harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{\delta\in \Delta}m_\delta^{11/2}= \sum_{j=0}^{2\log n}\sum_{\delta\in \Delta \atop 2^j \le m_\delta < 2^{j+1}}m_\delta^{11/2}< \sum_{j=0}^{2\log n}k_{2^j}(2^{j+1})^{11/2}= O\left(\sum_{j=0}^{2\log n}n^{15/2}\log n\right) &= O\left(n^{15/2}\log^2 n\right).
\| u_{0,n}\|_{\dot H^1}^2 = \sum_{j = 1}^J \| U^j_{L,n}(0) \|^2_{\dot H^1}+ \| w^J_{0,n}\|^2_{\dot H^1}+ o_n(1), \| u_{1,n}\|_{L^2}^2 = \sum_{j = 1}^J \| \partial_t U^j_{L,n}(0) \|^2_{L^2}+ \| w^J_{1,n}\|^2_{L^2}+ o_n(1)
|\theta(t)-\theta (u)| =\theta (u) =\dfrac{(2n+1)(u-a_{2n+1})}{b_{2n+1}-a_{2n+1}}=\dfrac{(2n+1)(u-a_{2n+1})}{(a_{2n+1})^{2/k}}&< \dfrac{(2n+1)t^{1/k}}{(a_{2n+1})^{2/k}}<\dfrac{(2n+1)(a_{2n+1})^{1/k}}{(a_{2n+1})^{2/k}}=\dfrac{2n+1}{(a_{2n+1})^{1/k}}&<\dfrac{2n+1}{2^{2n+1}}=\ell (2n+1).
q_M^\mathrm{mag}\left(\mathbf{x}_1,\mathbf{x}_2\right) = P_1^\mathrm{mag}\left(Q_{1,1},Q_{2,2},Q_{1,2}\right) \times f_{2,M}^{(3,1)}\left(\mathbf{x}_1,\mathbf{x}_2\right)+P_2^\mathrm{mag}\left(Q_{1,1},Q_{2,2},Q_{1,2}\right) \times f_{2,M}^{(3,2)}\left(\mathbf{x}_1,\mathbf{x}_2\right).
\tilde{\tau}_n^{(r)}= \dfrac{1}{2n}\biggl[\tilde{s}_{[2^4]}(n)+2\cdot\tilde{s}_{[2,4^2]}(n/2)+2\cdot\tilde{s}_{[3^3]}(2n/3)+2\cdot\tilde{s}_{[2,3,6]}(n/3)+\sum_{L|n}J_2(L)\cdot \tau^{(r)}(n/L)\biggr].
\{E_2^+,E_2^-\}= \sum_j1\otimes(x_jy_j-\tfrac{1}{2}) - \sum_{\alpha>0}c_\alpha s_\alpha\otimes(\alpha\alpha^\vee - 1) + \sum_{j}(x_jy_j + \tfrac{1}{2})\otimes 1 - \sum_{\alpha>0}c_\alpha s_\alpha\otimes 1 = Z_0+\Omega_c+H.
\begin{array}{ll}4 \; y_{22}+ 4 \; y_{23}+ y_{33}= 4 4 \; y_{22}- 4 \; y_{23}+ y_{33}= 4 y_{11}- 2 \; y_{13}+ y_{33}= 1 y_{11}+ 2 \; y_{13}+ y_{33}= 1 4 \; y_{11}= 4.\end{array}
|\partial_t^{1-\frac1{2s}}F_s(t)| \leq \int\frac{|F_s(r)-F_s(t)|}{|r-t|^{2-\frac1{2s}}}dr=\int_{|r|\leq|t|/2}\frac{|F_s(r)-F_s(t)|}{|r-t|^{2-\frac1{2s}}}dr &\quad\quad\quad+\int_{|t|/2<|r|\leq 2|t|}\frac{|F_s(r)-F_s(t)|}{|r-t|^{2-\frac1{2s}}}dr+\int_{|r|>2|t|}\frac{|F_s(r)-F_s(t)|}{|r-t|^{2-\frac1{2s}}}dr &=I_1+I_2+I_3.
h^{\prime}(t)= \frac{\boldsymbol{\eta}C e^{Ct}(-C) \mathbf{1}\,\,\boldsymbol{\eta}e^{Ct}\mathbf{1}- \boldsymbol{\eta}C e^{Ct}\mathbf{1}\,\, \boldsymbol{\eta}e^{Ct}(-C) \mathbf{1}}{(\boldsymbol{\eta}e^{Ct}\mathbf{1})^2}.
\sup_{x \in \mathbf{R}}\left( V_{\mathcal{D}_{d}}^{\mu_{N}}(x) + \log w(x) \right) \geq ``\sup_{x \in \mathbf{R}}\left( V_{\mathcal{D}_{d}}^{\mu_{N}}(x) + \log w(x) \right) & \geq -K_{N}^{\ast}= \sup_{x \in \mathbf{R}}\left( V_{\mathcal{D}_{d}}^{\mu_{N}^{\ast}}(x) + \log w(x) \right).
0 \leq -\frac{1}{2\gamma-1}\Delta_\phi J + \frac{(n+2-2\gamma)(n+2\gamma)}{(1-2\gamma)^2(n+1)}J^2, \quad, J = \frac{2\gamma-1}{4(n-1)(\gamma-1)}R_h, \quad, \lim_{y\to0}y^{m_1}\frac{\partial J}{\partial y}= -\frac{4\gamma(2\gamma-1)(\gamma-1)}{d_\gamma}Q_{2\gamma}.
S=\frac{i}{8\pi \alpha '}\int d\tau d\sigma (\tilde{\psi}\hat{\partial}\psi - \phi\hat{\partial}\phi - \rho\hat{\partial}\rho - \eta\hat{\partial}\eta + \theta\hat{\partial}\theta + \xi\hat{\partial}\xi + \chi\hat{\partial}\chi + \sigma_i\hat{\partial}\sigma_i + \sigma_{ij}\hat{\partial}\sigma_{ij}).
