0 0 \epsilon\end{smallmatrix}\right)+i\xi_1 c I+\left(\begin{smallmatrix}0 e^{-\kappa}0 -\kappa e^{-\kappa}\end{smallmatrix}\right)\right) &=0.
Op_h^w(x\xi +p(\xi))\theta_h(x) Op_h^w\Big(\chi(h^{\beta}\xi)\gamma(\frac{x+p'(\xi)}{\sqrt{h}})\Big)v^{\Sigma}= & = Op_h^w(x\xi +p(\xi))\theta_h(x) v_{\Lambda}^{\Sigma}- Op_h^w(x\xi +p(\xi))\theta_h(x) Op_h^w\Big((1-\chi(h^{\beta}\xi))\gamma(\frac{x+p'(\xi)}{\sqrt{h}})\Big)v^{\Sigma}\,,
\left[ \langle A_{m_{0}}\rangle \Phi^{2}\vert_{F}+ \langle A_{g_{0}}\rangle \Phi^{3}\vert_{F}+ h . c . \right]{}+ 2 R e ( \langle F_{m_{0}}\rangle A^{2}+ \langle F_{g_{0}}\rangle A^{3})
\gamma= \begin{cases}\dfrac{1-\rho_n\alpha^{-1}}{1-\rho_n}\hskip 7mm\alpha\geq\rho_n/\sigma_m \hskip 2mm \dfrac{1-\sigma_m\vphantom{1^{\textstyle|}}}{1-\sigma_m\alpha}\hskip 8.2mm\alpha\leq\rho_n/\sigma_m \end{cases}.
( k_{j_{1}}+ k_{j_{2}}+ k_{j_{3}})^{2}= X ( j_{1}, j_{2}) ( Q_{j_{1}}- Q_{j_{2}}) + X ( j_{1}, j_{3}) ( Q_{j_{1}}- Q_{j_{3}}) + X ( j_{2}, j_{3}) ( Q_{j_{2}}- Q_{j_{3}}) ,
\begin{array}{r c l r c l}{{Z_{1}}}&{{=}}&{{\partial_{\mu}Z_{\nu}- \partial_{\nu}Z_{\mu},}}&{{Z_{2}}}&{{=}}&{{[ Z_{\mu}, Z_{\nu}] ,}}\\{{A_{1}}}&{{=}}&{{\partial_{\mu}A_{\nu}- \partial_{\nu}A_{\mu},}}&{{A_{2}}}&{{=}}&{{[ A_{\mu}, A_{\nu}] ,}}\\{{M}}&{{=}}&{{[ Z_{\mu}, A_{\nu}] - [ Z_{\nu}, A_{\mu}] .}}&{{}}&{{}}&{{}}\\ \end{array}
= 1 - \frac{1}{2}+ \frac{1}{2}- \frac{1}{3}+ \cdots + \frac{1}{2 0 0 5}- \frac{1}{2 0 0 6}= - \frac{1}{2 0 0 6}= - \frac{2 0 0 5}{2 0 0 6}
\sum_{k=0}^{m}\frac{(aq;q^2)_k (bq;q^2)_k (x;q^2)_k q^{2k}}{(q^2;q^2)_k (abq^4;q^4)_k}\equiv (-1)^{(n-1)/2}\sum_{k=0}^{m}\frac{(aq;q^2)_k (bq;q^2)_k (-x;q^2)_k q^{2k}}{(q^2;q^2)_k (abq^4;q^4)_k}\pmod{(1-aq^n)(1-bq^n)}.
