harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{8}{P}+\frac{1}{2}(x^2+l^2-n^2-1)=x^2-1+\begin{array}{ccccccccc}1^2-n^2 1^2-l^2 3^2-n^2 3^2-l^2 \cline{1-1}\cline{3-3}\cline{5-5}\cline{7-7}\cline{9-9}1 + x^2-1 + 1 + x^2-1 +\cdots\end{array},
m^1_{x=V}=\int_{(\max\{0,\,(\tau_1)^{-1}(T)\},\,L_1]}\delta_{\tau_1(x)}\,dm_0^1(x) +\int_{[0,\,\max\{0,\,(\varsigma_1)^{-1}(T)\}]}\delta_{\varsigma_1(s)}\,d\sigma_0(s),
\vert\vert c_n \vert\vert^2 =\frac{1}{4}\bigg\vert\bigg\vert\int_{0}^{2\pi}e^{in\theta}\Big[\overline{\langle e_\ell, E(e^{i\theta})-E(e^{i\theta_n}) \rangle}E(e^{i\theta})+\overline{\langle e_\ell, E(e^{i\theta_n})\rangle}\big(E(e^{i\theta})-E(e^{i\theta_n})\big)\Big]\frac{d\theta}{2\pi}\bigg\vert\bigg\vert^2
\tilde L_F(\psi_R,\psi_L;A_+,A_-;S)=\bar\psi_L\bar\partial\psi_L + \bar\psi_R\partial\psi_R +A_-^aJ_L^a+A_+^{\bar a}J_R^{\bar a}+(S^{-1})_{a\bar a}A_-^aA_+^{\bar a}.
H^{\prime}( t ) = - i \hbar \sum_{m \neq n}\left[ \langle m ; t | e^{- i \alpha_{m}( t )}\right] \frac{d}{d t}\left[ e^{i \alpha_{n}( t )}| n ; t \rangle \right] \: | m ; 0 \rangle \langle n ; 0 | .
\Lambda^{\alpha+p}_{q}(x,y) =e(-ix;q^{2})j_{\alpha+p}(y;q^{2}) =\int_{-1}^{1}\int_{0}^{1}W_{p-1}\left(t_2 ; q^{2}\right)\Lambda^{\alpha}_{q}(t_1x,yt_2) \delta_{1}(t_{1}) d\mu_{\alpha,q}(t_1,t_2)
\sum_{\nu=2}^{n-1}t_{jk}^{-\nu+1}\sum_{\lambda=1}^\infty F^{kj}_{\nu\lambda}(z_j)\cdot \left(w_j+\sum_{\lambda=2}^{n-1}F^j_\lambda(z_j)\cdot w_j^\lambda\right)^\lambda\hskip-2mm\cdot w_j^\nu=\sum_{\nu=2}^{n}Q^{kj}_\nu(z_j)\cdot w_j^\nu+O(w_j^{n+1}).
J = \left(\begin{array}{rrrrr}1 -1 1 1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 1 -1 -1 1 1 -1 1 \end{array}\right), x = \left( \begin{array}{ccccc}x_1 x_2 x_3 x_4 x_5 \end{array}\right)^T.
s_\lambda = \sum_{\mu \vdash n}\frac{\chi^\lambda(\mu)}{z_\mu}p_\mu, h_\lambda = \sum_{\mu \vdash n}\frac{\eta^\lambda(\mu)}{z_\mu}p_\mu, e_\lambda = \sum_{\mu \vdash n}\frac{\epsilon^\lambda(\mu)}{z_\mu}p_\mu, m_\lambda = \sum_{\mu \vdash n}\frac{\phi^\lambda(\mu)}{z_\mu}p_\mu.
F [ L , ~ f ] ~ = ~ \sum_{j_{i}= 0 , 1 / 2 , 1 , . . . . k / 2}~ \mu_{j_{1}}~ \mu_{j_{2}}~ . . . . ~ \mu_{j_{r}}~ V [ L ; ~ n_{1}, ~ n_{2}, . . . . ~ n_{r}; ~ j_{1}, ~ j_{2}, . . . . ~ j_{r}]
I_1 = \Big(-\infty,\ \omega_{\xi}^{\ast}- \frac{\alpha \xi^{\alpha}}{2(1 - \alpha)}\Big], I_2 = \Big[\omega_{\xi}^{\ast}-\frac{\alpha \xi^{\alpha}}{2(1-\alpha)},\ \omega_{\xi}^{\ast}+ \frac{\alpha \xi^{\alpha}}{2(1-\alpha)}\Big], I_3 = \Big[\omega_{\xi}^{\ast}+ \frac{\alpha \xi^{\alpha}}{2(1 - \alpha)},\ 0\Big], \mbox{and}I_4 = [0,\ \infty).
0\leq\|f'\|_{L^2}^2 = -\int_{\mathbb R}q(t)|f(t)|^2\,\mathrm dt = -\int_{\mathbb R}\big(q_+(t)-q_-(t)\big)|f(t)|^2\,\mathrm dt \leq \int_{\mathbb R}q_-(t)|f(t)|^2\,\mathrm dt\leq \|q_-\|_{L^1}\|f\|_\infty^2.
\| \partial_{t}^{\ell}\nabla_{x}^{k}u(t) \|_{2}\ge |m_{1}|\| \partial_{t}^{\ell}\nabla_{x}^{k}\mathcal{F}^{-1}[\mathcal{G}_{\sigma, \nu}(t)] \|_{2}-\| \partial_{t}^{\ell}\nabla_{x}^{k}(u(t) - m_{1}\mathcal{F}^{-1}[\mathcal{G}_{\sigma, \nu}(t)] ) \|_{2}& = C t^{-\gamma_{\sigma,k}-\ell}-o(t^{-\gamma_{\sigma,k}-\ell})
I: =(|a - \bar a|^{p_1}\, |D_v u|^{p_1})^{1/p_1}_{Q_{2\nu r, 2^{k+1}(2\nu r)}}&\le (|a - \bar a|^{p_1 \alpha_1})^{1/(p_1 \alpha_1)}_{Q_{2 \nu r, 2^{k+ 1}(2\nu r)}}\, (|D_v u|^{p_1 \alpha})^{1/(p_1 \alpha)}_{Q_{2 \nu r, 2^{k+ 1}(2\nu r)}}=: I_1^{1/(p_1 \alpha_1)}I_2^{1/(p_1 \alpha)}.
