harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta S_{l i n}= \int d^{6}x \, \left( \frac 1{1 2}H_{\mu \nu \rho}^{*}- \frac 2 3 h_{\mu \nu \rho}^{+}\right) \delta H^{\mu \nu \rho}+ \frac 1{2 \sqrt{u^{2}}}\epsilon^{\mu \nu \rho \sigma \tau \phi}H_{\mu \nu}^{-}H_{\rho \sigma}^{-}v_{\tau}\partial_{\phi}\delta a \ .
{1 \over \omega}\left( -{9\over{20}}-{{z_0^2}\over l^2}\right)<{1 \over{32 \omega^2 l^2}}\left( -32 \omega z_0^2 + 9l^2+ 3 \sqrt{-64\omega z_0^2 l^2 + 9l^4 - 64 l^2 z_0^2 \omega^2}\right) \, ,
f_{n_1,n_2, \cdots, n_r}(\eta_1, \eta_2, \cdots, \eta_r) = \sqrt{\frac{(k-n_1-n_2\cdots - n_r)!}{k!n_1!n_2!\cdots n_r!}}{{\eta_1}}^{n_1}{{\eta_2}}^{n_2}\cdots{{\eta_r}}^{n_r}.
\begin{aligned}\tau_h(\boldsymbol{D}^{-\mu}u)=\boldsymbol{D}^{-\mu}(\tau_hu) ,\tau_h(\boldsymbol{D}^{-\mu*}u)=\boldsymbol{D}^{-\mu*}(\tau_hu) \Pi_\kappa(\boldsymbol{D}^{-\mu}u)=\kappa^\mu \boldsymbol{D}^{-\mu}(\Pi_\kappa u) ,\Pi_\kappa(\boldsymbol{D}^{-\mu*}u)=\kappa^\mu \boldsymbol{D}^{-\mu*}(\Pi_\kappa u).\end{aligned}
G(\mathbf{v};\tau,\lambda):= \int\limits_{\mathbb{R}^{3}}|\mathbf{v}- \mathbf{v}_{\ast}|^{-\lambda}\exp(-\tau{{\mathbf{v}_{\ast}}}^{2}) d\mathbf{v}_{\ast}; 0\leq \lambda <2, 0<\tau <\infty,
K_W(Z) =\frac{1}{W^{\dagger_3}}log\left(\frac{1}{1-W^{\dagger_3}Z}\right) \ =e_1\frac{1}{\overline{w}_1z_1}log\left(\frac{1}{1-\overline{w}_1z_1}\right)+e_2\frac{1}{\overline{w}_2z_2}log\left(\frac{1}{1-\overline{w}_2z_2}\right) =e_1K_{w_1}(z_1)+e_2K_{w_2}(z_2),
\frac{\delta}{\delta \{D_{\alpha}\}_{n}B_{\mu \nu}^{c}}\Delta_{0 b}^{\rho}( x ) = \{D_{\alpha}\}_{n}g F_{\mu \nu}^{c}( x ) X_{\rho}( A , g F )^{-}+ \{D_{\alpha}\}_{n}\epsilon^{\mu \nu \lambda \eta}g F_{\lambda \eta}^{c}( x ) X_{\rho}( A , g F )^{+}
\mathrm{\s l ~ S}_{E}={\frac{1}{\lambda}}\int d^{3}x \, \varepsilon^{\mu \nu \rho}\, E_{\mu}^{\underline{{{\alpha}}}}\, \omega_{\nu \rho}^{\underline{{{\beta}}}}\, \eta_{\underline{{{\alpha}}}\underline{{{\beta}}}},
\int_M \rho^{-\delta-\frac{n}{p}+j}|\hat\nabla^j u|dV_{\hat g}=\sum_{i=1}^p\int_{f_i^{-1}(E_i)}f_i^\ast\rho^{-\delta-\frac{n}{p}+j}|\partial_j f_i^\ast u|\sqrt{\det \hat g_{ij}}d\mathbf{x}
\bigg( \sum_{k=-\infty}^{M-1}2^{-k\frac{q}{p}}B(x_k, x_{k+1})^q a_k^q \bigg)^{\frac{1}{q}}\leq C'' \bigg(\sum_{k=-\infty}^{M-1}a_k^p \bigg)^{\frac{1}{p}}
\mathcal{I}^{\alpha, \beta, \bar{\alpha},\bar{\beta}}_{n}\equiv n^{\lambda}\int_{0}^{\infty}dt \L_{n}^{\alpha}(2t) t^{\beta}\exp(-t) \int_{0}^{t}du \L_{n}^{\bar{\alpha}}(2u) u^{\bar{\beta}}\exp(-u),
\zeta_{\mathrm int}(s) = - \zeta_{1/2 - |\alpha|}^{\mathrm (int)}(s) + \sum_{\nu = 1/2}^{\infty}\left[ \zeta_{\nu+\alpha}^{\mathrm (int)}(s) +\zeta_{\nu-\alpha}^{\mathrm (int)}(s) \right]
{\epsilon_j(\theta)\over T}=\phi_{3}(\theta-\theta')\star \ln\left(1+e^{\epsilon_j(\theta')/T}\right)-\sum_{l=0}^{t-3}\left(\delta_{j,l+1}+\delta_{j,l-1}\right)\phi_{3}(\theta-\theta')\star\ln\left(1+e^{-\epsilon_l(\theta')/T}\right)
\hat{x}_{1 2}= x_{1 2}+{\frac{2 i}{( 1 2 )}}\left[ ( 1^{-}2 ) \theta_{1}^{+}\bar{\theta}_{1}^{+}- ( 1 2^{-}) \theta_{2}^{+}\bar{\theta}_{2}^{+}+ \theta_{1}^{+}\bar{\theta}_{2}^{+}+ \theta_{2}^{+}\bar{\theta}_{1}^{+}\right] \ .