(\partial_t-L_t)(\chi^2h) = \left( 2\chi\partial_t\chi - 2\chi\Delta_{g_0(t)}\chi - 2|\nabla^{g_0(t)}\chi|^2 \right) h + \chi^2R_0[h] + \nabla^{g_0(t)}(\chi^2R_1[h]) - 2\chi \nabla^{g_0(t)}\chi * \nabla^{g_0(t)}h - 2\chi\nabla^{g_0(t)}\chi*R_1[h].
\tilde{f}\Bigl( \frac{i \varepsilon}{2 \pi}\Bigr) = \tilde{f}\Bigl( \frac{p i \varepsilon/2\pi + q}{ri\varepsilon/2\pi + s}\Bigr) =\tilde{f}\Bigl(\alpha + \frac{i h}{2 \pi m}\Bigr)
\mathbf{x}= \sum_{\mu = 1}^{8}\tilde{\sigma}_{\mu}x^{\mu}= \left( \begin{array}{c c}{{z_{2}}}&{{z_{1}}}\\{{- \bar{z}_{1}}}&{{\bar{z}_{2}}}\\ \end{array}\right) \, , \quad \mathbf{x}^{\prime}= \sum_{\mu = 1}^{8}\tilde{\sigma}_{\mu}^{\prime}x^{\mu}= \left( \begin{array}{c c}{{z_{4}}}&{{z_{3}}}\\{{- \bar{z}_{3}}}&{{\bar{z}_{4}}}\\ \end{array}\right) \, .
\smashoperator{\sum_{\substack{k_1,k_2,\dotsc,k_{2d},k_{2d+1}0\le k_1\le k_2\le\dotsb\le k_{2d}\le k_{2d+1}\le k}}}1 ~~~~~~~~~~= \binom{k+2d+1}{2d+1}= \frac{(k+1)(k+2)\dotsb(k+2d)(k+2d+1)}{(2d+1)!}<C_d^k,
{}_0\mathbf{F}_q(b;z)=\frac{1}{(2\pi i)^q}\int_{-\infty}^{(0+)}\cdots \int_{-\infty}^{(0+)}e^{\left(\sum_{j=1}^q w_j+\frac{z}{\prod_{j=1}^q w_j}\right)}\prod_{j=1}^q \frac{{\rm d}w_j}{w_j^{b_j}}
\left[ a\left( t\right) ,a^{\dag}\left( t\right) \right] =\left[\bar{\gamma}\left( t\right) a^{\dag}+\bar{\delta}\left( t\right)a,\gamma\left( t\right) a+\delta\left( t\right) a^{\dag}\right] & =\left( \left\vert \gamma\left( t\right) \right\vert^{2}-\left\vert\delta\left( t\right) \right\vert^{2}\right) I=I.
q^{-(\alpha_t, \xi)}[E_\xi, F_t]^* K_t= q^{-(\alpha_t, \xi)}(q^{(\alpha_t, \xi)}E_\xi E_t^* - E_t^* E_\xi)^*= E_t E_\xi^* - q^{-(\alpha_t, \xi)}E_\xi^* E_t.
\Lambda^{(h:h')}_{n,2,0}(\mathcal{W}_{k}(0) \setminus \mathcal S_k) = \frac{c\left(1+o(1) \right)}{n^{h+h'}\delta^4_n(k)}\cdot |\mathcal{W}_{k}(0) \setminus \mathcal S_k| \cdot |\mathcal{I}_h \times \mathcal{I}_{h'}|\cdot O(n^{-1}) =O(n^{-1})
\Psi_{V_0}=\mathcal{E}_{V_0}-\sum_{i=1}^2\dot{\mathfrak{q}}_i\sigma_i\partial_yR_i+\sum_{\substack{i,j=1, j\not=i}}^2\frac{a_{ij}\sigma_i}{(\mathfrak{q}_i-\mathfrak{q}_j)^3}\Lambda R_i+\sum_{i=1}^2\dot{\mathfrak{q}}_i(\mathcal{P}_i+\mathcal{Q}_i)
y' = \sigma_{x U(x) U^2(x) \cdots U^{k-1}(x)}(y) = \sigma_{kx}(y), U^{k-1}(x') = \tau_{y U(y) U^2(y) \cdots U^{l-1}(y)}U^{k-1}(x) = \tau_{ly}U^{k-1}(x),
\beta_1(z) :=\displaystyle \sum_{m=1}^{s_1}\frac{z^m}m, \beta_i(z) :=\displaystyle \beta_{i-1}(z)+\sum_{m=1}^{s_i}\frac{(1+(z-1)\exp(\beta_{i-1}(z)))^m}m, 2\le i\le q.
\mathcal{S}_{k}^{\operatorname*{Mw}}\left(\mbox{\boldmath$ \phi$}\right) =\mathcal{S}_{k}^{\operatorname*{Hh}}\left(\mbox{\boldmath$ \phi$}\right) +\frac{1}{k^{2}}\nabla\mathcal{S}_{k}^{\operatorname*{Hh}}\left(\operatorname{div}_{\Gamma}\mbox{\boldmath$ \phi$}\right) .