\int_0^T \tilde{\epsilon}_2(t)\,dt = M\int_0^T \int_0^t \frac{M}{f(F^{-1}(Mt))}\int_0^t M(t-u,t) \frac{f^2(F^{-1}(Mu))}{F^{-1}(Mu)}\,du\,dt &=M\int_1^{F^{-1}(MT)}K(v)\frac{1}{f^2(v)}\,dv.
P_{x,r}^U(A_k^c | U_{k-1}^- = z)= P_{x,r}^U\Big( |U_k^- - U_{k-1}^-| > \Big(\frac{\epsilon^{2/3}x^{2/3}}{z}\Big) \cdot z \Big|U_{k-1}^- = z \Big)\leq c_1 \exp\Big( -c_2 \epsilon^{2/3}x^{2/3}\Big).
\mathbb{E}(X(f_1)X(f_2)) = \frac{1}{8\pi}\int_{-\infty}^{\infty}\,dk\,|k|\hat{f_1}(k)\overline{\hat{f_2}(k)}|1-i\,\mathrm{sgn}(k)|^{2}\,dk+\mathrm{c.c.}&=\frac{1}{2\pi}\int_{-\infty}^{\infty}\,dk\,|k|\hat{f_1}(k)\overline{\hat{f_2}(k)}
\mathbb{E}[x^{V'_b};C_3]=\frac{1}{6}\Big{(}\sum_{k=0}^{\infty}\mathbb{P}(V_c=k)\sum_{j=0}^k \Big{(}\frac{1}{3}\Big{)}^j\Big{(}\frac{2}{3}\Big{)}^{k-j}\binom{k}{j}x^j\Big{)}^3=\frac{1}{6}f\Big{(}\frac{x+2}{3}\Big{)}^3
\overline{\Gamma}^\alpha_{00}=0, \ \overline{\Gamma}^0_{0i}=0, \ \overline{\Gamma}^k_{ij}=\widetilde{\Gamma}^k_{ij}, &\overline{\Gamma}^0_{ij}=f(t)f'(t)\widetilde{g}_{ij}, \ \overline{\Gamma}^k_{0i}=\frac{f(t)f'(t)}{f(t)^2}\delta^k_i.
\phi_{p}\left(x\right)=\prod_{j=1}^{d}e^{2\pi ip_{j}x_{j}},\quad\mu_{p}=\sum_{j=1}^{d}\left(2\pi p_{j}\right)^{2},p_{j}\in\mathbb{N},1\le j\le d,\; i=\sqrt{-1}.
ds^2 = d\chi^2 + \sinh^2\chi \, d\varphi^2 - \cosh^2\chi \, d\tau^2 =\left({2 \over 1- \rho^2}\right)^2 (d\rho^2 + \rho^2 d\varphi^2) -\left({1 + \rho^2 \over 1 - \rho^2}\right)^2 d\tau^2.
\displaystyle{\frac{m_{\pi}^{2}v j_{\pi}( q )}{m_{\pi}^{2}- q^{2}}( = \partial^{\mu}J_{\mu}( q ) ) = \frac{q^{2}v j_{\pi}( q )}{m_{\pi}^{2}- q^{2}}+ \partial^{\mu}\widehat J_{\mu}( q ) .}
{{\mathrm{\boldmath ~ \sigma ~}}\cdot{\bf p}}{\frac{f ( r )}{r}}{\phi^{l}}_{j m}= -{\frac{i}{r}}({\frac{d f}{d r}}+{\frac{\kappa f}{r}}){{\mathrm{\boldmath ~ \sigma ~}}\cdot{\bf n}}{\phi^{l}}_{j m},
\frac{4}{\pi}\int_0^1 \sqrt{1-x^2}\ln (x) \sum_{n = 0}^{\infty}\left(-\frac{x^2}{2}\right)^n \binom{-\frac{1}{2}}{n}(n+1) \, dx + \frac{1}{2}\sum_{n = 0}^{\infty}\left(\frac{1}{32}\right)^n \binom{2 n}{n}^2 \frac{1+2 (n + 1) \ln (2)}{n+1}.
\begin{array}{l}{L G_{2 ( 1 , Z )}= Z_{R}^{1 / 3}Z_{L}^{2 / 3}L G_{2}Z_{L}^{- 2 / 3}Z_{R}^{- 1 / 3},}\\{G_{2 ( 1 , Z )}= Z_{R}^{1 / 3}Z_{L}^{2 / 3}G_{2}Z_{L}^{- 2 / 3}Z_{R}^{- 1 / 3}.}\\ \end{array}
f_{11}f_{22}-f_{12}{}^2-\frac{a}{E}f_1f_{22}+bf_2f_{22}+\frac{2b}{E}f_1f_{12}+\frac{b^2}{E}f_2{}^2 = \frac{b^2c}{E}, & \frac{1}{E}f_1{}^2 + f_2{}^2 \leq c
ds^2=2dx^{+}dx^{-}+\left(m^{2}_{x}(x^{+}) x_{a}^{2}+m_{y}^{2}(x^{+}) y_{l}^2+m_{z}^2(x^{+})z^2\right)(dx^{+})^2+ dx_{a}^{2}+ dy_{l}^{2}+ dz^2.