- \int_{B_{R}\setminus \big(D \cup F^{-1}(D_{t}) \big)}\langle A \nabla u_0, \nabla u_0 \rangle + k^2 \int_{B_{R}}\Sigma |u_0|^2 + \int_{\partial B_R}\partial_r u_0 \bar u_0 + \int_{\big[\partial F^{-1}(D_t) \big]\setminus \Gamma}A \nabla u_0 \cdot \nu \; \bar u_0 = 0
G^{A B}( p ) = [ ( G_{0})^{- 1}+ i \Pi_{\mathrm{g h}}( p^{2}) ]_{A B}^{- 1}={\frac{1}{- i p^{2}+ i \Pi_{\mathrm{g h}}( p^{2})}}\delta^{A B}= ( - i ){\frac{G_{\mathrm{g h}}( - p^{2})}{- p^{2}}}\delta^{A B},
\bar{K}( s | \tau , \mathrm{\boldmath x}; \tau^{\prime}, \mathrm{\boldmath x}^{\prime}) = \frac{1}{( 4 \pi s )^{1 / 2}}\exp ({- \frac{( \tau -{\tau}^{\prime})^{2}}{4 s}}) \bar{K}^{( 3 )}( s | \mathrm{\boldmath x}; \mathrm{\boldmath x}^{\prime}) ,
\mathbb{P}[X^{h}(\omega^*) \in dy] =\mathbb{P}[S^{\downarrow}(\omega^{*}) \in d(y-r^h(\omega^{*}))] &=\mathbb{P}[S^{\downarrow}(\omega^{*}) \in d(y-x^{*})] &=\frac{J(\omega^{*},y-x^{*})}{j(\omega_1)}\partial_x f(\omega_1,0,\omega^{*},y-x^{*})dy
a_1^{67}=-v^{26}=-u_1^{25}=u_2^{35}=v^{37}=-a_4^{67}=:r_2, &-a_2^{67}=v^{36}=u_1^{35}=:r_1, &a_3^{67}=-v^{27}=-u_2^{25}=:r_3
Q=[\begin{matrix}1&0&0\\ 0&\frac{\sqrt{3}}{2}&\frac{1}{2}\\ 0&-\frac{1}{2}&\frac{\sqrt{3}}{2}\end{matrix}]=[\begin{matrix}1&0&0\\ 0&cos30^{\circ}&sin30^{\circ}\\ 0&-sin30^{\circ}&cos30^{\circ}\\ \end{matrix}]
\begin{cases}\|\xi_{\lambda^h_k}\|_{L^2(0,1)}\leq C, \|\xi_{\lambda^h_k}\|_{H^{-s}_{per}(0,1)}\leq C|k|^{-s}, \ \ 0<s<1 , \|\xi_{\lambda^h_k}\|_{H^{-s}_{per}(0,1)}\leq C|k|^{-1}, \ \s\geq 1, \|\eta_{\lambda^h_k}\|_{L^2(0,1)}\leq C|k|^{-1}.\end{cases}
e_{{\mathfrak{a}_{i+1}}}( s_i \cdot \mathfrak{c})= (t+t^{-1}) e_{{\mathfrak{a}_i}}( s_i \cdot \mathfrak{c})+ e_{{\mathfrak{a}_i}}( \mathfrak{c}) + (n'_{{\mathfrak{a}_{i+1}}}(s_i \cdot \mathfrak{c})|_{t=0})
S^{(N)}_{Loc.}=S_{Liou.}(\sigma)-\int d^2x\det(e)\Biggl[{g^2\over 2\pi}\phi_2\triangle\phi_2+{\pi\over 2}(N\lambda-{A\over \pi}\sigma)\triangle(N\lambda-{A\over \pi}\sigma)-g\phi_2\triangle(N\lambda-{A\over \pi}\sigma)\Biggr].
(\kappa_\delta+\sqrt{-1}\partial\bar\partial\bar\varphi)^n=\frac{1}{(\delta+|S|_0^2)^{1-\beta}}e^{h_{\kappa_\delta}-t\bar\varphi -(\mu-t)\Psi_{\delta}-(1-t)\Phi^\beta_\delta}\kappa_\delta^n,~~t\in (0,\mu_0).