\liminf_{n\to\infty}P^\zeta_n( \sigma_0^\zeta > n ) \geq \lim_{n\to\infty}P^\zeta_n( \sigma_{n/2}^\zeta > n ) = P( \sigma_{1/2}^{Z^{(2-2\gamma)}}> 1 \mid Z_0^{(2-2\gamma)}= 1 ) > 0,
\lambda = (n-\tfrac{3}{2}+p,\: n-\tfrac{1}{2}-p, \: \underbrace{2pq+\tfrac{1}{2},\: \ldots,\: 2pq+\tfrac{1}{2}}_{2p}, \: \ldots, \: \underbrace{2p+\tfrac{1}{2},\: \ldots,\: 2p+\tfrac{1}{2}}_{2p}, \: \underbrace{\tfrac{1}{2}\: \ldots,\: \tfrac{1}{2}}_{p-1}),
\frac{2Li_{(k_1,...,k_r)}(1-(ab)^{-t})}{(a^{-t}+b^t)^r}= \sum\limits_{n=0}^{\infty}\mathbf{E}_{n}^{{(k_1,...,k_r)}}(a,b){t^{n}\over n!}& = e^{rt\ln a}\frac{2Li_{(k_1,...,k_r)}(1-e^{-t\ln ab})}{(1+e^{t\ln ab})^r}
\sum_{n=0}^\infty a_n\frac{x^n}{n!}=e^{a_1x}\cosh (b_0\sqrt{p}x), \sum_{n=0}^{\infty}b_n \dfrac{x^n}{n!}= b_0e^{a_1 x}\left(\cosh\left( b_0\sqrt{p}x \right)+ \frac{a_1}{b_0\sqrt{p}}\sinh\left( b_0\sqrt{p}x \right)\right).
\widetilde{a}(e,v) - \int_{{\cal E}_h^0}[e]\cdot\{\nabla_h v\}\, ds - \int_{{\cal E}_h^i}\beta \cdot[e] [\nabla_h v]\, ds &=\widetilde{a}(u_h-z, v) -\int_{{\cal E}_h^0}[u_h-z]\cdot\{\nabla_h v\}\, ds{}- \int_{{\cal E}_h^i}\beta \cdot[u_h-z] [\nabla_h v]\, ds + \int_{\Gamma_2}g(\lambda-\lambda_h)v\,ds,
\sigma_{E}\, \longleftrightarrow \, ( - 1 )^{\sum J + j}\: \: \left\{\begin{array}{c c c}{{J_{1}}}&{{j}}&{{J_{2}}}\\{{J_{3}}}&{{J_{4}}}&{{J_{5}}}\\ \end{array}\right\}\, \doteq \, ( - 1 )^{\sum J + j}\: \{6 j \}( \sigma_{E}) ,
\int_{|J^{1/2}\xi|\geq\log n}e^{-\frac{1}{4}\xi^{\rm T}J\xi}d\xi \;=\; |J|^{-1/2}\int_{|\xi|\geq\log(n)}e^{-|\xi|^2/4}d\xi\;\lesssim \;n^{d\alpha/2}\int_{r\geq \log(n)}e^{-r^2/4}r^{d-1}dr \lesssim n^{d\alpha/2}\int_{r\geq \log(n)}e^{-r^2/8}dr=\mathcal{O}\big(n^{d\alpha/2}e^{-\frac{1}{8}\log^2(n)}\big) \;=\;\mathcal{O}(n^{-1/2})
4 \left\|u\right\|_{2}^2\geq & x_+^T\left(A^T \Sigma^{-1}+\Sigma^{-1}A+\sum_{k=1}^m N_k^T\Sigma^{-1}N_{k}+\sum_{i, j=1}^v H^T_{i}\Sigma^{-1}H_{j}k_{ij}\right)x_++4x_+^T\Sigma^{-1}Bu.
S(n-i+1) - S(n) = \frac{1}{6}((n-i+1)^3 - (n-i+1) - n^3+n) & = \frac{1}{6}((n-i+1-n)((n-i+1)^2 + n(n-i+1) + n^2) + i - 1) & = \frac{1}{6}(-(i-1)(n^2 -2(i-1)n + (i-1)^2 + n^2 - (i-1)n + n^2) + i-1) & = \frac{1}{6}(-3(i-1)n^2 + 3(i-1)^2n - (i-1)^3 + i-1).
{\frac{1}{{\cal A}_{c}}}\int_{0}^{1}d \epsilon \, a_{5 / 2}( f_{\epsilon}, P_{\epsilon}) = \frac{1}{4 \pi^{2}z_{+}^{4}}\left[ - \frac{2 7}{1 2 8}( \ln z_{+}+ \lambda^{4}\ln z_{-}) + \frac{5 3}{5 1 2 0}( 1 + \lambda^{4}) \right] .
\mu_{H_r,\epsilon}(\rho) = \int_{\epsilon}^{\rho}\frac{(nH_rJ_{n,r,\epsilon}(\xi))^{\frac{1}{r}}}{\sqrt{\cosh^{\frac{2(n-r)}{r}}(\xi)-(nH_rJ_{n,r,\epsilon}(\xi))^{\frac{2}{r}}}}\,d\xi, \rho \geq \epsilon.
\max\{u \ge y\colon u \perp_y x\}= \max\{u \ge u\colon (u] \cap (x] \cap [y) \subseteq \{y\}\}= \max\{u \ge y\colon (u] \cap ((x] \cup (y]) \cap [y) \subseteq \{y\}\}= \max\{u\colon (u] \cap ((x] \cup (y]) \cap [y) = \{y\}\}.
\begin{bmatrix}p^t \gamma\end{bmatrix}\begin{bmatrix}Df(x,\lambda_1,\lambda_2) \bar{p}\bar{q}^t 0\end{bmatrix}=\begin{bmatrix}0 1\end{bmatrix}, \begin{bmatrix}Df(x,\lambda_1,\lambda_2) \bar{p}\bar{q}^t 0\end{bmatrix}\begin{bmatrix}q \gamma\end{bmatrix}=\begin{bmatrix}0 1\end{bmatrix}.