\left\{\frac{2^{z_{k}+\sum\nolimits_{i=0}^{k+1}2b_{i}3^{i}}-\sum\nolimits_{j=0}^{k}3^{k-j}2^{j+2+\sum\nolimits_{i=0}^{j}2b_{i}3^{i}}-3^{k+1}}{3^{k+2}}\right\},
h(i,\boldsymbol{x}){= C_{0}+C_1x_{j_1}^{\tau-\zeta_i-d_{i,\bar{U}}-1}+\cdots +C_{l-1}x_{j_1}\cdots x_{j_{l-2}}x_{j_{l-1}}^{\tau-\zeta_i-l-d_{i,\bar{U}}+1}}&{+C_{l}x_{j_1}x_{j_2}\cdots x_{j_{l-1}}x_{j_l}^{\tau-\zeta_i-l-d_{i,\bar{U}}}}&{= : C_0h_0(i,\boldsymbol{x}) +C_1 h_1(i,\boldsymbol{x}) +\dots+C_{l}h_{l}(i,\boldsymbol{x})},
x_1=F_1(u)=u_1^6 - 10 u_1^3 u_2^3 - 10 u_1^3 u_3^3 + u_2^6 - 10 u_2^3 u_3^3 + u_3^6, x_2=F_2(u)=\big(u_1^3+u_2^3+u_3^3\big) \big(\big(u_1^3+u_2^3+u_3^3\big)^3+216 (u_1 u_2 u_3)^3\big), x_3=F_3(u)=\big(\big(u_2^3-u_3^3\big) \big(u_3^3-u_1^3\big) \big(u_1^3-u_2^3\big)\big)^2.
p(a_k, \theta_k)=\|f-a_k\cos \theta_k\|_{L^2}^2 = \|f\|_{L^2}^2+\|a_k\cos \theta_k\|_{L^2}^2-2\langle a_k\cos \theta_k+\sum_{k'\neq k}a_{k'}\cos\theta_{k'}+r(t), a_k\cos \theta_k\rangle = \|f\|_{L^2}^2-\|a_k\cos \theta_k\|_{L^2}^2-2\langle \sum_{k'\neq k}a_{k'}\cos\theta_{k'}+r(t), a_k\cos \theta_k\rangle.
0 =\nabla f(x_k)+\sum\limits_{i\in I^g(x_k)}\lambda_i^k\nabla g_i(x_k)+\sum\limits_{j\in\mathcal P}\rho_j^k\nabla h_j(x_k) &+\sum\limits_{l\in I^{\varphi^{t_k}_\textup{FB}}(x_k)}\xi^k_l\left(\alpha^k_l\nabla G_l(x_k)+\beta^k_l\nabla H_l(x_k)\right)
||\sqrt{\omega}\lambda(x)\left(K\left(K(y(x,\frac{\cdot}{\lambda(x)^{2}}))\right)(\omega \lambda(x)^{2})\right)||_{L^{2}(\rho(\omega \lambda(x)^{2}) d\omega)}&\leq C\left(||y(x)||_{L^{2}(\rho(\omega \lambda(x)^{2}) d\omega)}+||\sqrt{\omega}\lambda(x) y(x) ||_{L^{2}(\rho(\omega \lambda(x)^{2}) d\omega)}\right)
\int_{0}^{\infty}x^{\mu}J_{\nu}( a x ) d x = 2^{\mu}a^{- \mu - 1}\frac{\Gamma \left( \frac{1}{2}+ \frac{1}{2}\nu + \frac{1}{2}\mu \right)}{\Gamma \left( \frac{1}{2}+ \frac{1}{2}\nu - \frac{1}{2}\mu \right)}\quad - \Re \nu - 1 < \Re \mu < \frac{1}{2}, \quad a > 0
\mu\,\rho(x) = \lim_{n\to\infty}n\,x\,f(x|\tfrac{\mu}{n}){\rm and}\mu\, r_\alpha(x) = \lim_{n\to\infty}n\,x\,m_\alpha(x|\tfrac{\mu}{n}) = \phantom{\mu\,}x \int_0^\infty f_\alpha(x|y) \lim_{n\to\infty}n\,f(y|\tfrac{\mu}{n}) dy = \mu\, x \int_0^\infty f_\alpha(x|y)\,y^{-1}\rho(y) dy
\begin{aligned}C_6^{-1}\|\nabla^l (\varrho(t), u(t), \nabla n(t))\|_{H^{m-l}}^2\le \mathcal{E}^m_l(t)\le C_6 \|\nabla^l (\varrho(t), u(t), \nabla n(t))\|_{H^{m-l}}^2.\end{aligned}
w_{t}+\nabla\cdot\left(-a\left(x,t;w;\nabla w\right)\nabla w\right) =F\left(x,t;u;\nabla u\right)-F\left(x,t;v;\nabla v\right) & +\nabla\cdot\left(a\left(x,t;u;\nabla u\right)\nabla u\right)-\nabla\cdot\left(a\left(x,t;v;\nabla v\right)\nabla v\right) & -\nabla\cdot\left(a\left(x,t;w;\nabla w\right)\nabla w\right),
\arg\min_{\vec{X}}\Biggl\lbrace \dfrac{1}{2}\|\vec{Y}-\vec{X}\|_F^2 + \lambda_0\sum_{i=1}^{m}\phi(\sigma_i(\vec{\vec{Z}});a_0) &\qquad+ \lambda_1\sum_{i=1}^{m}\sum_{j=1}^{n}\phi(\vec{X}_{i,j};a_1) \Biggr\rbrace, \mbox{s.t.}\vec{X}= \vec{Z}.