{\hat{\cal H}}\, \psi ({\bf X}) = \int d^{3}x \left[ \,\frac{g^2}{2}{\hat X}^{ax}{\hat X}^{ax}+ \, \frac{1}{4g^2}\, (\partial_{[a}{\hat P}_{b]x})^2 \right] \, \psi({\bf X})
\widetilde{\Theta}_{g_\alpha,\rho}(F) = \sum_{j=1}^m \int_{2^{k_{j-1}}}^{2^{k_j}}\int_{\mathbb{R}^4}F(y,x') F(x,x')F(y,y')F(x,y') begin{align*}-1ex] \big(-t\partial_t(g_{\alpha t}(x'-y'))\big) \rho_{2^{k_{j-1}}}(x-y) \,dx dy dx' dy' \frac{dt}{t}.
\limsup\limits_{\delta\rightarrow0}\limsup_{n\rightarrow\infty}\limsup_{t\rightarrow\infty}\|P_\delta(W_{n,m;\textrm{far}})e^{-itH_0}\varphi\| \leq \limsup\limits_{\delta\rightarrow0}\limsup_{n\rightarrow\infty}\limsup_{t\rightarrow \infty}\|P_\delta(W_{n,m;\textrm{far}})e^{-itH}\psi\|=0
= \; C \; \times \operatorname{e x p}\left( \sum_{b , b^{\prime}= 1}^{N}\sum_{j = 1}^{n_{b}}\sum_{j^{\prime}= 1}^{m_{b^{\prime}}}\tilde{\tilde{V}}( x_{j}^{( b )}- y_{j^{\prime}}^{( b^{\prime})}) \right)
|\cos t - g_n(t)| = |(2\cos^2(\tfrac{t}{2})-1)-(2\cos(\tfrac{t}{2})g_{n-1}(\tfrac{t}{2})-1)| =2\cos(\tfrac{t}{2})|\cos(\tfrac{t}{2}) - g_{n-1}(\tfrac{t}{2})|=\dots &= 2^n\cos\tfrac{t}{2}\cos\tfrac{t}{4}\dots\cos(\tfrac{t}{2^{n-1}})(1-\cos\tfrac{t}{2^{n}})\leq 2^n(1-\cos\tfrac{t}{2^{n}}).
\widetilde{\mathbf{u}}_n\cdot\nabla\widetilde{c}_n=(1+|\widetilde{\mathbf{u}}_n|^{1/2})\frac{\widetilde{\mathbf{u}}_n}{1+|\widetilde{\mathbf{u}}_n|^{1/2}}\cdot\nabla\widetilde{c}_n
\int d x \left( \,{\frac{1}{2}}\, \tilde{g}^{i j}{\frac{\delta^{2}\Psi}{\delta \phi^{i}\delta \phi^{j}}}\, + \,{\frac{1}{2}}\, g_{i j}\partial_{x}\phi^{i}\partial_{x}\phi^{j}\Psi \, \right) \, = \, E \Psi \, .
\begin{array}{r@{}c@{}ll}\partial_t u_n -{\rm div}\, (a(x)\nabla u_n){}={}0 \mbox{in}(0,\infty ) \times \Omega_n , begin{align*}5pt] u_n = 0 \mbox{on}(0,\infty ) \times \partial\Omega_n , begin{align*}5pt] u_n (0,\cdot ) = u_{0,n}\mbox{in}\Omega_n ,\end{array}
C_{k-1}(N)\begin{bmatrix}B_{k-1}B_{k-2}\vdots B_0 \end{bmatrix}=\begin{bmatrix}Q_{n+k-1}Q_{n+k-2}\vdots Q_0\end{bmatrix}-A_{21}(N)\begin{bmatrix}B_{b}B_{b-1}\vdots B_k \end{bmatrix},
[p_i, x_j] = \delta_{ij}\, e^{- 2p_0/\kappa}-\frac{1}{\kappa^2}\left(\vec{p}\,{}^{2}\, \delta_{ij}- 2 p_{i}p_{j}\right),\quad [p_0, x_i] = -\frac{2}\kappa\, p_i
\tilde{h}_d(x) = x\log d + \log\left(\frac{\Gamma(x)}{\Gamma\left(x+\frac{1}{2}\right)\Gamma\left(x+\frac{d-1}{2}\right)}\right)+\log\left(\frac{2^{d-2}\Gamma\left(\frac{d}{2}\right)^2}{\sqrt{\pi}d^{\frac{d-1}{2}}}\right).
\begin{array}{l c r}{{D^{+ +}= u^{+ \alpha}{\frac{\partial}{\partial u^{- \alpha}}}; \qquad D^{--}= u^{- \alpha}{\frac{\partial}{\partial u^{+ \alpha}}}}}\\{{2 D^{+ +}= [ D^{0}, D^{+ +}] ; \qquad - 2 D^{--}= [ D^{0}, D^{--}]}}\\{{D^{0}= [ D^{+ +}, D^{--}] = u^{+ \alpha}{\frac{\partial}{\partial u^{+ \alpha}}}- u^{- \alpha}{\frac{\partial}{\partial u^{- \alpha}}}}}\\ \end{array}
\mathrm{Re}\left[\big(V\overline{\Psi}_{h}^{k},\partial_{\tau}\Psi_{h}^{k}\big)\right]=\frac{1}{2}\partial_{\tau}\big(V\Psi_{h}^{k},\Psi_{h}^{k}\big),\mathrm{Re}\left[\big(\overline{\phi}_{h}^{k}\overline{\Psi}_{h}^{k},\partial_{\tau}\Psi_{h}^{k}\big)\right]=\frac{1}{2}\big(\overline{\phi}_{h}^{k}, \,\partial_{\tau}|\Psi_{h}^{k}|^{2}\big).