2 \dot{\Delta}_{P, n}^{( \alpha, \beta; \,\ell)}= (n + \alpha + \beta + 1) \Delta_{P, n- 1}^{( \alpha+ 1, \beta + 1; \ell)}+ \ell \cdot \Delta_{P, n}^{( \alpha + 1, \beta + 1; \ell-1)}.
\AA^Y(\Lambda v_l) = (\AA^Y \Lambda) v_l + \Lambda (\AA^Y v_l) + \sum_{i, j = 1}^n \rho_{ij}\partial_{y_i}\Lambda \partial_{y_j}v_l = (\AA^Y \Lambda) v_l + \Lambda (\AA^Y v_l) + \Lambda \sum_{i, j = 1}^n \rho_{ij}\partial_{y_i}[\log \Lambda] \partial_{y_j}v_l,
X := \big \{(q,n) : \| q \|_{L^{2}(0,T_{0};H^{1}(\Omega))}+ \| n \|_{L^{2}(0,T_{0};L^{2}(\Omega))}\leq R, \; 0 \leq n \leq 1 \Omega \times (0,T_{0}) \big \},
[S_{n-1}\partial_n\theta](h_1, \ldots, h_{n-1})(\xi) &=[S_{n-1}f](h_1, \ldots, h_{n-1})(\xi) &=\sum_{t_1,\ldots, t_{n-1}}f(g_{t_1},\ldots, g_{t_{n-1}})(\pi(\xi)|_{[t_1,\ldots, t_{n-1}]}).\quad(\mbox{def. of $S_{n-1}$})
S_{f}~ = ~ \int d^{4}x ~ h \psi_{f}^{\dagger}\left[ \left( \frac{1}{2}m v^{\mu}v_{\mu}+ V - \frac{1}{2 m}\partial^{\mu}\partial_{\mu}- \mu \right) - i \left( \frac{1}{2}\partial^{\mu}v_{\mu}+ v^{\mu}\partial_{\mu}\right) \right] \psi_{f}.
\psi_{n,k}(z) =O(z^{-N(n,k)})=O(z^{-Z(n,k-1)+Z(n,k)-1}), \psi_{n+1,k}(z) =O(z^{-N(n+1,k)})=O(z^{-Z(n+1,k-1)+Z(n+1,k)}),
\begin{array}{l}\Phi_{j}\bar{\Phi}_{i}=\delta_{ij}+qR_{kijl}\bar{\Phi}_{l}\Phi_{k}, \Phi_{l}\Phi_{k}=q^{-1}R_{jikl}\Phi_{j}\Phi_{i},\qquad i,j=1,2,...,N,\end{array}
c_{n=k+m,k,m}\left(C\mathcal{Z},\mathcal{Z}\right)=\begin{cases}\frac{2k+m-1}{m+1}{\frac{k+m-2}{2}\choose \frac{m-1}{2}}{\rm for\;}k\;{\rm odd}, 2{\frac{k+m-1}{2}\choose \frac{m+1}{2}}{\rm for\;}k\;{\rm even.}\end{cases}
S =\{s, 2s, ..., l_0s\}&\cup \{(l_0+2)s,(l_0+3)s,\ldots, (l_0+l_1+1)s\}&\cup \{(l_0+l_1+3)s,(l_0+l_1+4)s,...,(l_0+l_1+l_2+2)s \}
1-\xi_q^k = 1-\left(1-\lambda\right)^k=\lambda k \sum_{j=0}^{k-1}\binom{k-1}j\frac{(-\lambda)^j}{j+1}= \lambda k \Big( 1-\frac{k-1}{2}\lambda + \frac{(k-1)(k-2)}{6}\lambda^2-\frac{(k-1)(k-2)(k-3)}{24}\lambda^3+\ldots \,.
\langle \psi_{1}^{I_{1}}( x_{1}) \cdots \psi_{n}^{I_{n}}( x_{n}) \rangle = e^{i ( \kappa_{1}+ \cdots + \kappa_{n}) \Omega}\langle \psi_{1}^{I_{1}}( x_{1}) \cdots \psi_{n}^{I_{n}}( x_{n}) \rangle \, ,
{\cal P}\left[ \begin{array}{c}u v \end{array}\right] =\left[ \begin{array}{c}v u \end{array}\right],{\cal T}\left[ \begin{array}{c}u(t) v(t) \end{array}\right] =\left[ \begin{array}{c}\bar{u}(-t) \bar{v}(-t) \end{array}\right].
\begin{matrix}T_{A}&=&0&:&csc^{2}\frac{B}{2}&:&csc^{2}\frac{C}{2}\\T_{B}&=&csc^{2}\frac{A}{2}&:&0&:&csc^{2}\frac{C}{2}\\T_{C}&=&csc^{2}\frac{A}{2}&:&csc^{2}\frac{B}{2}&:&0\end{matrix}
\Gamma (h_{\gamma_{-}})\left( \rho \right) =f_{\Phi +\Phi^{-\mathfrak{b}_{-},\gamma_{-}}}\left( \rho \right) +\int_{\mathbb{S}}\left\vert \rho\left( \mathfrak{e}_{\Psi}\right) \right\vert^{2}\mathfrak{a}_{\mathfrak{b}_{+}}\left( \mathrm{d}\Psi \right) \ ,\rho \in E_{1}\ .
