harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{{k}_{1}}\left({{u}_{y}}+{{v}_{x}}\right)+{{k}_{2}}\left({{u}_{x}}-{{v}_{y}}\right)+u\left({{k}_{1y}}+{{k}_{2x}}\right)+v\left({{k}_{1x}}-{{k}_{2y}}\right)=0{{k}_{1}}\left({{u}_{x}}-{{v}_{y}}\right)-{{k}_{2}}\left({{u}_{y}}+{{v}_{x}}\right)+u\left({{k}_{1x}}-{{k}_{2y}}\right)-v\left({{k}_{1y}}+{{k}_{2x}}\right)=0
g\rhd(g\rhd k)=g\rhd(k^{-1}gk\chi(g,k)) =\chi(g,k)^{-1}k^{-1}g^{-1}kgk^{-1}gk\chi(g,k)\chi(g,(k^{-1}gk\chi(g,k)) &=k^{-1}g^{-1}kgk^{-1}gk\chi(g,(k^{-1}gk\chi(g,k)) &= k^{-1}g^{-1}kgk^{-1}gk,
Z^\mathbb{R}_{|f|^2}=\sum_{i,j=1}^nP_\mathbb{R}^{ij}\frac{\partial|f|^2}{\partial x^i}\frac{\partial}{\partial x^j}+\sum_{i,j=1}^nP_\mathbb{R}^{in+j}\frac{\partial|f|^2}{\partial x^i}\frac{\partial}{\partial x^{n+j}}
\epsilon_{a}^{i}( \theta ) \approx r \, a \, m_{i}\cosh \theta - \sum_{b = 1}^{k - 1}\left( \delta_{a b}- 2 \left( K^{A_{k - 1}}\right)_{\, \, \, a b}^{- 1}\right) L_{b}^{i}( \theta ) - \sum_{b = 1}^{k - 1}\left( K^{A_{k - 1}}\right)_{\, \, \, a b}^{- 1}L_{b}^{j}( \theta - \sigma ) \; .
\|R_Ax -R_Ay\|^2 &=4\|J_Ax-J_Ay\|^2-4\langle J_Ax-J_Ay,x-y\rangle+\|x-y\|^2 &\leq (4(2(1-\alpha)-1)\langle J_Ax-J_Ay,x-y\rangle+(1-4(1-2\alpha)))\|x-y\|^2 &\leq \left(4(1-2\alpha)\left(\tfrac{\delta}{2}+\tfrac{(1-\alpha-\delta/2)^2-\alpha^2+(\delta/2)^2}{2(1-\alpha-\delta/2)}\right)+1-4+8\alpha\right)\|x-y\|^2,
\frac{\partial^{2}\varphi(\omega)}{\partial\omega^{2}}=\frac{\partial}{\partial\omega}(\frac{\partial\varphi(\omega)}{\partial\omega})={(\frac{\lambda}{2\pi c})}^{2}(2\lambda\frac{\partial\varphi(\lambda)}{\partial\lambda}+\lambda^{2}\frac{\partial^{2}\varphi(\lambda)}{\partial\lambda^{2}})
\intertext{By using all result above, we reach}j_K(x)+O\left(\frac{x^3}{T}\right) =cx+O\left(\frac{x^2}{T^{1-\varepsilon}}\right)+O(x^{1/2}T^{\varepsilon}\log T)+O\left(\frac{x^2}{T^{1-\varepsilon}}\right). \intertext{When we select $T=x^3$, this becomes}j_K(x) =cx+O(x^{1/2+\varepsilon}).\intertext{This proves the theorem.}
\max_{\theta}~ U_{L}(\theta,\lambda^{\textrm{opt}}) = \mu (1-\theta) \bigg[ \log\bigg(1+ t_{1}\frac{\theta}{1-\theta}\bigg) &~~~~~~~~~~~~~~~~~~ - \log\bigg(1+t_{2}\frac{\theta}{1-\theta}\bigg) \bigg] - \theta t_{3}, s.t. ~ 0<\theta<1,
