harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_X\Psi(M_u(f(x)))d\mu =\int_X\Psi(\frac{1}{x})d\mu &=\int_X(1+\frac{1}{x})log(1+\frac{1}{x})-\frac{1}{x}d\mu &>\int_A log(1+\frac{1}{x})-\frac{1}{x}d\mu+\int_BPAR log(1+\frac{1}{x})-\frac{1}{x}d\mu &=xlog(1+\frac{1}{x})+ln(1+\frac{1}{x})\mid_0^{a}+\sum\limits_{n>a}lnn.log(1+\frac{1}{lnn})+ln(1+\frac{1}{lnn}) &=\infty
-i \int_{\partial D_R}\!\!\big( (i\nabla +A_{\mathbf e})\tilde \varphi_{a_{n_\ell}}\!\cdot \nu \big) \overline{\varphi_{a_{n_\ell}}}\,ds \to -i \int_{\partial D_R}\!\!\big( (i\nabla +A_{\mathbf e})\tilde \Phi \cdot \nu \big) \overline{\tilde \Phi}\,ds = \int_{D_R}|(i\nabla + A_{\mathbf e})\tilde\Phi|^2dx
w_{j_l}(X_{j_l},Y_{j_l},t_{j_l},\tilde X,\tilde Y,\tilde t) =u(X_{j_l},Y_{j_l},t_{j_l})-(\phi(\tilde X,\tilde Y,\tilde t)-\varphi_{j_l}(\tilde X,\tilde Y,\tilde t)),
0 = (\kappa_1 + \kappa_2 + 2 - p)\| (y_1, y_2) \|_{1,p}+ (\kappa_1 + \kappa_2 + 2 - q)\| (\nabla{y_1}, \nabla y_2) \|_{q,\eta}- (\kappa_1 + \kappa_2 + \nu + 1)\int_{\Omega}\left[a_1\vert y_1 \vert^{1 - \nu}+ a_2\vert y_2 \vert^{1 - \nu}\right]dz.
d s^{2}= g_{M N}^{0}\, d x^{M}\otimes d x^{N}= \mathrm{e}^{2 \Delta ( y )}\left( d x^{\mu}\otimes d x^{\nu}\, \hat{g}_{\mu \nu}( x ) + d y^{m}\otimes d y^{n}\, \hat{g}_{m n}( y ) \right) \, .
\delta^{ab}\, \delta_{xy}\, =\, \langle \, \frac{\delta S}{\delta\bar c^a_x}\, \bar c^b_y\, \rangle\, =\, \widetilde Z_3 \, \langle \, i (\partial D_r c)^a_x \, \bar c^b_y\, \rangle \, - \,\frac{Z_\lambda \lambda}{Z_3}\,\langle \, (s_r B^a_x ) \, \bar c^b_y\, \rangle \; .
\|F\|_{H^*}=\sup_{\begin{subarray}{c}g\in H \|g\|_{H}\le 1\end{subarray}}\langle F, g \rangle_{H^*,H}=\sup_{\begin{subarray}{c}g\in H \|g\|_{H}\le 1\end{subarray}}\sum_{k=1}^{\infty}\hat{F}_k\,\hat{g}_k\,,
(\mathcal{J}\mathbf{Q})_{\pi_{\overline{\theta}^{*}}}(\mathbf{S}_t, \mathbf{A}_t, \pmb{\Omega}) = \mathbf{R}_t(\mathbf{S}_t, \mathbf{A}_t)+\gamma\sum_{\mathbf{S}^{'}\subseteq \overline{\mathbf{S}}}p(\mathbf{S}_t, \mathbf{A}_t, \mathbf{S}^{'}) (\mathcal{H}\mathbf{Q})_{\pi_{\overline{\theta}^{*}}}(\mathbf{S}^{'}, \mathbf{A}^{'}, \overline{\pmb{\omega}}).
Kh(L)(q,t) = q^{-\sigma(L)}\left ( (q + q^{-1}) \left (\sum_{E \subset \{2,\dots n\}}(t q^2)^{\sum_{j \in E, k \notin E}2 l_{jk}}\right) + \left ( q^{-1}+ t q^2 \cdot q \right ) Kh'(L)(t q^2) \right )
\varphi^c(M';\bigoplus_{i=1}^n L_i,0) =\langle Q_1(TM')\prod_{i=1}^n x_i Q_2'(\bigoplus_{i=1}^nL_i) \hat{A}(M'),[M']\rangle &= \langle \prod_{i=1}^n x_i,[M']\rangle\neq 0.
\left(I_{0+}^{\alpha,\alpha',\beta,\beta',\gamma}f\right)(x)=\frac{x^{-\alpha}}{\Gamma(\gamma)}\int_0^x(x-t)^{\gamma-1}t^{-\alpha'}F_3\left(\alpha,\alpha',\beta,\beta';\gamma;1-\frac{t}{x},1-\frac{x}{t}\right)f(t)\,dt
\lim_{n \rightarrow + \infty}\mathbb{E}\left\Vert N X_{s_n}\left( \xi \right) - N X_{t}\left( \xi \right) \right\Vert^2=\lim_{n \rightarrow + \infty}\mathbb{E}\left\Vert Y_{s_n}- Y_{t}\right\Vert^2= 0
\frac{T_1}{\surd{(\Delta_{1})}}=\frac{f^{(2)}(\tau)}{|f^{(1)}(\tau)|}\frac{\gamma^{-1}\|\Sigma_1-\Sigma_2\|_{\rm F}^2}{\surd{\{\frac{2}{n(n-1)}(\Sigma_{1}^{2})+\frac{2}{m(m-1)}(\Sigma_{2}^{2})+\frac{4}{nm}(\Sigma_{1}\Sigma_{2})\}}}\{1+o(1)\}.