\mathcal{E}_{n}^{\left( \alpha+2\right)}\left( 2;\lambda\right)=\frac{\left( 2\lambda^{-1}\right)^{2}}{2!}\binom{\alpha}{2}^{-1}\left\{\genfrac{[}{]}{0pt}{}{2}{2}\mathcal{E}_{n+2}^{\left( \alpha\right)}\left( \lambda\right) -\genfrac{[}{]}{0pt}{}{2}{1}\mathcal{E}_{n+1}^{\left( \alpha\right)}\left( \lambda\right) \right\}.
I_{1}= \bar{M}^{\bar{g}}M_{\bar{g}}\, , \; \, I_{2}= \frac 1 2 \bar{B}^{\bar{g}f}B_{\bar{g}f}\, , \; \, I_{3}= \bar{M}_{\bar{g}}\bar{B}^{\bar{g}f}B_{\bar{k}f}M^{\bar{k}}\, , \; \, I_{4}= \frac 1 2 \bar{B}^{\bar{k}h}\bar{B}^{\bar{g}f}B_{\bar{g}h}B_{\bar{k}f}\, .
S_b={-1\over 16\pi G}\int d^4 x \sqrt{-\tilde g}\left\{(\varphi_+^2-\varphi_-^2)\ \tilde{\cal R}-{6q\over q+1/2}\left[(\tilde\partial\varphi_+)^2-(\tilde\partial\varphi_-)^2\right]\right\}.
\|\varphi_s(a)\varphi_t(b)-\varphi_{st}(ab)\| =\|\varphi(\lambda_s(a))\varphi(\lambda_t(b))-\varphi(\lambda_{st}(ab))\| &=\|\varphi(\lambda_s(a))\varphi(\lambda_t(b))-\varphi(\lambda_{s}(a)\lambda_t(b))\|<\varepsilon.
\frac{\sqrt{2}}{2 \pi}T_{2}K |_{M^{1 0}}= T_{2}\frac{\kappa^{2}}{\lambda^{2}}\frac{1}{2 \pi}\hat{I}_{4}= \frac{1}{m^{\frac{1}{3}}}\frac{1}{1 6 \pi^{2}}\hat{I}_{4}
\frac{1}{N}\sum_{b_1=1}^{[a_1(N)]-1}\left| (a_1^{-1})''(b_1) \right|\ll \frac{1}{N}\int_\ast^{[a_1(N)]}\left| (a_1^{-1})''(t) \right|\;dt\ll \frac{\left|(a_1^{-1})'([a_1(N)])\right|}{N}
N \left(-\Delta^N_\Omega + \lambda V\right) = (2 \pi)^{-d}\left|B_1(0)\right| \lambda^{\frac{d}{2}}\int_\Omega |V|^{\frac{d}{2}}+ o \left(\lambda^{\frac{d}{2}}\right) \mathrm{\as\}\lambda \rightarrow \infty .
H=\frac{1}{2}L_{1}\frac{dQ_{1}}{dt}^{2}+\frac{1}{2}L_{2}\frac{dQ_{2}}{dt}^{2}+m\frac{dQ_{1}}{dt}\frac{dQ_{2}}{dt}+\frac{Q_{1}^{2}}{2C_{1}}+\frac{Q_{2}^{2}}{2C_{2}}
f^{s+1}= \Sigma(f^{s}) = \Sigma (\Delta^{\overline{-s}}f - \sum_{j=0}^{s-1}[e_{\overline{j-s}}]^{j}) = \Delta^{\overline{-s-1}}f - [e_{\overline{-s-1}}] - \sum_{j=1}^{s}[e_{\overline{j-1-s}}]^{j}= \Delta^{\overline{-(s+1)}}f - \sum_{j=0}^{s}[e_{\overline{j-(s+1)}}]^{j}.