\int_{C}{d^{n}}{\eta}D_{c}^{( n )}( \eta ,{\eta}) = - \sum_{N = 0}^{N_{0}}\sum_{\alpha = 0}^{N}\frac{1}{\lambda_{N}}Y_{\alpha}^{N}({\eta}) Y_{\alpha}^{N}({\eta}) \int_{C}{d^{n}}{\eta}
\frac{1}{2}m_{1}(v_{0}-\frac{p}{m_{1}})^{2}+\frac{1}{2}m_{2}(v_{0}+\frac{p}{m_{2}})^{2}-\frac{1}{2}m_{1}v_{0}^{2}-\frac{1}{2}m_{2}v_{0}^{2}
(\mathbb{A'}^{-1}_L)_{12}=(\Delta')^{-1}\left( a-([b,c]-[a,c]a^{-1}b)(d-ca^{-1}b)^{-1}\right) &= (\Delta')^{-1}\left( a(d-ca^{-1}b)-([b,c]-[a,c]a^{-1}b)\right)(d-ca^{-1}b)^{-1}&= (\Delta')^{-1}\left( ad - bc \right)(d-ca^{-1}b)^{-1}&= (d-ca^{-1}b)^{-1}
g(\Delta u, e^2) = \partial_r\{\frac 1s(\partial_r(s u_\theta)-\partial_\theta u_r)\}&= \partial_r\{\frac cs u_\theta+\partial_r u_\theta-\frac 1s\partial_\theta u_r\}&=-\frac{1}{s^2}u_\theta+\frac cs \partial_r u_\theta +\partial_r^2 u_\theta+\frac{c}{s^2}\partial_\theta u_r-\frac 1s \partial_r\partial_\theta u_r
\sum_{d<x}\frac{\phi(d)}d\pi(x;d,1) =\sum_{k<x}\frac{\mu(k)}k\sum_{\substack{{p\leq x}{p\equiv 1 (k)}}}\tau\left(\frac{p-1}k\right) =\sum_{k\leq \log^{A+2}x}+ \sum_{\log^{A+2}x<k<x}=\sum_{k\leq \log^{A+2}x}+ O\left(\frac{x}{\log^A x}\right).
\mathrm{Tr}(\Omega(c)(1+D^2)^{1-\frac{z+p}{2}}) = \frac{2}{\Gamma\left(\frac{z}{2}\right)}\int_0^1 s^{z-1}(s^{-1}h(s)+\frac{p}{2}\mathrm{Ch}(c)s^{-2})\,ds +\frac{2}{\Gamma\left(\frac{z}{2}\right)}\int_1^\infty s^{z-1}e^{-s}h(s)\,ds.
\mathcal{Z}_{M}:= \sum_{p=1}^{M}c_{p}V_{N}(z_{p}) = \sum_{p=1}^{M}c_{p}\frac{1}{d_{N}}(\mathrm{Tr}(G(E+z_{p}/d_{N}))-\mathbb{E}\mathrm{Tr}(G(E+z_{p}/d_{N})))
(X,Y)=(-y^{\prime}\frac{{x^{\prime}}^{2}+{y^{\prime}}^{2}}{x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime}},x^{\prime}\frac{{x^{\prime}}^{2}+{y^{\prime}}^{2}}{x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime}}).
b(1-\eta)\leq w\leq c_0^{-1}(1-\eta)\sqrt{-\ln(\mu(1-\eta))}, -C_1(1-\eta)^{\alpha_0}\leq\partial_\xi w+\partial_\tau w\leq \frac{\delta}{2}w, & w\partial_\eta^2 w\leq 2\delta , -C_1(1-\eta)^{\alpha_0}\leq \partial_\tau w\leq b\delta(1-\eta)^{\alpha_0}in D.