{\cal S}_\psi=i\int d^4x \int d\phi~[{\it det}~V]~\bar{\psi}~[\gamma^{a}v^\mu_a~(\partial_\mu-\frac{i}{2}G_{LN}\sigma^{ab}v^\nu_a\partial_\mu v^\lambda_b\delta^N_\nu\delta^L_\lambda-G_{AD}\sigma^{ab}v^\beta_a v^\delta_b{\bar{\Gamma}_{MB}}^A\delta^M_\mu\delta^B_\beta\delta^D_\delta)]~\psi~\delta(\phi-\pi)
\Delta_{N}^{+}= \frac{\pi}{\Gamma^{2}( 1 - \epsilon )}\left( 2 \int_{0}^{a}\frac{d r}{a}\left( \frac{r}{a}\right)^{2 N + 1}e^{2 e \varphi}+ \frac{1}{\epsilon - 1}\right) \left( \frac{k a}{2}\right)^{2 - 2 \epsilon}+ O ( k a )^{4 - 4 \epsilon},
\sum_{l=1}^{j+1}Z_l+\sum_{l=j+2}^kB_l^{(j+1)}=\prod_{l=1}^{j+1}y\big(\bigl\{X_{ni}^{(l)}\bigr\}\big) =\Bigg(\sum_{l=1}^{j}Z_l+\sum_{l=j+1}^kB_l^{(j)}\Bigg)\Bigg(Z_{j+1}+\sum_{l=j+2}^kA_l\Bigg).
\langle V({\bf x})V({\bf y})\rangle=gM^2\int\frac{d^2p}{(2\pi)^2}\,\frac{e^{i{\bf p}\cdot({\bf x}-{\bf y})}}{p^2+M^2}=\frac{gM^2}{4\pi}\int_0^{\infty}\frac{ds}{s}\,e^{-M^2s-({\bf x}-{\bf y})^2/4s}\; .
||{\tilde H}_{t,M}||_{L^2}^2 =||{\tilde H}_{t,M}-{\tilde H}_t||_{L^2}^2+||{\tilde H}_t||_{L^2}^2 =||H_{t,M}-H_t||_{L^2}^2+||{\tilde H}_t||_{L^2}^2 \leq ||H_{t,M}-H_t||_{L^2}^2+||H_t||_{L^2}^2
I = -{\frac{1}{1 6 \pi G_{5}}}\int d^{5}x \sqrt{- g^{5}}\left[ R +{\frac{1 2}{l^{2}}}-{l^{2}}F^{2}-{\frac{l^{3}}{6 \sqrt{3}}}\epsilon^{\mu \alpha \beta \gamma \delta}A_{\mu}F_{\alpha \beta}F_{\gamma \delta}\right] .
{\displaystyle 2\mathrm{i}(\partial \theta^{k}_{\Psi},\overline{\theta}_{\Psi}^{k})-B\left(\overline{\mathbf{A}}^{k}_{h};\overline{\theta}_{\Psi}^{k},\overline{\theta}_{\Psi}^{k}\right) =I_1^{(k)}+I_2^{(k)}+I_3^{(k)}+I_4^{(k)},}
\hat{\rm v}^a =\hat{\rm u}^a, \hat{E}^{\rm V}_a(R^\mu) = 0, v^i =\dot{x}^i,{X}^{\rm V}_i(R^\mu)=p_i, \dot{p}_i ={X}^{\rm C}_i(R^\mu) - \mu_a B^a_{ij}v^j.
A^{-1}\langle\xi\rangle^{-2}\le \pm x +1 \le A \langle\xi\rangle^{-2}\, , \xi \rightarrow +\infty A^{-1}\langle\xi\rangle^{-2}\le \mp x +1 \le A \langle\xi\rangle^{-2}\, , \xi \rightarrow -\infty
C_{hk\ell}(N) = e(h/k)^\ell \operatornamewithlimits{Res}_{z=0}\frac{e^z(e^z-1)^{\ell-1}}{(1-e(h/k) e^z)(1-e(h/k)^2 e^{2z}) \cdots(1-e(h/k)^N e^{N z})}.
Z = \int D \varphi^{i}J ( \varphi ) \exp \left\{i \int d x \, \frac{1}{2}\partial_{\mu}\varphi^{i}g_{i j}\partial_{\mu}\varphi^{j}\right\}\equiv \int D \varphi^{i}J ( \varphi ) \exp \left\{i S_{\mathrm{c h}}( \varphi ) \right\}.
\|\mathcal{R}\|^2\|f\|_{\dot{H}^s}^2 - \|\mathcal{R}f\|_{L^2(\mathbb{S}^{n-1})}^2 =\|\mathcal{R}\|^2(\|v\|_{\dot{H}^s}^2+\|f_\ast\|_{\dot{H}^s}^2) - \|\mathcal{R}(v+f_\ast)\|_{L^2(\mathbb{S}^{n-1})}^2 & \leq \|\mathcal{R}\|^2\|v\|_{\dot{H}^s}^2-2\mathrm{Re}\langle\mathcal{R}v,\mathcal{R}f_\ast\rangle_{L^2}.
-2 \int_{0}^{\frac{t}{2}}\frac{\sin(u)(b-1) du}{t^{3}u \log^{b}(\frac{t}{u})}= \frac{-2 (b-1)}{t^{3}}\frac{\pi}{2}\left(\frac{1}{\log^{b}(t)}+ O\left(\frac{1}{\log^{b+1}(t)}\right)\right)
Q^H \left[\begin{array}{cc}x \beta +\j z \beta - \j z y \end{array}\right]Q= Q^H \left[\begin{array}{cc}x \j z - \j z y \end{array}\right]Q + \left[\begin{array}{cc}0 \beta \beta 0 \end{array}\right]= \left[\begin{array}{cc}\alpha \beta \beta \gamma \end{array}\right].
0=\delta S=\frac{1}{2\kappa}\frac{\delta R}{\delta g^{\mu\nu}}+\frac{R}{2\kappa}\frac{1}{\sqrt{-g}}\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}}-\frac{\Lambda}{\kappa}\frac{1}{\sqrt{-g}}\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}}+\frac{\delta L_{M}}{\delta g^{\mu\nu}}+\frac{L_{M}}{\sqrt{-g}}\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}}
\widehat{Y}_{i}^{[K,L]}(qx) =\biggl\{1 + \frac{\big(1 - q^{\lambda}\big)b_{i}}{x - b_{i}}\biggr\}\widehat{Y}_{i}^{[K,L]}(x) \hphantom{\widehat{Y}_{i}^{[K,L]}(qx) =}{}+ \frac{(1-q) x}{x - b_{i}}\sum_{n=K}^{L}(P_{\lambda}(x, s/q) - P_{\lambda}(x, s)) Y(s)|_{s= q^n \xi}.