\int_X f\cdot T_hg\> d\omega_X =\int_G\int_G f(xH) g(xyH) \> d(\omega_H*\delta_h*\omega_H)(y) \> d\omega_G(x) &=\int_G\int_G f(xyH) g(xH)\> d(\omega_H*\delta_{h^{-1}}*\omega_H)(y) \> d\omega_G(x) &=\int_X T_{\bar h}f\cdot g\> d\omega_X.
|T(0,x) - F_m| = \left| T(0,x)-\frac{1}{\sharp B_m}\sum_{z\in B_m}T(z,z+x)\right| \le \frac{1}{\sharp B_m}\sum_{z\in B_m}|T(0,x)-T(z,z+x)| &\le \frac{1}{\sharp B_m}\sum_{z\in B_m}(T(0,z)+T(x,x+z))\ .
S_{n}\left( x\right)_{s,t}:=1+\sum_{k=1}^{n}x_{s,t}^{k}x_{s,t}^{k}:=\idotsint\nolimits_{s<u_{1}<\cdots <u_{k}<t}dx_{u_{1}}\otimes\cdots \otimes dx_{u_{k}}\forall 0\leq s\leq t\leq T
\frac{\partial x}{\partial q^{i}}\frac{\partial q^{i}}{\partial s}=\sum_{k}(\sum_{i=1}^{3}h_{ki}\frac{\partial q^{i}}{\partial s})e_{k};\frac{\partial x}{\partial q^{j}}\frac{\partial q^{j}}{\partial t}=\sum_{m}(\sum_{j=1}^{3}h_{mj}\frac{\partial q^{j}}{\partial t})e_{m}
[ \tilde{C}^{2}] ~ ~ W ={\frac{1}{2}}\sin{3 \tilde{\theta}}+ \sin{\tilde{\theta}}({\frac{1}{2}}- 4 \tilde{k}^{2}\tilde{\gamma}^{2}) - 2 \tilde{k}\tilde{\gamma}\cos{\tilde{\theta}}- 4 8 \tilde{k}^{3}t \cos{\tilde{\theta}}~ ~ ,
\lim\limits_{\epsilon\to0}\int_{B_{\delta_0}(0^n)}\frac{|((f_{\epsilon})^{\tau}-f^{\tau})(x+y)+((f_{\epsilon})^{\tau}-f^{\tau})(x-y)-2((f_{\epsilon})^{\tau}-f^{\tau})(x)|}{|y|^{n+2\alpha}}dy=0.
\mathrm{T}y := \underbrace{k_m}_{>0}\langle y , \phi_m \rangle + \sum_{i=m+1}^{M-1}\underbrace{k_i}_{\ge 0}\langle y , \phi_i \rangle+ \underbrace{k_M}_{>0}\langle y , \phi_M \rangle, 1\le m <M, m,M \in \mathbb{N}.
B_{\mathbf{m}}^{(d)}(\mathbf{z}) = \sum_{\mathbf{n}\subset \mathbf{m}}B_{|\mathbf{m}|-|\mathbf{n}|}\binom{\mathbf{m}}{\mathbf{n}}^{(d)}\Phi_{\mathbf{n}}^{(d)}(\mathbf{z}).