0 =\sum_{i=0}^\infty u((m+j+1)^2+j-i^2,-j-1+i) &= \sum_{i=0}^\infty u((m+j+1)^2+j-(i+j+1)^2,i) &= \sum_{i=0}^m u((m+j+1)^2+j-(i+j+1)^2,i).
\phi( \alpha \cdot P) = \bigg[ K(Z_i, Z_j) \big( \sum_{k=1}^N \alpha_{ik}P_{kj}\big) \bigg]_{1 \le i,j \le N}, & L_\alpha \phi(P) = \bigg[ \sum_{k=1}^N \alpha_{ik}K(Z_k, Z_j) (P_{kj}) \bigg]_{1 \le i,j \le N}.
\prod_{t=1}^{\eta}\left(1-\frac{\mathcal{O}(\varphi^{n/5})}{u_t^2-(1-2\varepsilon (-\varphi)^{-n})^2}\right) > 1- \mathcal{O}(\varphi^{n/5})\sum_{t=1}^{\infty}\frac{1}{u_t^2-(1-2\varepsilon (-\varphi)^{-n})^2}&= 1- \mathcal{O}(\varphi^{n/5}).
\sum_{[\xi]\in \widehat{G}:|\xi|\leq \lambda}d_\xi\smallint\limits_G\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx=(2\pi)^{-n}\left( \smallint\limits_{T^*\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi')d\mu_L(x,\xi')\right)\lambda^n+O(\lambda^{n-1}).
W_n(\lambda)=\frac{1}{nd_n}\sum\limits_{i=1}^{\lfloor n\lambda\rfloor}\sum\limits_{j=\lfloor n\lambda\rfloor+1}^n\left(1_{\left\{X_i\leq X_j\right\}}-\frac{1}{2}\right), \ 0\leq \lambda\leq 1.
\sup_{t\in[0,1]}\max_{i,i'}\bigl|(X_{2,t})_i^k(\hat\lambda)-(X_{2,t})_{i'}^k(\hat \lambda)\bigr|\le C\,(\log N)^2\,N^{2/(\sigma-17)}|\lambda_i^k-\lambda_{i'}^{k}|\forall\,k=1,\dots,d,
\sum_{i=1}^{k-1}w(\Gamma_i) + w(\Lambda_{L+1}) >\sum_{i=1}^{k}w(\Gamma_i)\implies\sum_{i=1}^{k}w(\Gamma^+_i) >\sum_{i=1}^{k}w(\Gamma_i)\geq \sum_{i=1}^{k}w(\Lambda_i)= \sum_{i=1}^{k}w(\Lambda^+_i).
U^{\ast}(\varphi,\theta)=\bar{m}^{5/2}\sum_{i=1}^{3}\frac{(\mu_{j}\mu_{k})^{3/2}(\mu_{i}^{\ast})^{-1/2}}{\sqrt{(1-\sin\varphi\cos(\theta-\theta_{i})}}\mu_{i}^{\ast}=\frac{1}{2}(1-\mu_{i})
S_{( f ) \sigma}( z , z^{\prime}) = G_{( \sigma )}( z , z^{\prime}) - \sum_{r = 1}^{n}\sum_{N_{r}= \mu_{r}, \nu_{r}}\int_{C_{b}^{( r )}}G_{( \sigma )}( z , z^{\prime \prime}) \frac{d \theta^{\prime \prime}d z^{\prime \prime}}{2 \pi i}\tilde{Y}_{\sigma , N_{r}}^{( 1 )}( t^{\prime \prime}) \Psi_{\sigma , N_{r}}^{( 0 )}( z^{\prime})
U(x)=\alpha_n \frac{1}{(1+|x|^2)^{\frac{n-2}{2}}},U_{\mu,\xi}(x)=\frac{\alpha_n \mu^{\frac{n-2}{2}}}{( \mu^2 +|x-\xi|^2)^{\frac{n-2}{2}}}, \mbox{with}\ \alpha_n=(n(n-2))^{\frac{n-2}{4}},
\begin{aligned}s_b(N_k^{2n}) =n(b-1)(2^{k}-1) &+(b-C_{2n}^1)+(b-C_{2n}^3)+\cdots + (b-C_{2n}^{2n-1}) &+(C_{2n}^2-1)+(C_{2n}^4-1)+\cdots +(C_{2n}^{2n-2}-1) &+1,\end{aligned}
\mathbb{P}(Z_1+Z_2=u+2) \ \mathbb{P}(Z_3=0) \ \mathbb{P}\Biggl(\bigcap \limits_{n=4}^{\infty}\Biggl\{n-2-\sum_{i=4}^{n}Z_i>0\Biggr\}\Biggr) &=c_0\varphi(1)\sum_{k=0}^{u+2}a_kb_{u+2-k},
f_{n_2}\leq (1-g)^{n_2-n_1}f_{n_1}+ h\sum_{n=n_1}^{n_2-1}(1-g)^{n_2-1-n}&\leq (1-g)^{n_2-n_1}f_{n_1}+ h\sum_{k \in \mathbb{N}_0}(1-g)^k &\leq f_{n_1}\exp\left[-g(n_2-n_1)\right] + g^{-1}h\,.
\widetilde{L}_k^r\big(h\big)=\sum_{a+b\le 2r}P_{a,b}(\gamma)\,\frac{\partial_\tau^a h}{\gamma^b}\,\,\,\,\,\,\,\,\,\,\,\left(h\in \mathcal{C}^\infty(U_\gamma)\right),
\displaystyle\int_{0}^{\frac{\alpha|\mathrm{y}-\mathrm{v}|^2}{4\tau}}\textbf{exp}(-\mathrm{m}^2) d(\mathrm{m}^2) = 1- \textbf{exp}\Big(\frac{\alpha|\mathrm{y}-\mathrm{v}|^2}{4\tau}\Big) = 1- \Big[1-\frac{\alpha|\mathrm{y}-\mathrm{v}|^2}{4\tau}+ \frac{1}{2!}\Big(\frac{\alpha|\mathrm{y}-\mathrm{v}|^2}{4\tau}\Big)^2-... \Big].