x^{5/2}(1-x)^{N/2}\frac{d\phi}{dx}=x^{5/2}(1-x)^{N/2}\frac{d\phi}{dx}\Big|_{x=1/2}+ &-\int_{1/2}^x\Lambda\phi(x')x'^{3/2}(1-x')^{N/2-1}dx'
\sum_{i=1}^k [K_i: \mathbb{Q}] n_i^2 \log n_i + \sum_{i = 1}^k O([K_i: \mathbb{Q}]^2 n_i^2) &= \sum_{i=1}^k [K_i: \mathbb{Q}] n_i^2 \log n_i + O(g^2),
\mu(x) =\frac{1}{2\pi \i}(R(x-\i 0)-R(x +\i 0))=|\alpha_x)(\beta_x| |\alpha_x\rangle = (x^2+\pi^2)^{-1/2}\left(\binom{1}{-\frac{\mathcal{P}}{x-\Omega}|E\rangle}+ x\binom{0}{|\delta_x\rangle}\right) \langle\beta_x| = (x^2+\pi^2)^{-1/2}\bigg (-(1,\langle E|\frac{\mathcal{P}}{x-\Omega})+ x(0,\langle\delta_x|)\bigg)
(1+D^2)^{-1/2}[D^2,T] = |D_0|^{-1}[|D_0|^2,T] = 2[|D_0|,T]-|D_0|^{-1}[|D_0|,[|D_0|,T]] = 2\delta_0(T)-|D_0|^{-1}\delta_0^2(T).
g^{(n)}(0) = \sum\limits_{m=0}^n \binom{n}{m}B^{(f)}_{n-m}(q)\frac{d^{m+r}}{dt^{m+r}}\Big\vert_{t=0}\bigl[e^{-b_0 t}\prod\limits_{j=1}^{M-1}(1-e^{-b_j t})\bigr], & = (-1)^r \sum\limits_{p=0}^{M-1}(-1)^p\sum\limits_{k_1<\cdots<k_p=1}^{M-1}\bigl(b_0+\sum b_{k_j}\bigr)^r \,B^{(f)}_n\bigl(q+b_0+\sum b_{k_j}\bigr).
\hat\Phi \cdot [A] = [B] \Longleftrightarrow \big(\Phi_B\ \hat\Phi\ \Phi_A^{-1}\big)\cdot [\Phi_A \cdot A]= [\Phi_B \cdot B] & \Longleftrightarrow \hat\Phi' \cdot [A'] = [B'],
\Pi_{x, x'}(f,g) = \sup_{y \in [x a_n, x' a_n]}\P_y\left[ \frac{S_j}{a_n}\in [f(j/n), g(j/n)], 0 \leq j \leq n \right] \leq \P\left[ \frac{S_j}{a_n}\in [f(j/n)-x', g(j/n)-x], 0 \leq j \leq n \right].
[T(z),E(w)]=-qa\delta(q^{n+1}s_1w/z):W'(q^{n+1}s_1w)\frac{\Lambda'(w)}{\Lambda'(q^2w)}E(q^2w):\,, &[T(z),F(w)]=s_3a\delta(q^{-n-1}w/z):W'(q^{-n-1}w)\frac{\Lambda'(w)}{\Lambda'(q^{-2}w)}F(q^{-2}w):\,,
e(p,q)= \begin{cases}\sum\limits_{k=1}^p(-1)^{k-1}e(p-k,q){[a_2-cq]_++k-1\brack k}_{v^b}ca_1q\le ba_2p; \sum\limits_{\ell=1}^q(-1)^{\ell-1}e(p,q-\ell){[a_1-bp]_++\ell-1\brack \ell}_{v^c}ca_1q\ge ba_2p. \end{cases}
\begin{aligned}\int_\mathbb C\log(|z-w|)d\mathrm{arg}(z)d\mathrm{arg}(z-1)= \int_{|z|< 1}\left(\log(|z-\overline w|)-\log(|1-z\overline w|)\right)d\mathrm{arg}(z)d\mathrm{arg}(z-1)+ &+\int_{|z|< 1}\log(|z|))d\mathrm{arg}(z)d\mathrm{arg}(z-1).\end{aligned}
E_{C}( R , T = 0 , m ) = = - \frac{R}{\pi}\int_{0}^{\infty}\frac{\omega d \omega}{g ( \omega ) - 1}\left( 1 - \frac{2 m M_{0}^{2}}{R ( \omega^{2}m^{2}- M_{0}^{4})}\right){,}
h \cdot X_{\alpha}\cdot f - X_{\alpha}\cdot h \cdot f = h(X_{\alpha}\cdot 1) \sigma_{\alpha}(f) - (X_{\alpha}\cdot 1) \sigma_{\alpha}(h)\sigma_{\alpha}(f) = (h - \sigma_{\alpha}(h))(X_{\alpha}\cdot 1) \sigma_{\alpha}(f) = \alpha(h)(X_{\alpha}\cdot 1) \sigma_{\alpha}(f) = \alpha(h)X_{\alpha}\cdot f = [h,X_{\alpha}] \cdot f.
d \Theta^{1 \underline{\mu}}= v_{( \underline{\alpha})}^{\underline{\mu}}\pi^{1 ( \underline{\alpha})}= v_{q}^{- \underline{\mu}}\pi_{q}^{1 +}+ v_{\dot{q}}^{+ \underline{\mu}}\pi_{\dot{q}}^{1 -},
z\Phi[n] =\|v^*\mathbf n\|^{2\dagger}\langle v^*\mathbf n,M_0\Phi\rangle\mathbf n [n]-\langle v^*\mathbf n ,\Phi\rangle\mathbf 1[n] &+\sum_{m=n}^\infty (m-n)(v\Phi)[m].
y^h_\lambda(t) - v^h_\lambda(t) = \int_0^t S(t-s) (-f'_\lambda(u_\lambda))v^h_\lambda\,ds + \int_0^t S(t-s) \sigma'(u_\lambda) (y^h_\lambda(t) - v^h_\lambda(t)) B\,dW(s),
e(N_{\{1\}\times C^n}Z_{\mu,\lambda}^w) = e(T_{(F_e,F_w)}\Omega_w)e\left(N_{C^n}\!\left( \widetilde{Q}_\mu\cap w.\widetilde{Q}_\lambda \right)\right) = \prod_{(i,j)\in N_\mu\cup gN_\lambda}(x_j-x_i)\prod_{(i,j)\in I_\mu \cap gI_\lambda}(x_i-x_j+\Delta_{ij}).