\rho'_x([u,v]) =\frac{d}{dt}\big|_{t=0}[\rho_{\mbox{exp}(tx)}(u),\rho_{\mbox{exp}(tx)}(v)] &=\frac{d}{dt}\big|_{t=0}\sum_{i,j}a_i(t)b_j(t)[e_i,e_j] &=\sum_{i,j}(\dot{a}_i(0)b_j(0)[e_i,e_j]+a_i(0)\dot{b}_j(0)[e_i,e_j]) &=[\rho'_x(u),v]+[u,\rho'_x(v)],
\Phi ( \vec{k}, t ) = \int d^{3}k{\hat{\Phi}}( \vec{k}, t ) e^{i{\vec{k}}\cdot{\vec{x}}}, \quad \mathrm{w i t h}\quad{\overline{{{\hat{\Phi}}}}}( \vec{k}, t ) ={\hat{\Phi}}( - \vec{k}, t ) , \nonumber
\rho_t(x)\rightarrow i \begin{cases}1 \rho_t(x)=2i=-1, &1+\delta \rho_t(x)=1i=-1, &\gamma \rho_t(x)=1i=2, &\lambda\sum_{y:y\sim x}1_{\{\rho_t(y)=2\}}\rho_t(x)=0 i=1, &0 ,\end{cases}
a^2 \int_0^t\int_\mathbb{R}w_{pxx}w_p g^2 dxd\tau= - a^2 \int_0^t\int_\mathbb{R}\big( w_{px}^2 g^2 + 2 g g_x w_p w_{px}\big)dxd\tau \leqslant \frac{1}{2}\int_0^t \int_\mathbb{R}g_x^2 w_p^2 dxd\tau+O(1) \int_0^t \|w_{px}\|^2d\tau.
S_{0}\left\vert \begin{array}{c}{N}\\{( p , q )}\\ \end{array}, k \right> = \left( k + \displaystyle \frac{1}{2}\left( \frac{2 p - 1}{2 m}- \frac{2 q - 1}{2 n}- N \right) \right) \left\vert \begin{array}{c}{N}\\{( p , q )}\\ \end{array}, k \right> ,
k_{1}e\pi^{T}\begin{bmatrix}1 0 0\end{bmatrix}+ k_{2}e\pi^{T}\begin{bmatrix}0 1 0\end{bmatrix}+ k_{3}e\pi^{T}\begin{bmatrix}0 0 1\end{bmatrix}= (k_{1}\pi_{1}+k_{2}\pi_{2}+k_{3}\pi_{3})e
V_{1}= i \left( \begin{array}{c c}{\operatorname{c o s}\frac{\vartheta}{2}}{e^{- i \varphi}\operatorname{s i n}\frac{\vartheta}{2}}{e^{i \varphi}\operatorname{s i n}\frac{\vartheta}{2}}{- \operatorname{c o s}\frac{\vartheta}{2}}\end{array}\right) ,
\int_{\Gamma_k \backslash V_{m,k}}| \Omega^n W| d\mu \leq \int_{V_{m,k}}| \Omega^n w| d\mu \leq \sup| \Omega^n w| \int_{X_1\leq x\leq 2X_1}\int_{X_2\leq y\leq 2X_2}d\mu \ll \sup| \Omega^n w| \frac{X}{d_1d_2\sqrt{m}}.
\epsilon <d(g_{N+1}x_j,g_{N+1}x_k)=d(\tilde{r}_{N+1}(\omega_j),\tilde{r}_{N+1}(\omega_k))\leq 8\delta+d(r(\omega_j),r(\omega_k)) & \leq 16\delta+d(\tilde{r}_N(\omega_j),\tilde{r}_N(\omega_k))=\epsilon+d(g_Nx_i,g_Nx_i)=\epsilon,
\dim\left(\bigcap_{i=1}^\ell \Im(W_{[k], A_{i}})\right) =\max_{P_1 \sqcup P_2 \sqcup \dots \sqcup P_s=[\ell]}\left(\sum_{i=1}^s\Big|\bigcap_{j \in P_i}A_j\Big|-(s-1) k\right),
B\widehat{\otimes}^{D-D}(D\otimes_{\sigma h}^{(I,1)}D) =[CB(B,(D\otimes_{\sigma h}^{(I,1)}D)^\natural)]_\natural &\to [CB(B,(D\otimes_{\sigma h}^{(I,1)}D)^\natural)]^\natural &\to [CB(B,(D\otimes_{\sigma h}^{(I,1)}D)_\natural)]^\natural
\phi_{( a )}= \sqrt{{\frac{N \, V}{g_{0}^{2}}}\, \mathbf{F^{( 0 )}}{}_{c d \, \mu \nu}\mathbf{F}^{( 0 )}{}^{c d \, \mu \nu}}\equiv \sqrt{\,{\frac{N \, V}{g_{0}^{2}}}\, \mathrm{T r}\, \mathbf{F^{( 0 )}}{}_{\mu \nu}\mathbf{F}^{( 0 )}{}^{\mu \nu}}
\lambda_{2,K^4}-\lambda_{1,K^4}= \frac{(n-1)^2(n-3)^2}{24^2}\left(\frac{3(750\pi^2-8\pi^4-6615)}{640}\right) + \frac{(n-1)(n-3)}{24}\left(\frac{51(2\pi^4-50\pi^2+315)}{640}\right).