\widetilde{\Gamma}^i_{\: \: jk}= \sum_{a,b,c}\frac{\partial y^i}{\partial x^a}\frac{\partial x^b}{\partial y^j}\frac{\partial x^c}{\partial y^k}\Gamma^a_{\: \: bc}+\sum_a \frac{\partial y^i}{\partial x^a}\cdot\frac{\partial^2 x^a}{\partial y^j \partial y^k}
\frac{\rho_{a}^{r}( \theta )}{\rho_{a}( \theta )}= \frac{\sum_{k = 1}^{G_{a}}k \exp \{( G_{a}- k ) \epsilon_{a}( \theta ) \}}{\sum_{k = 0}^{G_{a}}\exp \{k \epsilon_{a}( \theta ) \}}\, .
\hat L=\frac{1}{nh_n}\sum_{i=1}^n w_n(x_i/h_n)y_i=\frac{1}{nh_n}\sum_{i=1}^n w_{n}(x_i/h_n)f(x_i)+\frac{1}{nh_n}\sum_{i=1}^n w_n(x_i/h_n)u_i
N^{\mu \nu}={\frac{L_{*}{}^{\prime}}{\kappa_{1}}}\eta^{[ \mu}{}_{2}X^{\nu ]}{}^{\prime}-{\frac{L_{*}\kappa_{3}}{\kappa_{1}}}\eta^{[ \mu}{}_{3}X^{\nu ]}{}^{\prime}+ L_{*}\eta^{[ \mu}{}_{1}\eta^{\nu ]}{}_{2}\, ,
= (2 \pi)^3 N_0 + \int \frac{d^3 k}{2 k^0}a_0 (\vec{k})^\dagger \tilde{q}_0 (t, \vec{k})+ \int \frac{d^3 k}{2 k^0}a_0 (\vec{k}) \tilde{q}_0 (t, \vec{k})^*+ \int \frac{d^3 k}{2 k^0}|\tilde{q}_0 (t, \vec{k})|^2 \, .
x_i =\begin{cases}\frac{1}{t+2}\pm 30t\epsilon^{1/2}i\in [t], \frac{\alpha}{t+2}\pm 30t\epsilon^{1/2}i\in \{t+1, t+4\}, \frac{1-\alpha}{t+2}\pm 30t\epsilon^{1/2}i\in\{t+2, t+3\}.\end{cases}
G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|wz)=w^{b_{1}}\sum_{h=0}^{\infty}\frac{(1-w)^{h}}{h!}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{1}+h,b_{2},...,b_{q}\end{matrix}|z),m\ge1,
dg_{SU(N)}=C_N\cos(\psi_{N-1})\sin^{2(N-1)-1}(\psi_{N-1}) \left[\prod_{j=1}^{N-2}\cos^{2j-1}(\psi_{j})\sin(\psi_j)\right]\cdot &d\phi_1\ldots d\phi_{N-1}d\psi_1\ldots d\psi_{N-1}dg_{SU(N-1)}d\omega_N,
{\bf G}_{\{j\}}{\bf \Sigma}_{\{j\}}= \big( 1 + \sum \limits_{i \neq j}\frac{r_{ij}^2 \sigma_{i}^2}{\sigma_j^2}\big ) \sigma_{{j}}^2 = \sigma_{{j}}^2 + \sum \limits_{i \neq j}r_{ij}^2 \sigma_{i}^2
e^{i W}=\int dg_{\mu \nu}~ d\bar{\Gamma}^\sigma_{~\mu \nu}~d \bar{c}^\mu ~dc^\nu ~d \bar{\chi}^\mu ~d\chi^\nu~d \bar{\eta}^\mu ~d\eta^{\sigma \lambda \nu}e^{i S_{quan}}\left(det \omega_{\mu \nu}\right)^{\frac{1}{2}}\left(det \zeta_{\mu \nu}\right)^{\frac{1}{2}}\left(det \varsigma_{\mu \nu}\right)^{\frac{1}{2}}
h_{1}= \left({g_{1}( r )}\atop{f_{1}( r )}\right) \, e^{- i \, \theta}\mathrm{~ , ~}h_{2}= \left({g_{2}( r )}\atop{f_{2}( r )}\right) \, e^{\, i \, \Delta}e^{\, i \, \theta}\, ,
C_0^\vartheta(t,X) =X\cup\{0\}, C_{n+1}^\vartheta(t,X) =\begin{aligned}[t]C_{n}^\vartheta(t,X) \cup\{s\in\varepsilon(S_{\omega^\alpha}^u)\,|\,s^*\in C_{n}^\vartheta(t,X)\}\cup\{\vartheta(s)\,|\,s\in C_{n}^\vartheta(t,X)s<t\}\cup\{s\,|\,s<s's'\in C_{n}^\vartheta(t,X)\cap\Omega\}.\end{aligned}
\int \prod_{i}d Y_{i}\delta ( \sum_{i = 1}^{6}w_{i}Y_{i}- t ) e^{- \sum_{i}\mu_{i}Y_{i}}\ \stackrel{Y_{i}\rightarrow Y_{i}+ l n \mu_{i}}{\longrightarrow}\int \prod_{i}d Y_{i}\delta ( \sum_{i = 1}^{6}w_{i}Y_{i}- t^{\prime}) e^{- \sum_{i}Y_{i}},
\rho(v^-(\cdot,-n),v^+(\cdot,-n)) =\rho(v^-(\cdot,-(n+\tau)+\tau),v^+(\cdot,-(n+\tau)+\tau)) &<\rho(v^-(\cdot,-(n+\tau)),v^+(\cdot,-(n+\tau)))-\delta &<\rho(v^-(\cdot,-(n+k\tau)),v^+(\cdot,-(n+k\tau)))-k\delta
\begin{array}{rl}\displaystyle \mathrm{tr}(A_jX') = \displaystyle \mathrm{tr}(A_j\frac{1}{|G|}\sum_{g \in G}M_g X M_g^T) &= \displaystyle \mathrm{tr}(\frac{1}{|G|}\sum_{g \in G}M_g A_jX M_g^T) &= \displaystyle \mathrm{tr}(\frac{1}{|G|}\sum_{g \in G}A_jX ) &= \displaystyle \mathrm{tr}(A_jX),\end{array}
\Upsilon_{a \alpha}( \vec{k}) = \left[ \begin{array}{c}{\Upsilon_{1 1}}\\{\Upsilon_{2 1}}\\{\Upsilon_{1 2}}\\{\Upsilon_{2 2}}\\ \end{array}\right] = \left[ \begin{array}{c}{c_{k \uparrow}^{\vphantom \dagger}}\\{c_{- k \downarrow}^{\dagger}}\\{c_{k \downarrow}^{\vphantom \dagger}}\\{- c_{- k \uparrow}^{\dagger}}\\ \end{array}\right] .