\omega_N \big(\tau_{\lambda,N}^t (A)\big) = \omega_n \big(A(t)\big) + \sum_{r=1}^{\infty}(i\lambda)^r \int_0^t dt_1 \cdots \int_0^{t_{r-1}}dt_r \, \omega_N \big( P_{r,N}(A(t))\big),
1 = \sqrt{1 + k^{2}}\cdot \vert x_{4}- x_{2}\vert = \sqrt{1 + k^{2}}\cdot \sqrt{( x_{1}+ x_{2})^{2}- 4 x_{4}x_{2}}= \sqrt{2}\cdot \sqrt{4 + 4 \times 3}= 4 \sqrt{2}
F(t,q)\equiv-\partial_t\psi(t,q)-\nabla_qV(t,q)+\tilde F(t,q), &\frac{1}{2}\left(\tilde\gamma^{-1}-(\tilde\gamma^T)^{-1}\right)^{ij}F_j= (\tilde\gamma^{-1})^{ki}H_{k\ell}(\tilde\gamma^{-1})^{\ell j}F_j, &\frac{1}{2}(S-S^\prime)^i=\beta^{-1}\partial_{q^j}\left((\tilde\gamma^{-1})^{ki}H_{k\ell}(\tilde\gamma^{-1})^{\ell j}\right).
\partial_m\partial_n\gamma^{ij}=-\partial_m\Gamma_{nj}^i-\partial_m\Gamma_{ni}^j+\Gamma_{im}^h\Gamma_{hn}^j+\Gamma_{jm}^h\Gamma_{hn}^i+\Gamma_{mh}^i\Gamma_{nh}^j+\Gamma_{mh}^j\Gamma_{nh}^i
\mathcal{C}_i({\bf v}) |_{\lambda}= (t_1 t_2)^{-1}\bigl( (t_1 + t_2 - t_1 t_2 - 1)V_{\lambda}+ W \bigr)_i &= \sum_{(x,y) \in A_{\lambda,i}}t_1^{x-1}t_2^{y-1}- \sum_{(x,y) \in R_{\lambda,i}}t_1^{x}t_2^{y},
\int^{2\pi}_0 d \sigma \left( - \frac{1}{2\pi}p^+ \partial_\sigma X^- +{\cal P}^I \partial_\sigma X^I + \frac{i}{2}\psi^A_+ \partial_\sigma \psi^A_+ + \frac{i}{2}\psi^A_- \partial_\sigma \psi^A_- \right) = 0 ~.
\vec{y}( \omega ) ={\frac{- e \vec{E}^{e}( \omega ) / \omega^{2}}{m_{0}+{\frac{2}{3}}e^{2}\int{\frac{d \vec{k}| f ( \vec{k}) |^{2}}{( 2 \pi )^{3}( k^{2}- \omega^{2})}}}}{}.
M(\chi)=\begin{cases}p^{e-1}(p+1) s=e, \frac{p-\chi(-1)}{\gcd(2,p-1,e)}\varphi(p^{e-1}) s<e\le2, \frac{p^2-1}{\gcd(2,p-1,e)}\varphi(p^{e-2}) e>\max\{s,2\}.\end{cases}
n \int_0^1 \frac{(f_0(x) - \widehat f_{2n}(x))^4}{\widehat f_{2n}(x)^3}dx \leq C_1 n^{\frac{1-2\beta}{2\beta+1}}\int_0^1 \widehat f_{2n}(x)^{-\frac{2\beta+3}{2\beta+1}}dx+ 2^{16}\sum_{i=1}^{\widehat m}\omega_i \big(\eta_i^4-E\big[\eta_i^4\big]\big).
\Gamma^{(PV)}(A)=\frac{1}{2}\left.\frac{d}{d~s}( \zeta(s)-\zeta^P(s))\right|_{s=0}=i\frac{\pi}{2}\eta(0)-i \sum_{k=0}^\infty (-1)^k\frac{m^{(2 k+1)}}{2k+1}\eta (2 k+1),
S_{f i}= \overline{{u}}_{R \alpha^{^{\prime}}}^{( 0 )}( p_{1}^{\prime}) \Gamma_{R}^{( 0 ) \mu}( p_{1}^{\prime}, p_{1}) u_{R \alpha}^{( 0 )}( p_{1}) i D_{R \mu \nu}^{( 0 )}( k )
\hat{\mathrm{H}}_{\pm}^{s}= \int_{- \mathrm{L}/ 2}^{\mathrm{L}/ 2}d x \hat{\cal H}_{\pm}^{s}( x ) = \frac{1}{2}\hbar \int_{- \mathrm{L}/ 2}^{\mathrm{L}/ 2}d x ( \psi_{\pm}^{\dagger s}d_{\pm}\psi_{\pm}^{s}- \psi_{\pm}^{s}d_{\pm}^{\star}\psi_{\pm}^{\dagger s}) .
f(2n,r)= (-1)^r\binom{n-1}{r-1}-(-1)^r\binom{n-1}{r}+2(-1)^r \sum_{j=1}^{n-1}\binom{n-1-j}{r-1}= (-1)^r\binom{n-1}{r-1}+(-1)^r \binom{n-1}{r}= (-1)^r\binom{n}{r}.
L = \exp \left( \begin{array}{c c}{{}}&{{\theta_{N}}}\\{{\overline{{{\theta}}}^{N}}}&{{}}\\ \end{array}\right) \exp \left( \begin{array}{c c}{{i \frac{1 + i \gamma^{4}}{2}\gamma_{p}x^{p}}}&{{}}\\{{}}&{{0_{4}}}\\ \end{array}\right) \exp \left( \begin{array}{c c}{{i \frac{\gamma^{4}}{2}\rho}}&{{}}\\{{}}&{{0_{4}}}\\ \end{array}\right)
P_{b_0}\big( |Z_T(f)| \geq \sqrt{T}x \big) = P_{b_0}\big( |Z_T(f)|\geq \sqrt{T}x, [Z_\cdot(f)]_T \leq Td_L^2(f,0) \big) \leq 2e^{-\frac{x^2}{2d_L^2(f,0)}}.