\left[{\mathcal J}_{\ell}^{0}\left( t , \underline{{{x}}}\right) , ~{\mathcal J}_{\ell}^{0}\left( t , \underline{{{y}}}\right) \right] \; = \; \frac{i \, \sigma_{H}}{2 \pi}\; \delta^{\prime}\left( \underline{{{x}}}- \underline{{{y}}}\right) ~ ,
T_{\mu, \Phi, \delta}(f,g)(x) = \iint_{\max(|x-y|, |x-z|) > 2\Phi(x)}K_{\Phi}(x,y,z) f(y) g(z)\,d\mu(y)\,d\mu(z) &+ \iint_{\delta < \max(|x-y|, |x-z|) \le 2\Phi(x)}K_{\Phi}(x,y,z) f(y) g(z)\,d\mu(y)\,d\mu(z).
\frac{4}{3z^3}\left[4z^2 +(z^3-3z)\log\left(\frac{1+z}{1-z}\right) -2\log\left((1+z)(1-z)\right)\right] = 8\sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)(2k+3)(k+2)}
\begin{aligned}\ m_{1,\ell}([a_1\otimes\dots\otimes a_n], y_1\otimes\dots \otimes y_\ell)= &\sum_{\substack{1\le i_1<i_2<\dots<i_\ell\le n}}\pm[a_1\otimes\dots\otimes a_{i_1-1}\otimes (a_{i_1}\lozenge y_1)\otimes a_{i_1+1}\otimes\dots \otimes a_{i_\ell-1}\otimes (a_{i_\ell}\lozenge y_\ell)\otimes a_{i_\ell+1}\otimes\dots\otimes a_n]\end{aligned}
\left({\sum\limits_{i = 1}^k{\sum\limits_{p = 1}^n{{{({\lambda_{k + 1}}-{\lambda_i})}^{2\alpha - \beta - 1}}\left({\left\|{{X_p}{u_i}}\right\|_{{L^2}}^2 + \left\|{{Y_p}{u_i}}\right\|_{{L^2}}^2}\right)}}}\right) &= \left({\sum\limits_{i = 1}^k{{{({\lambda_{k + 1}}-{\lambda_i})}^{2\alpha - \beta - 1}}\lambda_i^{\frac{1}{2}}}}\right).
C \geq \sum_{a \in A_t}(r - d_H(a)) + \sum_{b \in B_s}(r - d_H(b)) \geq \sum_{a \in A_t}(r - d_H(a)) + |B_s|(r - \Delta) \geq \sum_{a \in A_t}(r - d_H(a)) + 2|A_s|\times \frac{r}{2}\geq \sum_{a\in A}(r - d_H(a)) = V.
R_{u,v}^{J,q}(q)= R_{us_j,vs_j}^{J,q}(q) begin{align*}5pt]= (-1)^{\ell(vs_j)-\ell(us_j)}(1-q)^2 \prod_{k\in D(us_j,vs_j)}\left(1-q^{a_k(us_j,vs_j)}\right) begin{align*}5pt]= (-1)^{\ell(v)-\ell(u)}(1-q)^2\frac{1-q^{a_{j+1}(us_j,vs_j)}}{1-q^{a_{j}(u,v)}}\prod_{k\in D(u,v)}\left(1-q^{a_k(u,v)}\right),
\langle\partial_t u,v\rangle +\int_{Q_T}\mathsf{A}\nabla u\cdot \nabla v\mathrm{dx}\mathrm{dt}+\int_{\Sigma_T}b(u)u v\mathrm{ds}\mathrm{dt}= =\int_{Q_T}u{\bf E}\cdot \nabla v\mathrm{dx}\mathrm{dt}+\int_{\Sigma_T}h v\mathrm{ds}\mathrm{dt},
{\cal S}[ \psi ] = \int d^{5}x \sqrt{g}\left\{\frac{i}{2}\left[ \overline{{\psi}}\tilde{\gamma}^{\hat{\mu}}\tilde{\nabla}_{\hat{\mu}}\psi - ( \overline{{\tilde{\nabla}_{\hat{\mu}}\psi}}) \tilde{\gamma}^{\hat{\mu}}\psi \right] - M \overline{{\psi}}\psi \right\}
ds^{2}=-\frac{L^{2}-(M^{2}-Q^{2})}{(L+M)^{2}}dt^{2}+\frac{(L+M)^{2}}{l_{+}l_{-}}(d\rho^{2}+dz^{2})+\frac{(L+M)^{2}}{L^{2}-(M^{2}-Q^{2})}\rho^{2}d\phi^{2},
\begin{aligned}\mathcal{C}^\perp(\lambda) =\bigoplus_{N^i\in\mathbb{Z}}\bigoplus_{w\in S_q}\mathcal{F}^{\perp}\left(P_{\lambda^w-pp'N^ie_i},\lambda^w-pp'N^ie_i\right)\ , P_{\lambda^w-pp'N^ie_i}^2 =\frac{q-2}{12}-\frac{1}{pp'}\left(\lambda^w-pp'N^ie_i\right)^2\ .\end{aligned}
a_2=2a_1-a_0,\; a_3=\left( \frac{\nu a_0-1}{2}\right) (2a_1-a_0) -a_1, \; a_4=\left( \frac{\nu a_0-1}{2}\right) (2a_1-a_0) -a_0, d_1=\left( \frac{\nu a_0-1}{2}\right) (2a_1-a_0),\; d_2=\left( \frac{\nu a_0+1}{2}\right) (2a_1-a_0) -a_1,\; \mbox{where}\; \gcd\left(\frac{\nu a_0-3}{2},a_1-a_0\right) =1.