4\eta(\eta-1)f''(\eta) +\left[-4\alpha\eta^2+(6+4\alpha+4\beta+8s)\eta-4(1+\beta)\right]f'(\eta) &+\left[(\alpha(\alpha-2\beta-4s-3)\eta-\alpha^2 + 2 \alpha (1 + \beta) + (1 + \beta) (\beta + 4 s)\right]f(\eta)=0
\frac{\kappa}{2i\pi}\int_{\gamma_{\kappa}}\frac{z^{m-1}(1-z^2)^j}{(z-m)^{m+1}}P_{n-j}^{0,2j}(1-2z^2)dz = \sum_{k=0}^{n-j}(-1)^k\binom{n-j}{k}\binom{n+j+k}{k}\frac{\kappa}{m!}\partial_z^m[z^{2k+m-1}(1-z^2)^{j}]_{|z=\kappa}.
\left[ \begin{array}{c}G_k(x) Q_k(x) \end{array}\right]= \left[\begin{array}{cc}\alpha_k \beta_k \gamma_k 1 \end{array}\right]\left[\begin{array}{c}G_{k-1}(x) (\delta_k x + \theta_k) Q_{k-1}(x) \end{array}\right],
\widetilde{F}({\bf x}) =\sum_{j=1}^{N}\big( -\tfrac{Z}{2}|x_j|+\tfrac{Z}{2}\sqrt{|x_j|^2+1}\big) &\quad+\sum_{1\le j<k\le N}\big(\tfrac{1}{4}|x_j-x_k|-\tfrac{1}{4}\sqrt{|x_j-x_k|^2+1}\big)\,.
\left( \sum_{j_{1}=1}^{\infty}\left( \sum_{j_{2}=1}^{\infty}\left(\sum_{j_{3}=1}^{\infty}\left\| T\left( x_{\mathbf{j}}\right) \right\|^{s_{3}}\right)^{\frac{s_{2}}{s_{3}}}\right)^{\frac{s_{1}}{s_{2}}}\right)^{\frac{1}{s_{1}}}\leq C \prod_{k=1}^{3}\left\| x^{k}\right\|_{w,q_{k}},
\theta_{0}= \frac{1}{2}i \sum_{k = 1}^{K}L_{k}\ln \left[ \frac{z - z_{k}}{z^{*}- z_{k}^{*}}\right] \ + \, \frac{1}{4}\sum_{k = 1}^{K}\sum_{n_{k}= 1}^{N_{k}}\left[ \theta_{k , n_{k}}( z ) + \theta_{k , n_{k}}^{*}( z^{*}) \right] .
d s^{2}= \left[ H ( r ) \right]^{- \frac{\widetilde{d}}{D - 2}}d x^{\mu}\otimes d x^{\nu}\eta_{\mu \nu}+ \left[ H ( r ) \right]^{\frac{d}{D - 2}}\left( d r^{2}+ r^{2}d s_{S^{D - p - 2}}^{2}\right)
\frac{d}{dt}\sum_{|\alpha|=0}^{m}\langle A_{3}^{0}\partial_{x}^{\alpha}u,\partial_{x}^{\alpha}u\rangle\leq\frac{C\epsilon}{2}\|\nabla v\|_{m}^{2}+C\left\lbrace\|f_{3}\|_{m}^{2}+(\mu_{0}(t)+\mu_{1}(t))\left(\|w\|_{m}^{2}+\|v\|_{m}^{2}\right)\right\rbrace.
x_{n+1}= x_n - \frac{f(x_n)}{f'(x_n)}= x_n - \frac{(x_n-a)^2 + b^2}{2(x_n-a)}= \frac{b}{2}\left( \frac{x_n-a}{b}- \frac{1}{\frac{x_n-a}{b}}\right) + a
P(x) = \det \begin{pmatrix}x_{11}x_{12}/\sqrt{2}x_{13}/\sqrt{2}x_{12}/\sqrt{2}x_{22}x_{23}/\sqrt{2}x_{13}/\sqrt{2}x_{23}/\sqrt{2}x_{33}\end{pmatrix}= x_{11}x_{22}x_{33}- \tfrac{1}{2}\left(x_{11}x_{23}^{2}+ x_{22}x_{13}^{2}+ x_{33}x_{12}^{2}\right) + \tfrac{1}{\sqrt{2}}x_{12}x_{23}x_{13},
\times\frac{[1 + A{\Bigl(\frac{q}{|m|}\Bigr)}^{-2(1-F)}]K^{2}_{F}(qr) - [1 +A^{-1}{\Bigl(\frac{q}{|m|}\Bigr)}^{-2F}]K^{2}_{1-F}(qr)}{A{\Bigl(\frac{q}{|m|}\Bigr)}^{2F}+ 2 + A^{-1}{\Bigl(\frac{q}{|m|}\Bigr)}^{2(1-F)}}.