V_{i j}={\frac{N ( x^{\prime})}{N ( x ) N_{1}( x^{\prime})}}\sum_{i^{\prime}j^{\prime}}c_{i^{\prime}i}( R_{\perp}) K_{i^{\prime}j^{\prime}}c_{j^{\prime}j}( R_{\perp}^{\prime}) , \qquad N_{1}( x^{\prime}) ={\frac{\alpha \pi}{m^{2}}}x^{\prime}( 1 - x^{\prime}) .
\int_{0}^{\infty}d k k^{{\mu}- 1}\mathrm{s i n}( a k - \frac{b}{k}) = 2 \left( \frac{b}{a}\right)^{\frac{\mu}{2}}\mathrm{s i n}\frac{{\mu}{\pi}}{2}K_{\mu}( 2 \sqrt{a b}) ,
\frac{d\omega(\Phi_t^1(x_1^0,x_2),x_2,t)}{dt}= \frac{\rho(\Phi_t^1(x_1^0,x_2),x_2)}{\Phi_t^1(x_1^0,x_2)}=\frac{\rho_0(x_1^0,x_2)}{\Phi_t^1(x_1^0,x_2)}.
A = n I_{m}, D = (10^4 n + m) I_n - 10^4 \boldsymbol{1}_{n\times n}, B_l = \boldsymbol{1}_{m}, B_r=\boldsymbol{1}_{n}, C_l=\boldsymbol{1}_{n}, C_r=\boldsymbol{1}_{m},
\int_1^\infty \biggl| \frac{e^{(1-a)x}}{e^x-1}\biggr| \bigl| x^{s-1}\bigr| dx \ll\int_1^\infty \frac{e^{(1-a)x}}{e^x-1}x^{\sigma -1}dx < \infty , &\int_1^\infty \bigl| x^{s-2}\bigr| dx = \int_1^\infty x^{\sigma-2}dx = \frac{1}{1-\sigma}< \infty.
\Omega=2i\left(1-\frac{1}{2}\frac{1+z\bar{z}}{\zeta}\theta\bar{\theta}\right)\frac{{\rm d}z\wedge{\rm d}\bar{z}}{\zeta^2}+i\left(\frac{{\rm d}\theta \wedge{\rm d}\bar{\theta}}{\zeta}-\frac{\bar{z}\theta}{\zeta^2}{\rm d}z \wedge{\rm d}\bar{\theta}-\frac{z\bar{\theta}}{\zeta^2}{\rm d}\bar{z}\wedge{\rm d}\theta\right).
{\cal A}={\cal A}_{0}+{\cal A}^{\mathrm{i n t}},{\cal A}_{0}= \int_{t_{a}}^{t_{b}}d t \left[ \frac{M}{2}\dot{x}^{2}- \frac{\Omega^{2}}{2}x^{2}\right] , ~{\cal A}^{\mathrm{i n t}}= \int_{t_{a}}^{t_{b}}d t \, V_{\Omega}^{\mathrm{i n t}}\equiv \int_{t_{a}}^{t_{b}}d t \left[ - \frac{( \omega^{2}- \Omega^{2})}{2}x^{2}- V^{\mathrm{i n t}}( x ) \right] ,
C^l_{pol,m}:=\lbrace f:{\mathbb R}^D\rightarrow{\mathbb R}:\exists c>0~ \forall |x|\geq 1~ \forall 0\leq |\gamma|\leq m~{\Big |}D^{\gamma}_xf(x){\Big |}\leq \frac{c}{1+|x|^l}\rbrace
\hat{G}_{M N}= \left( \begin{matrix}{{e^{a \varphi}g_{{\mu}{\nu}}+ G_{{m}{n}}A_{{\mu}}^{( 1 ) \, m}A_{{\nu}}^{( 1 ) \, n}}}&{{A_{{\mu}}^{( 1 ) \, m}G_{{m}{n}}}}\\{{A_{{\nu}}^{( 1 ) \, n}G_{{m}{n}}}}&{{G_{{m}{n}}}}\\ \end{matrix}\right) ,
\begin{array}{c}{{K = h^{\mu \nu}\nabla_{\mu}u_{\nu}= - h^{\mu \nu}\Gamma_{\mu \nu}^{\lambda}u_{\lambda}= - \frac{1}{2 \sqrt{f}}h^{\mu \nu}\left( \partial_{\nu}h_{\mu \tau}+ \partial_{\mu}h_{\nu \tau}- \partial_{\tau}h_{\mu \nu}\right)}}\\{{= \frac{1}{2 \sqrt{f}}h^{\mu \nu}\partial_{\tau}h_{\mu \nu}}}\\ \end{array}
Z_{0}\big |_{\beta \omega \gg 1}\sim e^{- \frac{5 \pi^{2}}{1 2 \beta \omega}- \frac{2 \pi^{2}}{3 \sinh \frac{2 \pi^{2}}{\beta \omega}}}\cosh \biggl ( \frac{\pi^{2}}{\beta \omega}\biggr ) .
p_{-+}= \pi(\{(x,y) | x <0, y>0\}), p_{0+}= \pi(\{(0,y) | y>0\}), p_{++}= \pi(\{(x,y) | x >0, y>0\}) & p_{-0}\,\, = \pi(\{(x,0) | x <0\}), \, \, \, \, p_{00}\, \, = \pi(\{(0,0)\}), \, \, \, \, p_{+0}\, \, = \pi(\{(x,0) | x >0\}), & \, p_{--}= \pi(\{(x,y) | x <0, y<0\}), p_{0-}= \pi(\{(0,y) | y<0\}), p_{+-}= \pi(\{(x,y) | x >0, y<0\}).