E= \{w\in C((\bar t,\infty); L^6_\sigma(\Omega)\cap L^\infty(\Omega));\,\nabla w\in C((\bar t,\infty); L^3(\Omega)), &\qquad\qquad\|w\|_E:=\sup_{t\in (\bar t,\infty)}\phi_w(t)<\infty,\,\lim_{t\to\bar t+0}\phi_w(t)=0\}
(\nabla_A^* \nabla_A+2)(\phi_A-\phi_B) = 2(F_B^+ -F_A^+) + 2(F_B^+\cdot\phi_B - F_A^+\cdot\phi_A) + 2\left\{(d_B^*\phi_B\wedge d_B^*\phi_B)^+ -(d_A\phi_A^*\wedge d_A^*\phi_A)^+\right\}+ a*\nabla_B\phi_B + (\nabla_A a)*\phi_B + a*a*\phi_B.
B_{i,j}=\begin{pmatrix}a_{1,4}a_{1,5}a_{1,6}a_{1,j}a_{2,4}a_{2,5}a_{2,6}a_{2,j}a_{3,4}a_{3,5}a_{3,6}a_{3,j}a_{i,4}a_{i,5}a_{i,6}a_{i,j}\end{pmatrix}=\begin{pmatrix}\eta_1^{T}\eta_2^{T}\eta_3^{T}\eta_4^{T}\end{pmatrix}.
\tilde M_{F_s,F_r'}(\{\sigma_q\})=\int\limits_{C_{v_r}}\Phi_{\sigma,F_s}^{(0)}(z)\frac{dz}{2\pi i}P_{F_r'}(z)-\frac{[1-(-1)^{2l_{1_r}}]}{2}\int\limits_{\tilde C_r}\Phi_{\sigma,F_s}^{(0)}(z)\frac{dz}{2\pi i}P_{F_r'}^{(a)}(z).
(Z_{1/\Omega}\circ H)g(t,\nu) & \hskip.1in = \Omega\,\sum_{q=0}^{L-1}c_{-q}\,\sum_{m=0}^{L-1}\sum_{k}\eta^{QP}_H(t+qT,\nu+m\Omega) & \hskip1in e^{-2\pi i\nu qT}\,e^{-2\pi i\nu mq/L}
\psi(y_1\dots y_{t-1}y_{t}) = \psi(y_1\dots y_{t-1})\psi(y_{t}) = b(y_1\dots y_{t-1})p_{\gamma_{t-1}}b(y_{t})p_{\gamma_{t}}= b(y_1\dots y_{t-1})(b(y_{t}))_{\gamma_{t-1}\gamma_{t}^{-1}}p_{\gamma_{t}}=b(y_1\dots y_t)p_{\gamma_t}
A(j_1,k_1)=\begin{cases}2\overline{\mathbf{a}}_\mathbf{1}\mathbf{a_2}(2c)^{2t_1+k_1+1}t_1! (k_1+t_1)! \frac{\pi^{m/2+1}}{\Gamma(m/2)}j_1=2t_1, 2\overline{\mathbf{a}}_\mathbf{1}\mathbf{a_2}(2c)^{2t_1+k_1+2}t_1! (k_1+t_1+1)! \frac{\pi^{m/2+1}}{\Gamma(m/2)}j_1=2t_1+1.\end{cases}
\| Q\|_2^2= \lim_{n\to \infty}\| u_n \|_2^2 \ge \sum_{j=1}^l \| U^j \|_2^2, \| \nabla Q \|_2^2= \lim_{n \to \infty}\| \nabla u_n \|_2^2 \ge \sum_{j=1}^l \| \nabla U^j \|_2^2.
\big(\mathfrak{N}^{\zeta}P_{\lambda}\big)(z_1,\ldots, z_N) = \prod_{1\leq i\leq N}\frac{\big(q^{\lambda_i}t^{N+1-i}\zeta;q\big)_{\infty}}{\big(q^{\lambda_i}t^{N-i}\zeta;q\big)_{\infty}}P_{\lambda}(z_1,\ldots, z_N).
h(v, lp+x)h(u, lq+y)e^{-ax}=h(v, lp+x, a)e^{\frac{va^2}{2}+alp}h(u, lq+y) &=e^{\frac{va^2}{2}+\frac{u}{2}\left(\frac{ap}{q}\right)^2-ay\frac{p}{q}}h(v, lp+x, a)h\left(u, lq+y,-apq^{-1}\right)
\int_{R^{8 + 1}}{\hat{C}}_{{\hat{\mu}}_{1}{\hat{\mu}}_{2}{\hat{\mu}}_{3}}D{\hat{X}}^{{\hat{\mu}}_{1}}D{\hat{X}}^{{\hat{\mu}}_{2}}D{\hat{X}}^{{\hat{\mu}}_{3}}\wedge \mathrm{T r}({\hat{F}}\wedge{\hat{F}}\wedge{\hat{F}}) \, .
\left\{{\frac{\partial^{2}}{\partial r^{2}}}+{\frac{5}{r}}{\frac{\partial}{\partial r}}+ \tilde{\omega}^{2}\left( 1 +{\frac{\tilde{R}_{\frac{\pi}{2}}^{4}}{r^{4}}}\right) -{\frac{2 R_{\frac{\pi}{2}}^{8}}{r^{1 0}f^{2}}}\right\}\hat{\phi}= 0 .