\|\theta\|_{{B}_{4,1}^{1-\beta}(\mathbb{R}^{2})}\leq C\|\theta\|_{{B}_{2,2}^{s}(\mathbb{R}^{2})}^{\lambda}\|\theta\|_{L^{\infty}(\mathbb{R}^{2})}^{1-\lambda},\lambda=\frac{2\beta-1}{2-2s}\in (0,\,1).
i_{\partial B_{2k-1}}^*\left(\gamma^{\partial B_{2k-2}}(\nu^{\partial B_{2k-1}})Q^{\partial B_{2k-2}}\zeta^{\partial B_{2k-1}}\gamma^{\partial B_{2k-3}}(\nu^{\partial B_{2k-2}})Q^{\partial B_{2k-3}}\zeta^{\partial B_{2k-2}}\omega^{2k-2}\right)= &\quad=\mu_{\partial B_{2k-1}}.
V^{}_{\{\epsilon j,\epsilon^\prime j^\prime \}}(\xi ;g) = w(\epsilon \xi_j)w(\epsilon^\prime \xi_{j^\prime})v(\epsilon \xi_j+\epsilon^\prime\xi_{j^\prime})v(1+\epsilon \xi_j+\epsilon^\prime\xi_{j^\prime}) & \times \prod_{\substack{1\leq k\leq n k\neq j,j^\prime}}v(\epsilon \xi_j+\xi_k) v(\epsilon \xi_j-\xi_k) ,
H_\gamma(u(t)) = 2 E_\gamma(u(t)) + \frac{4}{p+1}\|u(t)\|^{p+1}_{L^{p+1}}&\leq 2E_\gamma(u(t)) + B (H_\gamma(u(t)))^{\frac{N(p-1)}{4}}(M(u(t)))^{\frac{4-(N-2)(p-1)}{4}}&\leq 2|E(u_0)| + B (H_\gamma(u(t)))^{\frac{N(p-1)}{4}}(M(u_0))^{\frac{4-(N-2)(p-1)}{4}}.
\int_0^\infty\left|(u_{n,R}^0)'\right|^pr^{p-1}\mathrm dr =\int_0^\infty\left|\varphi^0_Ru_n'+(\varphi_R^0)'u_n\right|^pr^{p-1}\mathrm dr &=\int_0^\infty|u_n'|^p|\varphi_R^0|^pr^{p-1}\mathrm dr+\mathcal R_{n,R},
B_{k+1}^{-1}=\left[\begin{array}{cc}B_{k}^{-1}-w_{k}\frac{\beta_{k}}{\alpha_{k+1}}\frac{1}{\alpha_{k+1}}\end{array}\right],w_{k+1}=\frac{1}{\alpha_{k+1}}\left[\begin{array}{c}-w_{k}\beta_{k}1\end{array}\right],
H = \sum_{i = 1}^{N}\lbrace \sqrt{m_{i}^{2}+{({\vec{p}}_{i}- e_{i}\vec{A}({\vec{x}}_{i}, x^{o}) )}^{2}}+ V ({\vec{x}}_{i}, x^{o}) \rbrace +{\frac{1}{4}}\int d^{3}z [{\vec{\pi}}^{2}( \vec{z}, x^{o}) +{\vec{B}}^{2}( \vec{z}, x^{o}) ] ,
S = \frac{1}{2 \lambda_{s}^{2}}\int d t e^{\alpha + 2 \beta - \phi}\biggl \{\frac{1}{N}\biggl [ - \dot{\phi}^{2}- 2 \dot{\beta}^{2}- 4 \dot{\alpha}\dot{\beta}+ 2 \dot{\alpha}\dot{\phi}+ 4 \dot{\beta}\dot{\phi}\biggr ] + \frac{w \lambda_{s}^{2}}{4 N^{3}}\biggl [ 8 \dot{\phi}~ \dot{\alpha}~ \dot{\beta}^{2}- \dot{\phi}^{4}\biggr ] \biggr \}
(B_{6-n}A_{4-n}A_{2-n}+A_{6-n}B_{4-n}A_{2-n}+A_{6-n}A_{4-n}B_{2-n})|_{r^{n-6}\mathcal{H}_2}=8(n+2)n(n-2)\neq 0; &(B_{6-n}A_{4-n}A_{2-n}+A_{6-n}B_{4-n}A_{2-n}+A_{6-n}A_{4-n}B_{2-n})|_{r^{n-4}\mathcal{H}_0}=-4n(n-2)(n-4)\neq 0
\{\, u\in[\dot{\mathrm{H}}^{s_0,p_0}\cap\dot{\mathrm{H}}^{s_1,p_1}](\Omega) \,|\, u_{|_{\partial\Omega}}=0 \,\}\simeq [\dot{\mathrm{H}}^{s_0,p_0}_0\cap\dot{\mathrm{H}}^{s_1,p_1}_0](\Omega)
\mathcal{E}(\overline{D}_l'+\frac{1}{k}\overline{A})-\mathcal{E}(\overline{D}'+\frac{1}{k}\phi_A)=-\int_{\Delta_D+\frac{1}{k}\Delta_A}(\check{g}_{k,l}(x)-\check{g}_{{\overline{D}_0+\frac{1}{k}\overline{A}}}(x))dx.