E_\infty^{*,*}(\tilde{B}A_{n-1})^{S_n}=(H^{*}({B}A_{n-1}) \otimes H^{*}({\Omega BSO_n}))^{S_n}=H^{*}({B}A_{n-1})^{S_n}\otimes H^{*}({\Omega BSO_n}).
\sigma^*\colon\xi_j'=M_j(\xi\eta)\xi_j, \eta_j'=M_j^{-1}(\xi\eta)\eta_j, 1\leq j\leq p, \tau_i^*=\Lambda_{ij}(\xi\eta)\eta_j, \eta_j'=\Lambda_{ij}^{-1}(\xi\eta)\xi_j.
\lambda^{(\downarrow),\,\mathrm{new}}_m = \begin{cases}\lambda^{(\downarrow),\,\mathrm{old}}_m -1\, , m<n \, , N-2M-1-j+m \, , m=n \, , \lambda^{(\downarrow),\,\mathrm{old}}_{m-1}\, , m> n \, .\end{cases}
\times \left| H_{n - k}\left( \frac{( T + 1 ) z + \left[ \sigma_{p p}- \sigma_{q q}- 2 i \sigma_{p q}\right] z^{\star}}{\left\{( 2 T + 4 d + 1 ) \left[ \sigma_{p p}- \sigma_{q q}- 2 i \sigma_{p q}\right] \right\}^{\frac 1 2}}\right) \right|^{2}.
|c_{l}^{\pm}| \leq |f(t)| \begin{cases}\frac{\xi}{k^2}l\geq k+1, \sqrt{\frac{\xi}{k^2}}l \in \{k-1,k+1\}, (\frac{\xi}{k^2})^{-1/2}l=k, \frac{\xi}{k^2}+ \xi^{-1}l\leq k-2. \end{cases}
A A^\top=\Sigma:=\Big(\frac{T}{n}\min(j,k)\Big)_{j,k=1}^n=\frac{T}{n}\left(\begin{array}{cccccccc}1 1 1 \ldots 1 1 2 2 \ldots 2 1 2 3 \ldots 3 \vdots \vdots \vdots \ddots \vdots 1 2 3 \ldots n\end{array}\right)\,.
R_{(kl)}= \sum_i \left( \partial_i \Gamma^i_{\: \: kl}- \frac12 \partial_k\Gamma^i_{\: \: il}- \frac12 \partial_l \Gamma^i_{\: \: ik}\right) +\sum_{i,j}\left( \Gamma^i_{\: \: kl}\Gamma^j_{\: \: ji}-\Gamma^j_{\: \: ki}\Gamma^i_{\: \: lj}\right)
|<l_{1,0}w,w>_{L^2}| = \left|<l_{1,\gamma}w,w>_{L^2}+ <w^2,\psi_{1,\gamma}-Q_{1}>_{L^2}- \gamma||\partial_x^{-1}w||_{L^2}^2\right| &\lesssim \epsilon(\gamma) + \gamma.
S_1(z_4) =\sum_{\substack{z_1<q\le p\le z_2 q\le (X/p)^{1/2}}}S_{p q}(q)+\sum_{\substack{z_3<p\le z_4 z_1<q\le (X/p)^{1/2}}}S_{p q}(q)+o\Bigl(\frac{\#\mathcal{A}{}}{\log{X}}\Bigr).
f_1(x, y)+\phi_1([xy])=[x\phi_1(y)]+[\phi_1(x)y]+f_1'(x, y), g_1(x, y, z)+\phi_1(\{xyz\})=\{\phi_1(x)yz\}+\{x\phi_1(y)z\}+\{xy\phi_1(z)\}+g_1'(x, y, z).
\sup_{a\in\mathcal{M}_1}\Bigl|S_{\mathcal{A}}\Bigl(\frac{a}{X}\Bigr)\Bigr|=\#\mathcal{A}\sup_{a\in\mathcal{M}_1}F_X\Bigl(\frac{a}{X}\Bigr)\ll \#\mathcal{A}\exp(-\sqrt{\log{X}}).
\biggl(- \frac{d^2}{dx^2}+ 2(x^3 - 3ax^2 - 2bx - c) \frac{d}{dx}+ (r - 4b^2 - 6ac + 3)x^2 + (q -4bc - 6a)x -c^2 - 2b\biggr) \phi_n(x) = E_n \phi_n(x),
\alpha_{A}( \varphi ) = - \left[ \ln{\frac{\widehat \Lambda_{s}}{m_{A}}}+{\frac{1}{2}}\right]{\frac{\partial \ln B ( \varphi )}{\partial \varphi}}= + \ln{\frac{\widehat \Lambda_{s}^{\prime}}{m_{A}}}\,{\frac{\partial \ln B^{- 1}( \varphi )}{\partial \varphi}}\ ,
\begin{binom}n{k-1}\end{binom}\left(\prod_{l=k-1}^{n-1}(m-l)\right)+\begin{binom}n{k}\end{binom}\left( \prod_{l=k}^{n-1}(m-l) \right) (m-2k){}-\begin{binom}n{k+1}\end{binom}\left( \prod_{l=k+1}^{n-1}(m-l) \right) (k+1)(m-k).
\sum_{n\geq 1}\frac{(-1)^n 2 q^n}{1+q^n}\frac{(-q;q)_{n-1}}{(q;q)_{n-1}}= 2\frac{(q;q)_\infty}{(-q;q)_\infty}\left(-1 + \sum_{n\geq 0}(-1)^nq^{n^2}\right).