J_{\alpha}= \frac{\pi}{2 e}\frac{\delta S}{\delta \bar{\chi}^{\alpha}}= \rho^{\beta}\rho_{\alpha}\bar{\Psi}^{\mu}\partial_{\beta}X_{\mu}- \frac{1}{4}\bar{\Psi}^{\mu}\Psi_{\mu}\rho_{\alpha}\rho^{\beta}\chi_{\beta}= 0
\begin{array}{c}d_{C_1Cn}(M, \, Y_q(\Omega)) \underset{\mathfrak{Z}_0}{\lesssim}\sum \limits_{t=0}^{\hat t(n)}\sum \limits_{m=0}^\infty d_{k_{t,m}}(W_{t,m}, \, l_q^{\nu_{t,m}})+ +\sum \limits_{t=0}^{\hat t(n)}d_{s_t}(2^{‐\alpha_*k_*t}B_{p_0}^{\nu_{t,0}}, \, l_q^{\nu_{t,0}}) + \sup_{f\in M}\|f\|_{Y_q(\tilde \Omega_{\hat t(n)+1})}.\end{array}
\pi_{\mu\nu}= -\frac{1}{4}T_{\mu\alpha}T^{\;\alpha}_{\nu}+\frac{1}{12}TT_{\mu\nu}+\frac{1}{8}g_{\mu\nu}T^{\alpha}_{\;\beta}T^{\;\beta}_{\alpha}-\frac{1}{24}g_{\mu\nu}T^2
H_{2,2}= U(N_{a,1}+N_{a,2}-N_{b,1}-N_{b,1})^2+\mu(N_{a,1}+N_{a,2}-N_{b,1}-N_{b,2}) &\quad+t_{1,1}(a_{1}b_{1}^\dagger+a_{1}^\dagger b_{1})+t_{1,2}(a_{1}b_{2}^\dagger+a_{1}^\dagger b_{2}) &\quad+ t_{2,1}(a_{2}b_{1}^\dagger+a_{2}^\dagger b_{1})+t_{2,2}(a_{2}b_{2}^\dagger+a_{2}^\dagger b_{2})
\frac{\partial}{\partial z_{\rho}}\langle J_{\mu}(x)J_{\nu}(y)J_{\rho}(z)\rangle_{UV}=-\frac{1}{12\pi^{2}}\epsilon_{\mu \nu\rho \sigma}\frac{\partial}{\partial x_{\rho}}\frac{\partial}{\partial y_{\sigma}}\delta(x-z)\delta (y-z)\equiv\frac{16}{9}{\cal A}_{\mu \nu}(x,y,z)
t_a : E_{A/v^nI}= E_{A/v^nI}\underset{\mathcal{M}_{A/v^nI}}{\times}\mathcal{M}_{A/v^nI}\xrightarrow{\mathrm{id}\times\mathrm{lev}_a}E_{A/v^nI}\underset{\mathcal{M}_{A/v^nI}}{\times}E_{A/v^nI}\xrightarrow{}E_{A/v^nI}
\langle v(t),z \rangle + \int_0^t \langle A v(s) , z\rangle \,{\rm d}s+\int_0^t \langle B(v(s),v(s)), z\rangle\,{\rm d}s=\langle v_0,z\rangle+\langle \int_0^tG(v(s))\,{\rm d}W(s),z\rangle.
|f_j(\tau_k)f_k(\tau_j)| \leq \max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)|\max_{1\leq j,k \leq n, j\neq k}|f_k(\tau_j)| =\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)|\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)|=\left(\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)|\right)^2.
|(w)_*(W)|^2 \le \left(\frac{\sin(1-\eta)R}{\sin R}\right)^2 (|(u_{\frac{1}{2}})*(W)|^2 - |\nabla_W R|^2) + |\nabla_W ((1-\eta)R)|^2 &= \left( \cos \frac{d(x)}{2}\right)^2 (|(u_{\frac{1}{2}})_*(W)|^2 - |\nabla_W R|^2) + |\nabla_W ((1-\eta)R)|^2.
(g_m+\Delta_i) \circ u=g_m \circ u+\Delta_i \circ u \equiv g_m \circ u+J_0(u)^{m+1}\Delta_i~ (m+2), u \circ (g_m+\Delta_i) \equiv u \circ g_m + J(u)(g_m) \Delta_i \equiv u \circ g_m+ J_0(u) \Delta_i~ (m+2),
\widehat{y}_0(x) =(1-q)\alpha^{-\mu}q^{\mu}\frac{\big(q, q^{\mu+1}\beta/\alpha;q\big)_{\infty}}{(q\beta/\alpha, q^{\mu};q)_{\infty}}\,_2\phi_1 \biggl(\!\! \begin{array}{c}q^{\lambda},q^{\mu}q^{\mu+1}\beta/\alpha \end{array}\! ;q, \frac{q^2}{\alpha x}\biggr).
f_k(\tau_r)=\dfrac{\eta(2^{r+2}i)^{8k-2}\eta(2^{r+3}i)^4}{\eta(2^{r+1}i)^{4k}}=\dfrac{a^{4k+2}}{2^{2k(r+4)+r+5}}\cdot e^{\frac{\pi(2k+1)}{3}-2^{r+1}(k+1)\pi-\frac{2k+1}{\pi}\sum_{m=1}^{r+1}2^m L_m-\frac{2^{r+2}(4k+1)}{\pi}L_{r+2}-\frac{2^{r+4}}{\pi}L_{r+3}}.
\left.\partial_{w_{aa}}\left( G\left(W,Z\right)\right)\right\vert_{Z= O_{q\times N}\atop{W= O_{q\times q}}}=\begin{pmatrix}0 \dots 0 \dots 0 \vdots \ddots \vdots \ddots \vdots 0 \dots 0 \dots 0 \vdots \ddots \vdots \ddots \vdots 0 \dots 0 \dots 0 \end{pmatrix},