(I_i \cap [j_i +1, j_{i+1}]) \cup (I_{i+1}\cap [j_i +1, j_{i+1}]) \ = \ [j_i +1, j_{i+1}] \mbox{and}(I_i \cap [j_i +1, j_{i+1}]) \cap (I_{i+1}\cap [j_i +1, j_{i+1}]) \ = \ \emptyset.
2M^\dag J(k) MT= -ik{\rm diag}\{I_{n-n_D},0_{n_D}\}&+i{\rm diag}\{(X_1-I_{n-n_D})(X_1+I_{n-n_D})^{-1},-I_{n_D}\}&+Q'{\rm diag}\{I_{n-n_D},0_{n_D}\}+o(1) = -ik{\rm diag}\{I_{n-n_D},0_{n_D}\}+i{\rm diag}\{0_{n-n_D},-I_{n_D}\}&+Q{\rm diag}\{I_{n-n_D},0_{n_D}\}+o(1),
\left\{T(h), \Psi(f) \right\}_\ell^\Sigma [\varphi] = -\int_\mathcal{I}T(h)^{(1)}[\varphi](s) \frac{d \Psi(f)^{(1)}[\phi](s)}{ds}\, \mathrm{d}s = -\int_\mathcal{I}h(s) \varphi(s) \sqrt{\gamma'(s)}\frac{df}{ds}(s) \, \mathrm{d}s = -\Psi \left( h f' \right) [\varphi].
\begin{cases}\displaystyle M_x-L_y=\Gamma^2_{22}L-2\Gamma^2_{12}M+\Gamma^2_{11}N, \displaystyle N_x-M_y=-\Gamma^1_{22}L+2\Gamma^1_{12}M-\Gamma^1_{11}N, \displaystyle LN-M^2=K.\end{cases}
{\cal L}= - \frac{1}{1 2}( \partial_{\mu}A_{\nu \lambda}+ \partial_{\lambda}A_{\mu \nu}+ \partial_{\nu}A_{\lambda \mu})^{2}+ \frac{m^{2}}{4}A_{\mu \nu}^{2}- \overline{{{\psi}}}( \gamma_{\mu}\partial^{\mu}+ M ) \psi - \frac{1}{2}g \varepsilon_{\mu \nu \lambda \varrho}\overline{{{\psi}}}\gamma^{\nu}\psi \partial^{\mu}A^{\lambda \varrho}.
\int_{B_r(x)}\nabla(\rho^2_\delta(|u|)\overline{u}) \cdot A \nabla u \,dy =\delta^{p-2}\int_{B_{r}(x)\setminus E_\delta}\langle A \nabla u, \nabla u \rangle \,dy &+ \int_{E_\delta\cap{B_r(x)}}A\nabla u \cdot \nabla (|u|^{p-2}\overline{u})\,dy
Z_{1}Y_{2}Z_{3}=[\begin{matrix}c_{1}c_{2}c_{3}-s_{1}s_{3}&-c_{3}s_{1}-c_{1}c_{2}s_{3}&c_{1}s_{2}\\c_{1}s_{3}+c_{2}c_{3}s_{1}&c_{1}c_{3}-c_{2}s_{1}s_{3}&s_{1}s_{2}\\ -c_{3}s_{2}&s_{2}s_{3}&c_{2}\end{matrix}]
z_i(\tilde{\mathbf{r}}) =|\{\tilde{r}_j\in N\backslash (N_{-c}\cup N_0):1\leq j\leq i-1\}| &=|\{\tilde{r}_j\in N\backslash (N_{-c}\cup N_0):1\leq j\leq j^*\}| &=|S_1\cup S_2\cup \cdots \cup S_c|
\sum_{k=0}^{M}[4k-1]\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,cq^{-1},dq^{-1},q^{-1};q^2)_k}{(q^{2}/a,aq^2,bq^2,q^2/c,q^{2}/d,q^2;q^2)_k}\bigg(\frac{bq^7}{cd}\bigg)^k begin{align*}5pt]\: \:\equiv[n](bq)^{(n+1)/2}\frac{(q^{-2}/b;q^2)_{(n+1)/2}}{(bq^2;q^2)_{(n+1)/2}}\sum_{k=0}^{(n+1)/2}\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,q^3/cd;q^2)_k}{(q^2,q^{-2}/b,q^2/c,q^2/d;q^2)_k}q^{2k}.
|X_i|,|Y_i| \leq (1+\xi)\frac{n}{2\ell}+ 2\hat\ell \beta n=\left(1+\xi+ 4\ell\hat\ell\beta\right) \frac{n}{2\ell}=(1+2\xi) \frac{n}{2\ell}\leq|S_{\min}|.
\left\vert \sin\left( 2\pi R_{j}n\cdot\sigma\left( n\right) \right)\right\vert =\left\vert \sin\left( 2\pi\left( R_{j}n\cdot\sigma\left(n\right) -s_{n}\right) \right) \right\vert \leqslant2\pi j^{-1}.
arctan(z)=z-\frac{z^{3}}{3}+\frac{z^{5}}{5}-\frac{z^{7}}{7}+\cdot\cdot\cdot=\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{2n+1};|z|\le1z\ne i,-i
B_N = \frac{\displaystyle \min_{1\leq i \neq j \leq N}X_{ij}\cdot U_{ij}}{4XT + 6UT^2 + 2T^3}= \frac{\displaystyle \min_{1\leq i \neq j \leq N}\lambda \left|X_{ij}\right|^2}{4XT + 6UT^2 + 2T^3}.