\left( \begin{matrix}{{u}}\\{{v}}\\{{w}}\\ \end{matrix}\right)_{t_{1}}= \left( \begin{matrix}{{0}}&{{u}}&{{- u}}\\{{v}}&{{0}}&{{- v}}\\{{w}}&{{w}}&{{- 2 w}}\\ \end{matrix}\right) \left( \begin{matrix}{{u}}\\{{v}}\\{{w}}\\ \end{matrix}\right)_{x}
|A_{n_k}f(x)-A_{n_{k-1}}f(x)| =\bigg|\sum_{j\in J(k)}(A_{m_j}f(x)-A_{m_{j-1}}f(x))\bigg| &\leq \sum_{j\in J(k)}|A_{m_j}f(x)-A_{m_{j-1}}f(x)|.
C_1 :=\bigl\{\{1,2\},\{3,4\},\{1,3\},\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},\{1,2,3,4\}\bigr\}, C_2 :=\bigl\{\{1,4\},\{2,3\}, \{1,3\},\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},\{1,2,3,4\}\bigr\}.
(H_{n-1}(\Sigma),\mathbb{Z}_p) \cong(\mathbb{Z}_{q_{n-1,1}}\oplus\ldots\oplus\mathbb{Z}_{q_{n-1,l_{n-1}}},\mathbb{Z}_p) &\cong(\mathbb{Z}_{q_{n-1,1}},\mathbb{Z}_p)\oplus\ldots\oplus(\mathbb{Z}_{q_{n-1,l_{n-1}}},\mathbb{Z}_p)=0.
\mathcal{I}_{2}=\int_{\Omega^{\varepsilon}}\kappa^{\varepsilon}\nabla\left(\theta^{\varepsilon}-\theta_{1}^{\varepsilon}\right)\cdot\nabla\varphi dx +\int_{\Omega^{\varepsilon}}\left(\kappa^{\varepsilon}\nabla\theta_{1}^{\varepsilon}-\mathbb{K}\nabla\theta^{0}\right)\cdot\nabla\varphi dx.
h_{\tilde \omega}:=\sum_{j=1}^{n}\frac{p_j^2+\tilde \omega_j^2q_j^2}{2}\ , h_1(q):=\frac{1}{2}\sum_{j=1}^{n}(\omega_j^2-\tilde\omega_j^2)q_j^2\ ,f:=H_0+h_1
S \lesssim \sum\limits_{j=1}^\infty\int_{t2^{j-1}\leqslant d(x,y)<t2^j}\frac1{\omega(x,t+d(x,y))}\bigg(\frac{t}{t+d(x,y)}\bigg)^{\varepsilon}d\omega(x) +\int_{d(x,y)<t}\frac1{\omega(x,t+d(x,y))}d\omega(x).
T(M, \bar{u}) = \{v\in \mathcal{U}:\ \exists \{u_n\}\subset M,\ \{t_n\}\subset \mathbb{R}^+,\u_n \rightarrow \bar{u}, \t_n\rightarrow 0^+, \frac{1}{t_n}(u_n - \bar{u})\rightarrow v\},
S = \int d^4 x \sqrt{g}\left({{m^8 v_1 v_2 v_3}\over (2 \pi)^7 \lambda^2}R +{1 \over 4}{{m_s^2 v_2}\over{(2 \pi)^3 \lambda}}F^2+{1 \over 4}{{m_s^2 v_1}\over{(2 \pi)^3 \lambda}}\tilde{F}^2 +\dots \right)
|(D\Theta_3(\varphi_i).(v_1,v_2)\| \le C_s^3(\varphi_i)\left[\|v_1\|_{s_0+1}|v_2\|_{s+1}+\|v_1\|_{s+1}|v_2\|_{s_0+1}+\|v_1\|_{s+1}|v_2\|_{s+1}\right].
\begin{cases}u_t-u_{5x}+u_{3x}+u_x=F(t,x), \ \ (0,T)\times (0,2\pi), \partial_x^ju(t,0)=\partial_x^ju(t,2\pi), \ \ \ (0,T), u(0,x)=u_0(x), \ \ \ (0,2\pi),\end{cases}
S^\textnormal{coh}(\tau,z) ( \alpha ) = e^{-\tau_2/z}\alpha -\sum_{\substack{d \in H_2(X;\mathbb{Z})-\{0\}l \geq 0}}\, \sum_{k=0}^N \frac{1}{l!}\left\langle T_k, \tau ', \dots , \tau ', \frac{e^{-\tau_2/z}\alpha}{z+\psi}\right\rangle^\textnormal{coh}_{0,l+2,d}T^k e^{\tau_2(d)}\in QH(X) \otimes \mathbb{C}( \! (z) \! )
\left( \ref{claim:dificil}\right) u=t\exists Z_{u=t,3}^{\ast}\mathbf{Z}_{min}\backslash \left[ \mathbf{O}_{min}\cup \left\{Z_{u=t,1}^{\ast},Z_{u=t,2}^{\ast}\right\}\right] Z_{u=t,3}^{\ast}&\text{is a descendantof}Z_{u=t,2}^{\ast}.
\sqrt{\bigl(f'|_{V(u_1)}\bigr) + J\bigl(f'|_{V(u_1)}\bigr)}\; = \; \sqrt{\bigl(x_1',x_2'^2x_3',x_2'^3+3x_3'^2,x_3'^3\bigr)}\; = \; \bigl(x_1',x_2',x_3'\bigr)
{\cal X}{\bar{H}}_{\mathrm{\scriptsize ~ I}}= \Xi^{\dagger}{\cal A}^{\dagger}H_{\mathrm{\scriptsize ~ I}}+ \left( 1 -{\cal X}\right) H_{0}- \left( 1 - \Xi^{\dagger}{\cal A}^{\dagger}\right) H_{0}+{\cal X}{\bar{H}}\left( 1 - \Xi^{\dagger}{\cal A}^{\dagger}\right) .