\eta^{2}(\alpha \mathbf{z})=-\langle\alpha \mathbf{z},\overline{\alpha \mathbf{z}}\rangle=-\lvert\alpha\rvert^2 \langle\mathbf{z},\mathbf{\overline{z}}\rangle=\lvert\alpha\rvert^2 \eta^{2}(\mathbf{z}),
\begin{array}{l}b_{n-1}=-\alpha_1e_{n-1}+e_n~~~\,{\rm and}~~~\, b_{n+1}=-\alpha_1e_{n-3}+e_{n-2}~,~~~\,{\rm so}t_{n+1}=-\alpha_1e_{n-1}+e_n+\alpha_1e_{n-3}-e_{n-2}~.\end{array}
\zeta_{\widetilde{L}_p}^\mathrm{iso}(s) = \prod_{\mathfrak{p}\mid p}\int_{H_\mathfrak{p}^+}\lvert \det h \rvert_\mathfrak{p}^{\, s}\, \theta_0(h) \, \theta_1(h)^d \, \mathrm{d}\mu_{H_\mathfrak{p}}(h),
F''(t) = \left(\frac{\frac{1}{2}c^{3}(2+t)}{c^{3}de^{c(2+t)}+c^{2}t(2+t)+2c(1+t)+2}+\frac{\frac{1}{2}(t^{2}-2t-5)}{(1-t^{2})(t+2)}\right)
g(x,n) \ =\C(0,h_0(x,n+1)) begin{align*}12pt] =\ \bigg[\frac{0}{h_0(x,n) + 1}+ \frac{1}{2}\bigg] begin{align*}12pt] =\ \bigg[\frac{1}{2}\bigg] begin{align*}12pt] =\ 0.
{\cal L}=-{1\over 12}[2g^{\mu\nu}h_{ij}{\cal D}_\mu q^i{\cal D}_\nu q^j+F^{ab}_\Lambda F_\Lambda^{ab}+2g^2{\cal P}^u_\Lambda{\cal P}^u_\Lambda]\, \varepsilon_{cdef}e^c e^d e^e e^f
{{{\bf{\tilde h}}}^H}\left[ k \right] = \frac{1}{{\sqrt K}}\sum\limits_{t = 0}^{{n'_{{\rm{span}}}}}{\left({\sum\limits_{l' = 1}^{L'}{{\bf{g}}_{l'}^H\left[ t \right]{\bf{\bar H}}_{l'}^\bot{{{\bf{\bar X}}}_{l'}}}}\right){e^{- j\frac{{2\pi}}{K}kt}}}, \ \forall k,
A_{k}( 0 , x ) = \mathrm{s i n}( 2 \frac{k}{q}B ( x ) ) \ \mathrm{o r}\ \mathrm{c o s}( 2 \frac{k}{q}B ( x ) ) \ \ \ \mathrm{a n d}\ \ \, o m e g a_{k}^{2}( 0 ) = ( k^{2}+ m^{2})
\kappa_{0}\left( u,\xi \right) =\sup_{t\in \mathcal{L}_{j}}\left\{\frac{1}{\left\vert{1+\phi \left( t\right) /\left({2u}\right)}\right\vert}\right\},\ \kappa_{2}\left( \xi \right) =\sup_{t\in \mathcal{L}_{j}}\left\vert{\phi \left( t\right)}\right\vert .
\hat{\theta}_{ij}-\tilde{\theta}_{ij}= \begin{cases}M-\bar{\theta}_{ab}(\hat{z}), \bar{Y}_{ab}(\hat{z}) \geq M; \bar{Y}_{ab}(\hat{z})-\bar{\theta}_{ab}(\hat{z}), -M \le \bar{Y}_{ab}(\hat{z}) < M; -M-\bar{\theta}_{ab}(\hat{z}), \bar{Y}_{ab}(\hat{z}) < -M \end{cases}
(a\vee b)^{-1}dc^{-1}\geq(a\vee b)^{-1}acc^{-1}=a^{-1}acc^{-1}. (a\vee b)^{-1}dc^{-1}\geq(a\vee b)^{-1}bcc^{-1}=b^{-1}bcc^{-1}.
\left(\begin{array}{cc}A_1-\lambda_1 E_n A_2-\lambda_2 E_n \overline{A_2-\lambda_2 E_n}\overline{A_1-\lambda_1 E_n}\end{array}\right)\left(\begin{array}{c}z_1 \overline{z_2}\end{array}\right)=\left(\begin{array}{c}0 0 \end{array}\right),
\Delta E_{2}\equiv - \frac{1}{2 \mathrm{L}}\int_{- \mathrm{L}}^{\mathrm{L}}d z ( e_{1}{\phi}_{( 1 )}^{\mathrm{s e l f}}( z , \mathrm{L}) + e_{2}{\phi}_{( 2 )}^{\mathrm{s e l f}}( z , \mathrm{L}) ) .
\begin{array}{rl}{\cal L}= \frac{1}{g^2}\int \left( -\frac{1}{4}F_{mn}F^{mn}+ \frac{1}{2}D_m\Phi^{ab}D^m \Phi_{ab}+\frac{1}{4}\{\Phi^{ab},\Phi^{cd}\}_* \{\Phi_{ab},\Phi_{cd}\}_*\right.+ &\left. + i \lambda_a \sigma^m D_m\bar{\lambda}^a +i\{\lambda_a,\lambda_b\}_* \Phi^{ab}+i\{\bar\lambda^a,\bar\lambda^b\}_* \Phi_{ab}\right)\,.\end{array}
(2\pi h)^{-k}e^{-\frac{i\theta_r\cdot \alpha}{h}}|\det(A_r-I)|^{-1}\sum_{j=0}^{\infty}h^j\int \left(\left(-\frac{ih}{2}c(D)\right) a_{\rho,j}\right) (u, v, \hat{u},\hat{v}) \varphi^{}_{q,r,s}(u) dudvd\hat{u}.