\binom{N}{2}=(k-\alpha)\left(k-\alpha-\frac{1}{n}\right)\frac{n^2}{2}\leq k \left(k-\alpha-\frac{1}{4}\right)\frac{n^2}{2}\alpha^2-k\alpha+\frac{k}{4}-\frac{k-\alpha}{n}\leq 0\,.
\begin{array}{lll}\dot{\gamma}(t)g(\hat{W},\dot{\gamma})=\displaystyle{\left.\frac{d}{ds}\right|_{s=0}g\Big(W, (\operatorname{Ad}(\alpha(t+s)^{-1})X)_{\frak{m}}}\ & \ \ +\displaystyle{(\operatorname{Ad}(\exp (-t-s)Z)Y)_{\frak{m}}\Big)_o}.\end{array}
\displaystyle t_2-3\chi(M_g)=\sum_{j\ge3}(j-3)t_j+\sum_{k\ge3}(k-3)p_k, \displaystyle p_3-4\chi(M_g)=\sum_{j\ge2}2(j-2)t_j+\sum_{k\ge4}(k-4)p_k.
M(w) = \sup\left\{M_{p, \delta}^{(t,\lambda)}\left(\frac{\mu_{A,H}(u)}{m_{A,H}}, \frac{\mu_{B,H}(v)}{m_{B,H}}\right)\colon w= C_{p,\lambda,t}u + D_{p,\lambda,t}v, 0 < \lambda <1 \right\},
(\alpha_1, t_1) : \ \u_2^\ell u_{12}^m u^n_1 \to (\alpha_1^{-1}u_2)^\ell\left ( \alpha_1 u_{12}+t_1 \alpha_1^{-1}u_2\right )^m(\alpha_1 u_1\alpha_1+t_1)^n \alpha_1^{-\lambda_2}\;.
e_i \eta := ( x_1,\,\ldots,\,x_p,\,r_i x_p,\,r_i x_{p+1},\,\ldots,\, r_i x_q,\,x_{q+1},\,\ldots,\,x_s ; & a_0,\,\ldots,\,a_{p-1},\,t_0,\,a_p,\,\ldots,\,a_{q}=t_1,\,\ldots,\,a_s);
\rho \; F [ \phi ] \; = \; N^{2}\; \det \{X \}^{- 1 / 2}\; \exp \{- \textstyle{\frac{1}{4}}\phi c^{\ast}X^{- 1}c \phi \}\; F [ \frac{\delta}{\delta ( \phi c^{\ast})}+ \frac{\delta}{\delta ( c \phi )}] \; \exp \{+ \textstyle{\frac{1}{4}}\phi c^{\ast}X^{- 1}c \phi \}\; \; ,
( \theta^{i j}) = \left( \begin{array}{c c c c}{{\theta_{1}{\bf \epsilon}}}&{{0}}&{{\dots}}&{{0}}\\{{0}}&{{\theta_{2}{\bf \epsilon}}}&{{\dots}}&{{0}}\\{{\vdots}}&{{\vdots}}&{{\ddots}}&{{\vdots}}\\{{0}}&{{0}}&{{\dots}}&{{\theta_{p}{\bf \epsilon}}}\\ \end{array}\right)
\tilde{\mathcal{C}}:= \left\{[0,1] \times R_{(\omega_1 , \ldots , \omega_N)}\cap ( \Pi^{ss})^{-1}A : \omega_1 , \ldots , \omega_N \in \{0 , \ldots , \ell -1 \}, \, N \in \mathbb{N}\mbox{, and}A \in \mathcal{B}( [0,1] \times \mathbb{R}) \right\}
\hat{T}( \lambda ; g_{1}) = L^{a b}( \hat{G}) : \hat{J}_{a}^{( 0 )}\hat{J}_{b}^{( 0 )}: \hspace{. 0 5 i n}, \hspace{. 3 i n}\hat{c}( \lambda ; g_{1}) = 2 L^{a b}( \hat{G}) \hat{G}_{a b}, \hspace{. 3 i n}\hat{\Delta}_{0}( \lambda ; g_{1}) = 0
\begin{matrix}x(\varphi)&=&a(-cos(2\varphi)+2cos\varphi-1)&=&2a(1-cos\varphi)\cdot cos\varphi&&\\y(\varphi)&=&a(-sin(2\varphi)+2sin\varphi)&=&2a(1-cos\varphi)\cdot sin\varphi&.&\end{matrix}
g^{2}( t ) ={\frac{g^{2}}{\displaystyle 1 +{\frac{1 1 g^{2}t}{1 2 \pi^{2}}}}}, \hspace{5 m m}a_{1}( t ) = a_{1}, \hspace{5 m m}a_{3}( t ) = a_{3}-{\frac{6 2 t}{1 2 0 ( 4 \pi )^{2}}}.
{\rm d}s^2_f = \frac{r+2m}{r}\Big[{\rm d}r^2 +\frac{m^2r^2}{(r+m)^2}\left({\rm d}\theta^2 + \sin^2\theta\,{\rm d}\phi^2 \right) \Big] + \frac{4rm^2}{r+2m}\Big({\rm d}\psi + \cos\theta\,{\rm d}\phi\Big)^2
H(t) = \left[ E_\alpha\left[ -\dfrac{\alpha}{1-\alpha}(t-x)^{\alpha}\right]w(x,t) \right]^{x=t}_{x=t_0}&-\int^{t}_{t_{0}}\dfrac{\alpha^2(t-x)^{\alpha-1}}{1-\alpha}E_{\alpha,\alpha+1}^{2}\left[ -\dfrac{\alpha}{1-\alpha}(t-x)^{\alpha}\right]w(x,t) dx.