c_0=k\left( \sum_{i\in\overline{\mathcal{I}^-}}c_i\nabla\psi_{i,T}(F)\cdot\mathbf{v}(F)+\sum_{i\in\mathcal{I}^+}v_i\nabla\psi_{i,T}(F)\cdot\mathbf{v}(F) \right),
\sum_{p=0}^{q-1}(-1)^{q-p-1}\Big(_\mathbb{R}T^p_Z-_\mathbb{R}T^{p-1}_Z \Big) =\sum_{p=0}^{q-1}(-1)^{q-p-1}\Big(_\mathbb{R}G^{p+1}_Z-_\mathbb{R}G^{p}_Z \Big) &=\sum_{p=0}^{q}(-1)^{q-p}\Big(_\mathbb{R}G^p_Z-_\mathbb{R}G^{p-1}_Z \Big).
{{\mathsf{R}}^{\lambda}_{\mathcal{N}_{[1:N]}}}= \min_{\mathcal{A}_{\rm{F}}\subseteq{[1:N]}}\sum_{s\in [0:1]^N}\lambda_s \left( \max_{i \in\mathcal{A}_{\rm{F}}}\ell_{i,s}^{\prime}+ \max_{i \in{[1:N]}\backslash \mathcal{A}_{\rm{F}}}r_{i,s}^{\prime}\right),
\beta = \beta^{\prime}/ ( g_{Y M}^{2}N )^{1 / 3}\quad R = R^{\prime}( g_{Y M}^{2}N )^{1 / 3}\quad \Delta_{l}^{2}= ( g_{Y M}^{2}N )^{1 / 3}\Delta_{l}^{\prime}{}^{2}/ N \, .
\rho(2,(11);p) = \frac{1}{2}\rho(2,(2);p) = \frac{p^2-p+1}{2(p^2+p+1)}\rho(3,(111);p) = \frac{p^4+2p^2+1}{6(p^4+p^3+p^2+p+1)}\rho(3,(12);p) = \frac{p^4+1}{2(p^4+p^3+p^2+p+1)}.
W(0) = [1 - B_K] \left[ 1 + \int_{0^-}^K \lambda\alpha e^{\lambda \sigma}e^{\lambda (\alpha e^{\lambda \sigma}-1 )x}dx \right]^{-1}= [1 -B_K] \left[ \dfrac{\alpha_{\lambda}e^{\lambda \sigma}- 1}{\alpha_{\lambda}e^{\lambda(K \alpha_{\lambda}e^{\lambda \sigma}- K + \sigma)}-1}\right],
I_n(\mu)=-\left(\frac{\mu}{2}\right)^{4-D}\frac{M}{(4 \pi)^{3/2}}\frac{\Gamma(\frac{2n + D - 1}{2})}{\Gamma(\frac{3}{2})}(-M E)^{(2n + D - 3)/2}\Gamma\left(\frac{3 - D - 2n}{2}\right),
\langle{\cal A}_{a}\rangle_{0 n}={}^{\parallel}\nabla_{a}\omega_{\bar{I}J}\langle z^{\bar{I}}z^{J}\rangle_{0 n}- k_{\bar{I}b a}\langle \lambda^{\bar{I}}\lambda^{b}\rangle_{0 n}- a_{I \bar{J}a}\langle \lambda^{\bar{J}}\lambda^{I}\rangle_{0 n}+ a^{\prime}{}_{I^{\prime}\bar{J}^{\prime}a}\langle \nu^{I^{\prime}}\nu^{\bar{J}^{\prime}}\rangle_{0 n}
\overline{K^{\alpha \gamma}|_{\beta \delta}(x,y)}= K^{\beta \delta}|_{\alpha \gamma}(y,x) \overline{K^\alpha_{\;\;\,\gamma}|_\beta^{\;\;\,\delta}(x,y)}= K^\beta_{\;\;\,\delta}|_\alpha^{\;\;\,\gamma}(y,x) \:.