\begin{cases}\partial_t\rho+(u\partial_x+v\partial_y)\rho+\rho(\partial_xu+\partial_yv)=0, \rho\left(\partial_tu+(u\partial_x+v\partial_y)u\right)+\partial_x(p+\frac{1}{2}h^2)-(h\partial_x+g\partial_y)h-\mu\partial_y^2u=0, \partial_y(p+\frac{1}{2}h^2)=0, \partial_th+\partial_y(vh-ug)-\kappa\partial_y^2h=0, \partial_xh+\partial_yg=0.\end{cases}
\chi' \Delta_{H'}(f(x))=(f\otimes f)(\chi)(f\otimes f)(\Delta_H(x))=(f\otimes f)(\chi\Delta_H(x)) &=(f\otimes f)(\Delta_H(x)\chi)=(f\otimes f)(\Delta_H(x))(f\otimes f)(\chi)=\Delta_{H'}(f(x))\chi'
w :=v_1\Uparrow u=v_2\Uparrow u, w' :=v'_1\Uparrow u'=v'_2\Uparrow u', y_1 :=v_1\Uparrow x_1=v'_1\Uparrow x_1, y_2 :=v_2\Uparrow x_2=v'_2\Uparrow x_2.
\frac{c}{t}\sum_{v\in S}\delta_v^2 + \frac{.34c}{t}\sum_{v\in W}\delta_v^2 = \frac{.66c}{t}\sum_{v\in S}\delta_v^2 + \frac{.34c}{t}\sum_{v\in V(K)}\delta_v^2 &\geq \frac{.66c}{t|S|}\bigg(\sum_{v\in S}\delta_v \bigg)^2 + \frac{.34c}{t}\sum_{v\in V(K)}\delta_v^2.
\delta_T^{k}(K) =\limsup_{l\rightarrow\infty}\frac{M(K,k,l)|K|}{|lI^2|}&\leq\limsup_{l\rightarrow\infty}\frac{\widetilde{M}((1-\varepsilon)K,k,l)|K|}{|lI^2|}&=\frac{1}{(1-\varepsilon)^2}\widetilde{\delta}_T^{k}((1-\varepsilon)K) &=\frac{1}{(1-\varepsilon)^2}\widetilde{\delta}_T^{k}(K).
C = \begin{pmatrix}8 s_{13}^2 4 s_{13}(s_{23}-2s_{13}) 4 s_{13}(s_{23}-2s_{13}) 4 s_{13}(s_{23}-2 s_{13}) -4s_{13}^2-4 s_{23}s_{13}+s_{23}^2 (s_{23}-2 s_{13})^2 4 s_{13}(2 s_{13}-s_{23}) -(s_{23}-2 s_{13})^2 -12 s_{13}^2+4s_{23}s_{13}-s_{23}^2\end{pmatrix}.
\psi ( y ) \sim \frac{\sqrt{\nu}}{\sinh^{2}\left( \nu ( y_{0}- | y | ) \right)}{\,}_{2}F_{1}\left( 5 / 4 \mp 3 i \delta / 4 , \, 5 / 4 \pm 3 i \delta / 4 ; \, 3 ; \, - \sinh^{- 2}\left( \nu ( y_{0}- | y | ) \right) \right)
\begin{aligned}\lfloor\frac{j-1}{2}\rfloor-\lceil\frac{s-1}{2}\rceil+\frac{l}{2}+q+1 \geq \lceil\frac{j}{2}\rceil+\frac{l}{2}+\lfloor\frac{q}{2}\rfloor &\geq \lceil\frac{j}{2}\rceil+\frac{l}{2}+\lfloor\frac{j-l}{2}\rfloor &\geq j-\frac{1}{2}.\end{aligned}
\boldsymbol{F}_{(n+1)^2,n^2}(z,x) = F^{\kappa_1}\circ U^{-1}\circ \widetilde{\boldsymbol{F}}_{(n+1)^2-\kappa_1, n^2+\kappa_0}\circ U \circ \boldsymbol{F}_{n^2+\kappa_0,n^2}(z,x)+o(1) &=F^{\kappa_1}\circ U^{-1}\circ \widetilde{\boldsymbol{F}}_{(n+1)^2-\kappa_1, (n+1)^2-k_n}(z^o_n,x^o_n)+o(1).
\frac{\sum_{k=0}^n V_k - \frac{\mu_Q}{\mu_{\sigma}}n}{d(n)}= \left( \frac{d(m_n)}{d(n)}\cdot \frac{\sum_{i=1}^{m_n}Q_i - m_n \mu_Q}{d(m_n)}\right)+ \frac{\mu_Q(m_n - n/\mu_{\sigma})}{d(n)}+ \frac{\sum_{k=0}^n V_k - \sum_{i=1}^{m_n}Q_i}{d(n)}& \equiv (I) + (II) + (III).
| z > = e^{- \frac{1}{2}| z |^{2}}\, e^{z a^{\dagger}}| 0 > \ \ \ , \ \ \| p , q > = e^{- \frac{i}{\hbar}q \hat{p}}\, e^{\frac{i}{\hbar}p \hat{q}}\, | 0 > \ \ \ .