\inf_{\zeta\in\mathcal{K}_{t}}\sup_{P^{\theta}\in\mathcal{P}}E_{P^{\theta}}({x}_{t}-\zeta)^{2}=\inf_{\zeta\in\mathcal{K}_{t}}E_{P^{\theta^{\ast}}}({x}_{t}-\zeta)^{2}\geq\inf_{\zeta\in\bar{\mathcal{K}}_{t}}E_{P^{\theta^{\ast}}}({x}_{t}-\zeta)^{2}
D(p)\xi\cdot\xi =\frac{1}{m(m^2c^2+|p|^2)}\sum_{i,j=1}^d(m^2c^2\delta_{ij}+p_ip_j)\xi_i\xi_j &=\frac{1}{m(m^2c^2+|p|^2)}(m^2c^2\|\xi\|^2+(p\cdot \xi)^2)\geq 0.
m_{L}=\frac{m_{em}(1-\frac{1}{3}\frac{v^{2}}{c^{2}})}{(\sqrt{1-\frac{v^{2}}{c^{2}}})^{8/3}},m_{T}=\frac{m_{em}}{(\sqrt{1-\frac{v^{2}}{c^{2}}})^{2/3}}
g=f^{48}(f^{-48}g), g^{-1}=(f^{-48}g)f^{48}(f^{-48}g)f^{96}(f^{-48}g), &f^{-1}=f^{47}(f^{-48}g)(f^{-48}g)f^{48}(f^{-48}g)f^{96}(f^{-48}g)
\Bigr\langle{H_{2\lambda}\psi^\lambda_{f_1}\left(\frac{1-\lambda}{\lambda}\right),\psi^\lambda_{f_2}\left(\frac{1-\lambda}{\lambda}\right)\Bigr\rangle^\lambda_z=-\Bigr\langle\psi^\lambda_{f_1}\left(\frac{1-\lambda}{\lambda}\right),H_{2\lambda}}\psi^\lambda_{f_2}\left(\frac{1-\lambda}{\lambda}\right)\Bigr\rangle^\lambda_{z}
Y =(x_j,j\in J_1,x_j,j\in J_2,\ldots,x_j,j\in J_m), (\sigma(1),\ldots,\sigma(|Y|) =(\underbrace{1,\ldots,1}_{|J_1|},\underbrace{2,\ldots,2}_{|J_2|},\ldots,\underbrace{m,\ldots,m}_{|J_m|}).
L_{1/2}= \frac{1}{2}h_{ij}(y) \dot{y}^i \dot{y}^j + \frac{1}{2}L_{\alpha\beta}(y) (\dot{\psi}^\alpha + w^\alpha_i(y)\dot{y}^i) (\dot{\psi}^\beta + w^\beta_j(y)\dot{y}^j).
F = \left[ \left\{\sum_{i = 1}^{2 k}\left( \frac{d T_{i}}{d x^{i}}\right)^{2}- m^{2}T_{i}^{2}+ \lambda T_{i}^{4}+ \frac{m^{2}}{4 \lambda}\right\}+ \left\{\partial_{\mu}\mathcal{T}\partial^{\mu}\mathcal{T}- m^{2}\mathcal{T}^{2}+ \lambda \mathcal{T}^{4}+ f ( \mathcal{T}) \overline{{{\theta}}}_{R}\theta_{L}\right\}\right]
\begin{cases}\Delta v_i+\sum\limits_{j=1}^{2n}\hat{k}_{ij}e^{v_j}=4\pi\beta_i\delta_0 \ \B_1(0), \ \forall \i=1,\cdots,2n, v_1+v_2+\cdots+v_{2n}\equiv0,\end{cases}
\sum\psi_{j\bar\alpha}\psi^{\bar\alpha\beta}=-\frac{\rho^{\beta}\rho_{j}}{\rho-|\partial\rho|^2}+ \sum \left(\rho^{\bar\alpha\beta}\rho_{j\bar\alpha}+ \frac{\rho^{\bar\alpha}\rho^{\beta}\rho_{j\bar\alpha}}{\rho-|\partial\rho|^2}\right).
(\begin{matrix}\frac{\partial\Phi}{\partial x}\\ \frac{\partial\Phi}{\partial y}\\ \frac{\partial\Phi}{\partial z}\end{matrix})\approx(\begin{matrix}f\frac{\partial\varphi}{\partial x}\\f\frac{\partial\varphi}{\partial y}\\ \frac{\partial f}{\partial z}\varphi\end{matrix}).
|R_1(x,y)| \lesssim \sum\limits_{j=-\infty}^\infty\int_{r^{-j}}^{r^{-j+1}}|S_j(x,y)|\frac{dt}{t}&\lesssim (r-1)\sum\limits_{j=-\infty}^\infty\int_{r^{-j}}^{r^{-j+1}}\frac1{V(x,y,t+d(x,y))}\Big(\frac{t}{t+\|x-y\|}\Big)^{\varepsilon}\frac{dt}{t}&\lesssim (r-1)\int_{0}^\infty\frac1{V(x,y,t+d(x,y))}\Big(\frac{t}{t+\|x-y\|}\Big)^{\varepsilon}\frac{dt}{t}.
\int_{- \infty}^{\infty}\frac{d x}{\sqrt{\pi}}e^{i \lambda x^{2}}= \lim_{\epsilon \rightarrow 0}\int_{- \infty}^{\infty}\frac{d x}{\sqrt{\pi}}e^{i \lambda x^{2}}e^{- \epsilon x^{2}}= \frac{1}{| \sqrt{\lambda}|}\exp \left( \frac{i \pi}{4}\mathrm{s g n}\lambda \right) ,
\forall a\in{\mathbb{B}},\k_{a}(z)=\frac{1}{(1-\bar a\cdot z)^{n-2s}},\{\left\Vert{k_{a}}\right\Vert}_{H_{s}^{p}}\simeq (1-\left\vert{a}\right\vert^{2})^{s-n/p'}