\begin{cases}s \hat{v}(x,s)+ a\hat{v}_{xxx}(x,s)=0, \,\, (0,L)\times (0,T), \hat{v}(0,s) =\hat{h}_0(s),\,\,\hat{v}(L,s) = \hat{h}_1(s),\,\,\hat{v}_{x}(L,s) = \hat{h}_2(s), \,\, (0,T), \hat{v}(x,0)=0, \,\, (0,L).\end{cases}
Q^{\prime}\left( \begin{array}{c}{\phi}\\{a}\\{\psi^{\prime}}\\{\bar{\phi}}\\{\bar{\psi}}\\ \end{array}\right) = \left( \begin{array}{c}{0}\\{\psi^{\prime}}\\{- L ( \phi ) a}\\{\bar{\psi}}\\{- L ( \phi ) \bar{\phi}}\\ \end{array}\right)_{( \phi , a , \psi^{\prime}, \bar{\phi}, \bar{\psi})}.
\widetilde{\boldsymbol{\eta}}_{u,h}^n:=\widetilde{\boldsymbol{w}}_t^n - j_u\widetilde{\boldsymbol{w}}_t^n, \boldsymbol{\eta}_{u,h}^n:=\boldsymbol{w}_t^n - j_u \boldsymbol{w}_t^n, \eta_{p,h}^n:=r_t^n - j_p r_t^n
I_{\lambda}(u,v)=\frac{1}{2}\left(\|(u,v)\|_{E}^{2}-2\int_{\mathbb{R}^{N}}\lambda (x)uv\,\mathrm{d}x\right)-\int_{\mathbb{R}^{N}}\left(F_{1}(u)+F_{2}(v)\right)\,\mathrm{d}x.
Q(v) :=f\,[(4\delta^{2}+\delta \,\acute{}\,^{2})v^{4}+4\delta^{2}\, \delta \,\acute{}\, \lambda \,v^{3}+2\delta^{2\,}[2\delta^{2}(\lambda^{2}+1)-\delta \,\acute{}\,^{2}]v^{2}& -4\delta^{4}\, \delta \,\acute{}\, \lambda \,v+\delta^{4}\delta \,\acute{}\,^{2}]+4\delta^{2}\,v[k\,v^{3}+\delta^{2}(k-\lambda)v+\delta^{2\,}\delta \,\acute{}\,]=0.
\frac{1}{2 \pi i}\int_{C_2}\zeta^{\ell}(s_2+1) \mathcal{D}_{k,\ell}^{(i_2,0)}(0,s_2) (t+h)^{s_2}ds_2 = \sum_{\substack{j_1+j_2 +j_3 = \ell-1 j_1,j_2,j_3 \ge 0}}\frac{\alpha_{j_1,\ell}\mathcal{D}_{k,\ell}^{(i_2,j_2)}(0,0) (\log (t+h))^{j_3}}{j_{2}! j_{3}!}
S \sim \tilde{\psi}Q_{o}\tilde{\psi}+ g_{s}^{1 / 2}\tilde{\psi}^{3}+ \tilde{\phi}Q_{c}\tilde{\phi}+ g_{s}\tilde{\phi}^{3}+ \tilde{\phi}B ( g_{s}^{1 / 2}\tilde{\psi}) + \cdots .
|z^{-n}-w^{-n}|_p =|{w^n-z^n}|_p|{z^{-n}w^{-n}}|_p &\leq|w-z|_p|w|_p^{n-1}|z^{-n}w^{-n}|_p &=|z-w|_p|z|_p^{-n-1}.