= \sum_{i=1}^m \bar{\eta}_1 \wedge \cdots \wedge\big( (\sum_{k=1, k\neq i}^{m}c_{kk}) \bar{\eta}_i \big) \wedge \ldots \bar{\eta}_m = \sum_{i=1}^m (\sum_{k=1, k\neq i}^{m}c_{kk}) \bar{\eta}_1 \wedge \cdots \wedge \bar{\eta}_m = (m-1) (c_{11}+ \cdots + c_{mm}) \bar{\eta}_1 \wedge \cdots \wedge \bar{\eta}_m.
{\rm{\bf K}}{\rm d}_{t}\mathcal{E}+{\rm{\bf L}}_{1}\mathcal{E}_{x}{\rm d}t+{\rm{\bf L}}_{2}\mathcal{E}_{y}{\rm d}t+{\rm{\bf L}}_{3}\mathcal{E}_{z}{\rm d}t=\nabla S(\mathcal{E})\circ{\rm d}W(t),
l ( \sigma ) = \bar{\phi}+ \sqrt{2}\sum_{e = 2}^{\infty}l_{e}\left( \cos e \sigma - i^{e}\right) \, , \quad r ( \sigma ) = \bar{\phi}+ \sqrt{2}\sum_{e = 2}^{\infty}r_{e}\left( \cos e \sigma - i^{e}\right) \, . ~
f(\hat Z_{-}) = \Bigl( 1-{Z_{-}^{2}\over 2N}\Bigr)\rho +{\cal O}(1/N^{2})\, ,\quad f(\hat Z_{1})=\Bigl( 1-{Z_{1}^{2}\over 2N}\Bigr)\rho +{\cal O}(1/N^{2})\, .
T_{B}^{( i )}( z ) = \displaystyle \sum_{\mu = 0}^{n - 1}G_{a + \bar{\varepsilon}_{\mu}}\phi^{* ( a , a + \bar{\varepsilon}_{\mu})}( z ) K^{( i )}\left( \left. a + \bar{\varepsilon}_{\mu}\begin{array}{c}{a}\\{a}\\ \end{array}\right| z \right) \phi^{( a + \bar{\varepsilon}_{\mu}, a )}( z ) .
4N(X,T) =[JX,JT]-[X,T]-J[X,JT]-J[JX,T], =\frac{JX(\textbf{H})}{2}\left(T-JT \right)-\frac{X(\textbf{H})}{2}\left(T-JT \right)-\frac{X(\textbf{H})}{2}\left(JT+T \right)-\frac{JX(\textbf{H})}{2}\left(JT+T \right), =-X(\textbf{H})T-JX(\textbf{H}) JT.
\hat{\Pi}^{\underline{m}}=d\hat{X}^{\underline{m}}-i d\hat{\Theta}^{1}{\sigma}^{\underline{m}}\hat{\Theta}^1-i d\hat{\Theta}^2\tilde{\sigma}^{\underline{m}}\hat{\Theta}^2 = d\xi^m \hat{\Pi}^{~\underline{m}}_m (\xi )
\sum_{\substack{i\in \mathbb{Z}^3:|i|\geq1 |\mu-i|\geq 1}}\frac{1}{|\mu-i|^2|i|^{2+\epsilon}}\leq \sum_{\substack{i\in \mathbb{Z}^3: |i|\geq1}}\frac{1}{|i|^{4+\epsilon}}+ \sum_{\substack{i\in \mathbb{Z}^3: |\mu-i|\geq 1}}\frac{1}{|\mu-i|^{4+\epsilon}}\leq C
C_{NUT}=-\frac{(4\pi )^{(n-2)/2}\beta}{16\pi^{3/2}\ell^{2}}\Gamma \left(\frac{3-n}{2}\right) \Gamma \left( \frac{n}{2}\right) N^{n-3}\left[n(n-1)(n-2)N^{2}-(n-2)(n-3)\ell^{2}\right]
{\bf B}(z) = R \circ J \circ T_{e_N}\circ S_2 \circ J \circ T_{-e_N}(z), \,\,R= \begin{pmatrix}1 & \ddots & 1 & -1\end{pmatrix},\,e_N = \begin{pmatrix}0 \vdots 0 1\end{pmatrix}
\lim_{T\rightarrow\infty}\frac{1}{T^{nd-2\left(n-1\right)}}\mathrm{Vol}\left(P_{Q}^{n}\left(I_{1},\dots,I_{n-1}\right)\cap B\left(T\right)\right)=C_{Q,n}\prod_{i=1}^{n-1}\left|I_{i}\right|.
w_t+w_{xxx}=0,\;\; x\in(0,1), \;t>0; w(x,0)=0,\;\; x\in(0,1); &w(0,t)=w(1,t), \quad t>0; & \partial_xw(0,t)=\partial_xw(1,t)+h_1(t), \quad t>0; & \partial_{xx}w(0,t)=\partial_{xx}w(1,t)+h_2(t), t>0.
\| |D|^\sigma( f_1 \cdots f_n)\|_{L^r}\lesssim \sum_{k = 1}^n \| |D|^\sigma f_k\|_{L^{p_k^{(k)}}}\prod_{j \neq k}\|f_j\|_{L^{p_j^{(k)}}}, \frac{1}{r}= \sum \frac{1}{p_j^{(k)}}.