G(z;x) = \sum_{l=m}^\infty \frac{\rho_{l-m}}{(m+1)_{l-m}}\frac{x^{l-m}}{z^{l-m+1}}= \frac{m!}{x^m}\frac{z^{m-1}}{\rho_m'}\sum_{l=m}^\infty \frac{\rho_l(a-m,b-m)}{l!}\left(\frac{x}{z}\right)^l,
A_{0}( x ) = \left[ \begin{array}{c}{\frac{\pi T}g \exp \left\{\frac g{\sqrt{\pi}}( x - x_{0}) \right\}, \x \leq x_{0}}\\{\frac{\pi T}g \left[ 2 - \exp \left\{\frac g{\sqrt{\pi}}( x_{0}- x ) \right\}\right] , \x \geq x_{0}}\\ \end{array}\right.
*(dx^{i_{1}}\wedge...\wedge dx^{i_{k}})=\frac{\sqrt{|det[g_{ij}]|}}{(n-k)!}g^{i_{1}j_{1}}\cdot\cdot\cdot g^{i_{k}j_{k}}\epsilon_{j_{1}...j_{n}}dx^{j_{k+1}}\wedge...\wedge dx^{j_{n}}.
\kappa^4 - 6 \kappa^2 + 1 =\left( \kappa^2 - 1 \right)^2 - 4 \kappa^2 =\left( \kappa^2 + 2 \kappa - 1 \right)\left( \kappa^2 - 2 \kappa - 1 \right) =\left( \kappa + 1 + \sqrt{2}\right)\left( \kappa + 1 - \sqrt{2}\right)\left( \kappa - 1 + \sqrt{2}\right)\left( \kappa - 1 - \sqrt{2}\right),
G_n = \bigcup_{j=1}^n B \left( j\theta , n^{-\tau}\right) &\supset B \left( i\theta , \frac{1}{n^{\tau}}\right) \cup B\left ( (q_k + i) \theta , \frac{1}{n^{\tau}}\right) \cup \dots \cup B\left ( \Big ( \left\lfloor \frac{n-i}{q_k}\right \rfloor q_k + i \Big)\theta , \frac{1}{n^{\tau}}\right).
k_{\beta}( \mathrm{\bf 1}, e^{\varphi^{i}T_{i}}) = \frac{M}{( 2 \pi \beta )^{\frac{D}{2}}}\sum_{\nu}\prod_{\alpha > 0}\frac{< \varphi + 2 \pi \nu , \alpha > / 2}{\sin < \varphi + 2 \pi \nu , \alpha > / 2}e^{- \frac{1}{2 \beta}< \varphi + 2 \pi \nu , \varphi + 2 \pi \nu >}
\frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty}\sum_{n\geq 0}\frac{(-1)^n q^{n^2+n}}{(-q;q^2)_{n+1}}= \frac{(-q;q^2)_\infty}{(q^2;q^2)_\infty}\left(\sum_{j\geq 0}\frac{q^{4j^2 +2j}(1-q^{4j+2})}{(-q;q^2)_{2j+1}}+ \sum_{j\geq 1}\frac{q^{4j^2 + 2j-1}}{(-q;q^2)_{2j}}\right).
(\partial_t^2+ia\partial_{\alpha})(I-\mathfrak{H})u= -\frac{i}{\pi}\sum_{j=1}^N (\partial_t^2+ia\partial_{\alpha})\frac{\lambda_j}{z(\alpha,t)-z_j(t)}= \frac{i}{\pi}\sum_{j=1}^N \lambda_j\Big(\frac{2z_{tt}+i-\ddot{z}_j}{(z(\alpha,t)-z_j(t))^2}-2\frac{(z_t-\dot{z}_j(t))^2}{(z(\alpha,t)-z_j(t))^3}\Big)
\begin{aligned}\left(\frac{t^N/N!}{e^t-T_{N-1}(t)}\right)^{r}\frac{{\rm d}^p}{{\rm d}x^p}e^{xt}= \left(\frac{t^N/N!}{e^t-T_{N-1}(t)}\right)^{r}(e^{xt}t^p) =\left(\left(\frac{t^N/N!}{e^t-T_{N-1}(t)}\right)^{r}e^{xt}\right)t^p. \end{aligned}
0 = \int_{M_k\setminus\{p\}}| \nabla \psi' |^2 dv^{g'}+ \frac{n-1}{n-2}\int_{M_k\setminus\{p\}}|\psi|^{\frac{n-2}{n-1}}L_g(|\psi|^{\frac{n-2}{n-1}})\,dv^g.
C^{-\infty}_{{\cdot}/}(L;E\otimes\Omega^{-1}NL)\subset C^{-\infty}_{{\cdot}/}(M;E)\;, u\mapsto\delta_L^u\;,\quad\langle\delta_L^u,v\rangle=\langle u,v|_L\rangle\;,v\in C^\infty_{/{\cdot}}(M;E^*\otimes\Omega)\;.
\int_{\Omega}\rho(x){\rm d}x = \sum_{n=1}^N \int_{\Omega}|u_n(x)|^2 \,{\rm d}x = \sum_{n=1}^N \frac{1}{\lambda_n}&= \sum_{n=1}^N \frac{1}{n}\cdot \frac{|\Omega|}{4\pi}\left( 1 + o(1)_{n\to \infty}\right) = \ln(N) \cdot \frac{|\Omega|}{4\pi}\left( 1 + o(1)_{N\to \infty}\right).