Q(x)=\int_{x}^{\infty}\phi(u)du<\int_{x}^{\infty}\frac{u}{x}\phi(u)du=\int_{\frac{x^{2}}{2}}^{\infty}\frac{e^{-v}}{x\sqrt{2\pi}}dv=-\frac{e^{-v}}{x\sqrt{2\pi}}|_{\frac{x^{2}}{2}}^{\infty}=\frac{\phi(x)}{x}.
\tilde\psi(\bar z)=\psi_{\max(y_j)}\left(\begin{pmatrix}x_0 y_0\end{pmatrix},\ldots,\begin{pmatrix}x_{q-1}y_{q-1}\end{pmatrix}\right)+\sum_{t=1}^{\max(y_j)-1}d_{q-2}\chi_t\left(\begin{pmatrix}x_0 y_0\end{pmatrix},\ldots,\begin{pmatrix}x_{q-1}y_{q-1}\end{pmatrix}\right)
\tilde{\Phi}^{J}( x , t , \theta ) = \Phi^{J}\Bigl \vert_{\theta^{+}= \theta^{-}= \theta}\qquad \mathrm{a n d}\qquad \tilde{\Sigma}_{J}( x , t , \theta ) = \Sigma_{J}\Bigl \vert_{\theta^{+}= \theta^{-}= \theta}\ .
\quad\langle x-y,-(-A_n)^{\frac12}F_n((-A_n)^{\frac12}x)+(-A_n)^{\frac12}F_n((-A_n)^{\frac12}y)\rangle = -\langle (-A_n)^{\frac12}x-(-A_n)^{\frac12}y,F_n((-A_n)^{\frac12}x)-F_n((-A_n)^{\frac12}y)\rangle \le \|(-A_n)^{\frac12}(x-y)\|^2\le (n-1)^2\|x-y\|^2,
\left( \begin{array}{c c}{D^{2}e^{2 V}}&{\sqrt{2}\bar{\Phi}}\\{\sqrt{2}\Phi}&{\bar{D}^{2}e^{- 2 V}}\\ \end{array}\right) \left( \begin{array}{c}{Q}\\{\widetilde{Q}^{\dagger}}\\ \end{array}\right) = 0 .
\tilde{\delta}_{n}( p) = \frac{1}{m_p(\tilde{\alpha}_{n}(p))}\frac{1}{n}\sum_{i=2}^n n^{p/ \tilde{\alpha}_{n}(p)}\frac{\left| \Delta_i^n X - \Delta_{i-1}^n X\right|^{p}}{X_{\frac{i-2}{n}}^{p/ \tilde{\alpha}_{n}(p)}}.
\begin{cases}\rho_{n-j-1}(u_1,\ldots,u_{2n-1})=0, \rho_{n-j-1}(u_2,\ldots,u_{2n})=0, \rho_{n-j}(u_1,\ldots,u_{2n})=0, \rho_{n-j-1}(u_2,\ldots,u_{2n-1})=0, \end{cases}1\leq j\leq w,
Z^*_\psi(m,n;3/4)=\lim_{X\to \infty}\sum_{[Q]\in \mathcal Q_{N;\hat h,\hat v}(mn) / \Gamma_0(N;\hat h,m\hat v)}\frac{\chi_m(\varphi_{\hat h, m \hat v}Q)}{\omega_Q}&\sum_{\substack{r,Ns\in Z, (r,Ns/h)=1 0<c(Q,r,Ns)<X}}\frac{1}{4Nc(Q,r,Ns)}\exp\left(2\pi i \frac{\beta}{2Nc(Q,r,Ns)}\right).