M_{1}= -{\frac{1}{4 \pi}}\int d^{2}x \left\lbrace{\frac{( a - 1 )}{2}}\, \partial_{\mu}\theta \; \partial^{\mu}\theta + \theta \left\lbrack ( a - 1 ) \partial_{\mu}A^{\mu}+ \epsilon^{\mu \nu}\partial_{\mu}A_{\nu}\right\rbrack \right\rbrace
(\mathbf 1_{A_{\alpha^*}}\bar{m}'_{\tilde\xi_i(t)},\sigma_i\bar{m}_{\tilde\xi_i(t)}-\bar{m}_{\tilde\xi(t)})_{L^2(d\nu)}=(\mathbf 1_{A_{\alpha^*}}\bar{m}'_{\tilde\xi(t)},u)_{L^2(d\nu)},
M_{\alpha}:=\sum_{0< i_2<\cdots< i_n}x_0^{\alpha_1}x_{i_2}^{\alpha_2}\cdots x_{i_k}^{\alpha_k}F_{\alpha}:=\sum_{{0\leq i_1\leq i_2\leq \cdots\leq i_n}\atop{j\in D(\alpha)\Rightarrow i_j<i_{j+1}}}x_{i_1}x_{i_2}\cdots x_{i_n},
d s^{2}= - \left( \gamma + \frac{r^{2}}{l^{2}}- \frac{\mu}{r^{d - 3}}\right) d t^{2}+ \frac{d r^{2}}{\left( \gamma + \frac{r^{2}}{l^{2}}- \frac{\mu}{r^{d - 3}}\right)}+ r^{2}d \sigma_{\gamma}^{2},
( 0 , 1 , 1 , 1 ) , ( 0 , 0 , 1 , 1 ) , ( 1 , 0 , 1 , 1 ) , ( 0 , 0 , 1 , 0 ) , ( 0 , 0 , 0 , 1 ) , ( 0 , 1 , 0 , 1 ) .
\omega^{(2)}= \left(\omega^{(2)}- \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-i_1}\omega^{(2)}_{i_2}x^{i_2}\right) + \sum_{\beta \le i_2 < \alpha^\prime+N-m-1-i_1}\omega^{(2)}_{i_2}x^{i_2}
b_{0}^{\ast}( \overline{{K}}, t ) b_{0}^{\ast}( \overline{{P}}, t ) | 0 \rangle = \int \int \overline{{\kappa}}_{\overline{{K}}, \overline{{P}}}^{( + )}( K , P ) b^{\ast}( K , t ) b^{\ast}( P , t ) | 0 \rangle d K d P
\begin{aligned}\|G(f)-G(\bar{f})\| &=\left\|\int_0^\cdot e^{(\cdot-s)\Delta}\left(\Gamma^{-1}(s)[K(\Gamma(s)f(s)\cdot\nabla](\Gamma(s)f(s))-\Gamma^{-1}(s)[K(\Gamma(s)\bar{f}(s)\cdot\nabla](\Gamma(s)\bar{f}(s))\right)ds\right\| &\leq \mathcal{C}_1\eta_\infty R^*\|f-\bar{f}\|,\end{aligned}
\left( (m_{1}+m_{2})\left( \cot \frac{\vartheta}{2}\right)^{2}-m_{2}\right) \left( \left( \cot \frac{\vartheta}{2}\right)^{2}-1\right) =\left( \frac{m_{1}}{2}+ 6m_{2}\right) \left( \cot \frac{\vartheta}{2}\right)^{2}.
i_{-}\left( s\right) :=\max\left\{i\geq1:s\subset\mathcal{H}_{b_{i}}^{t,+}\ ;\b_{i}\in\mathbf{K}_{e}^{t}\left( 0,x\right) \right\}\ , i_{+}\left( s\right) :=\min\left\{i\geq i_{-}\left( s\right)+1:s\subset\mathcal{H}_{b_{i}}^{t,-}\ ;\b_{i}\in\mathbf{K}_{e}^{t}\left(0,x\right) \right\}\ .
y^2=x^3+\sum_{k=-4}^{4}\sum_{l=0}^{12-nk}f_{kl}z^{\prime l}z^{4+k}\ldots a^{4-6k}x+\sum_{k=-6}^{6}\sum_{l=0}^{12-nk}g_{kl}z^{\prime l}z^{6+k}\ldots a^{6-6k}
L(k,m,\lambda) =\frac{(n+\lambda)_{m}(n+\lambda+\epsilon_2km+(1-\delta_{\epsilon_2,1})m)_m^{\epsilon_2}}{(n+\lambda+\epsilon_2mk)_m}, M(k,m,\mu) =\frac{(n+\epsilon(\mu+\epsilon_1km))_m}{{(n+\epsilon\mu)_m(n+\mu+\epsilon_1 km+(1-\delta_{\epsilon_1,1})m)_m^{\epsilon\epsilon_1}}}.
\begin{array}{c}\displaystyle{Res\, \zeta_\nu(s,x)|_{s=1}= \frac{1}{\pi}\, ,}\displaystyle{Res\, \zeta_\nu(s,x)|_{s=0}= 0\, ,}\displaystyle{Res\, \zeta_\nu(s,x)|_{s=-1}= \frac{1 - 4\,{\nu}^2}{8\,\pi}\, ,}\end{array}
D =\{\omega\in\mathbb{C}_{2}:\frac{x_{1}+y_{1}}{2}+i_{1}\frac{x_{2}+y_{2}}{2}+i_{2}\frac{y_{2}-x_{2}}{2}+i_{1}i_{2}\frac{x_{1}-y_{1}}{2}, -\infty <x_{1},y_{1}<\infty,-\alpha<x_{2},y_{2}<\beta\}.
b_{--}(\xi,\eta) = - \frac{i}{4}\frac{|\xi| \widetilde{m}_2(\xi,\eta)}{|\xi-\eta|^{1/2}|\eta|}+ \frac{i}{4}\frac{|\xi|^{1/2}\widetilde{q}_2(\xi,\eta)}{|\xi-\eta|^{1/2}|\eta|^{1/2}},
e_0^a =\left(\begin{array}{c}C(\eta) 0 0 0\end{array}\right) \qquad e_1^a=\left(\begin{array}{c}0 C(\eta) 0 0\end{array}\right) \qquad e_2^a=\left(\begin{array}{c}0 0 C(\eta) 0\end{array}\right) \qquad e_3^a=\left(\begin{array}{c}0 0 0 C(\eta)\end{array}\right)