{\cal L}= \frac{1}{1 6 \pi \alpha^{\prime}}\sqrt{- g}\e^{\mathrm{- \phi}}\Biggl \{R + \nabla \phi \cdot \nabla \phi -{\frac{1}{3}}H^{2}- \frac{1}{4}\mathrm{T r}F^{2}+ \cdots \Biggr \}
B_{ii}(t_1:t_0) \geq \eta_i 0\leq t_0\leq t_1\leq t_L,\cr B_{jj}(t_1:t_0) \geq \eta_j 0\leq t_0\leq t_1\leq \sigma,\cr \sum_{t=0}^{\sigma-1}B_{ji}(t) \geq \delta\delta\in (0, \eta_j).
-\sin\psi\csc\eta\cdot\left(\frac{F[-q,p](i\mu_{0})w_{1}[-q,p](i\mu_{0})w_{3}[-q,p](i\mu_{0})}{w_{2}[-q,p](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{2}\partial_{\phi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\lim_{\substack{u\in H_r(B^n(1)) \|u\|_{H_1^2}\to 0}}\frac{\mathcal G_r(u)}{\|u\|_{H_1^2}^2}=\lim_{\substack{u\in H_r(B^n(1)) \|u\|_{H_1^2}\to \infty}}\frac{\mathcal G_r(u)}{\|u\|_{H_1^2}^2}=0.
|V_\phi f(x,\xi)| \leq C'_{N_0}(1+|\xi|^2)^{N_0}\sum_{|\alpha|\leq N_0}\sum_{\gamma\leq\alpha}\binom{\alpha}{\gamma}|| e^{-r_0|\cdot-x|^{1/s}}e^{r|\cdot|^{1/s}}||_2.
\overline{\Lambda}_{n}(\hat{T})-\underline{\Lambda}_{n}(\hat{T})<\overline{\Lambda}_{3}-\frac{1}{6}cot\frac{\pi}{8}+\frac{\pi}{64}\frac{1}{sin^{2}(3\pi/16)}-\frac{2}{\pi}(\gamma-log\pi)\approx0.201.
\frac{d}{d\left\|f_j\right\|_2^2}\left( \prod_{i=1}^d\left\|f_i\right\|_2^2 - \prod_{i=1}^d\left(\left\| f_i\right\|_2^2 - \frac{L^2}{12b^2}\right)\right) = \prod_{i\neq j}\left\|f_i\right\|_2^2 - \prod_{i\neq j}\left(\left\| f_i\right\|_2^2 - \frac{L^2}{12b^2}\right) \ge 0.
\int x^{4}r^{3}dx=\frac{x^{3}r^{5}}{8}-\frac{a^{2}xr^{5}}{16}+\frac{a^{4}xr^{3}}{64}+\frac{3a^{6}xr}{128}+\frac{3a^{8}}{128}ln(x+r)
u_M(x):=\mathbb{E}\left\{g(X_M^{x})+\sum\limits_{k= 1}^M \widetilde{r}^2(X^{x}_{k-1})K_0f(X^{x}_{k-1}+\widetilde{r}(X^{x}_{k-1})\cdot)\right\}\mbox{for all}x\in D,
R_{2}=\sqrt{\frac{a^{2}+\frac{b^{2}}{e}ln(\frac{1+e}{b/a})}{2}}=\sqrt{\frac{a^{2}}{2}+\frac{b^{2}}{2}\frac{tanh^{-1}e}{e}}=\sqrt{\frac{A}{4\pi}},
\begin{array}{ll}N_3= \int_{\mathbb R}(\zeta_k+\varphi_k)[(2 c_1a_k-\omega b_k) \xi_{n_k}+ (c_2b_k-\omega a_k) \nu_{n_k}] &+ ((\zeta_k \xi_{n_k})' + (\varphi_k \xi_{n_k})' )(2 c_3a'_k-\omega c_4 b'_k)+((\zeta_k \nu_{n_k})' +(\varphi_k \nu_{n_k})') (2 c_5b'_k-\omega c_4 a'_k)\end{array}
W_{\bf a}^{r e n}[ M^{4}] = \kappa_{0}^{r e n}A ( \Sigma ) S_{2}( e^{\frac{2 \pi}{{\bf a}_{r e n}}}) + B ( \Sigma ) ( \kappa_{1}^{r e n}+ S_{0}( e^{\frac{2 \pi}{{\bf a}_{r e n}}}) ) .
W_{C C}^{*}= k_{1}^{2}k_{2}^{2}W_{S S}, W_{S S}^{*}= k_{1}^{2}k_{2}^{2}W_{C C}, W_{C S}^{*}= k_{1}^{2}k_{2}^{2}W_{S C}, W_{S C}^{*}= k_{1}^{2}k_{2}^{2}W_{C S},
{\cal Z}_{\rm grand}^{\rm 3D}=\int{\cal D}{\bf j}\exp\left\{-\left[\frac{\pi\eta^2}{2}\int d^3xd^3yj_\mu({\bf x})\frac{{\rm e}^{-m|{\bf x}-{\bf y}|}}{|{\bf x}-{\bf y}|}j_\mu({\bf y})+V[2\pi{\bf j}]\right]\right\}.
\ \sum_{n_{r}= 1}^{\infty}\; \sum_{k_{r}= 1}^{n_{r}}\;{\binom{n_{r}}{k_{r}}}\; \frac{( - )^{k_{r}}}{n_{r}}\; k_{r}\; E_{r}^{k}\; = \; E_{r}\; \cdot \; \sum_{n^{\prime}= 0}^{\infty}\; ( E - E_{r})^{n^{\prime}}\; = \; E \; .
D ( j_{3}, j_{2}, j_{1}) = \frac{G ( j_{1}+ j_{2}+ j_{3}+ 1 ) G ( j_{1}+ j_{2}- j_{3}) G ( j_{1}+ j_{3}- j_{2}) G ( j_{2}+ j_{3}- j_{1})}{( \nu ( b ) )^{- j_{1}- j_{2}- j_{3}- 1}\; G_{0}\; G ( 2 j_{1}+ 1 ) G ( 2 j_{2}+ 1 ) G ( 2 j_{3}+ 1 )}.