e^{2i\pi A(0}=\left(\begin{array}{ccccc}1 i\pi X_1 -i\pi X_2 ... i\pi X_{23}0 1+2i\pi X_1 -2i\pi X_2 ... 2i\pi X_{23}0 2^2i\pi X_1 1 - 2^2i\pi X_2 ... 2^2i\pi X_{23}... ... ... ... ... 0 2^{23}i\pi X_1 2^{23}i\pi X_2 ... 1+2^{23}i\pi X_{23}\end{array}\right)
\left( \Delta_{r \theta}- \frac{{\hat{L}}^{2}}{r^{2}\sin^{2}\theta / 2}- \frac{{\hat{J}}^{2}}{r^{2}\cos^{2}\theta / 2}\right) \psi + \frac{2 M}{\hbar^{2}}\left( \epsilon + \frac{e^{2}}{r}\right) \psi = 0
6 \cdot \sum_{a=\frac d6}^{\frac{2d}{3}}\left(1- \frac{a^k + (2d-a)^k}{(2d)^k}\right) \sim 6\cdot \int_{a=\frac d{6}}^{\frac{2d}3}\left(1- \frac{a^k + (2d-a)^k}{(2d)^k}\right) \d a \sim 6\cdot \int_{\frac 1{12}}^{\frac13}\left( 1-x^k-(1-x)^k \right) \d x \cdot 2d
\sum_{m , n \geq 0}d^{m - \delta}z^{n - \delta}\langle \eta_{m}| U_{3}( \psi , w ) | \eta_{n}\rangle = \sum_{m , n \geq 0}d^{m - \delta}z^{n - \delta}\langle P ( \eta_{m}, \eta_{n}; w^{\ast}) | \psi \rangle .
\begin{array}{ll}\forall r>0~\forall x\in B_r~\exists R_0~\forall R\geq R_0 ~E^x\left(f^{R}\left(\sum_{i=1}^nc_iX^{R}_i(t)\right) \right)\leq E^x\left(f^{R}\left(\sum_{i=1}^nc_iY^{R}_i(t)\right) \right)\end{array}
S(p) = \frac{\phi(m_1)}{m_1}\Bigg(1+O\bigg(\frac{1}{y{\sqrt{\log y}}}\bigg)\Bigg)-\frac{\phi(m_2)}{m_2}\Bigg(1+O\bigg(\frac{1}{y{\sqrt{\log y}}}\bigg)\Bigg) & = \frac{\phi(m_1)}{m_1}-\frac{\phi(m_2)}{m_2}+O\left(\frac{1}{y{\sqrt{\log y}}}\right),
\left\Vert \left\{\xi_{j}-\sum_{i=1}^{m}f_{i}\left( z_{j}\right) \right\}_{j=1}^{J}\right\Vert_{\ell^{\infty}}<\delta^{m}, \left\Vert \left\{a_{i}^{m}\right\}_{i=1}^{J}\right\Vert_{\ell^{2}\left(\mu\right)},\left\Vert f_{m}\right\Vert_{B_{2}}\leq C\delta^{m-1}.
\pi (x) ~=~ \frac{\delta \,{\cal L}}{\delta\, \dot{\varphi}\, (x)}~=~\dot{\varphi}+ \lambda\,\frac{\delta\,{\cal L}_{int}}{\delta\,\dot{\varphi}}\, \left(\varphi, \vec{\bigtriangledown}\,\varphi,\dot{\varphi}\right)~=~\dot{\varphi}+\lambda\, \dot{\varphi}\, \left[ \left(\vec{\bigtriangledown}\,\varphi\right)^2 - \dot{\varphi}^2\right],
\sqrt{\gamma}\int^{\frac{1}{2}+ i\infty}_{\frac{1}{2}- i\infty}\frac{ds}{2\pi i}\frac{e^{\frac{1}{2}s^2 - ys}}{s\Gamma(\frac{s}{\sqrt{\gamma}})}= \int^{\frac{1}{2}+ i\infty}_{\frac{1}{2}- i\infty}\frac{ds}{2\pi i}\frac{e^{\frac{1}{2}s^2 - ys}}{\Gamma(\frac{s}{\sqrt{\gamma}}+ 1)},
D(\lambda) = \prod_{j=1}^{m}D_{j}(\lambda),D_{j}(\lambda) = \left( \frac{(\sqrt{\lambda}-i \sqrt{|\lambda_{j-1}|})(\sqrt{\lambda}+i \sqrt{|\lambda_{j}|})}{(\sqrt{\lambda}-i \sqrt{|\lambda_{j}|})(\sqrt{\lambda}+i \sqrt{|\lambda_{j-1}|})}\right)^{\frac{\log s_{j}}{2\pi i}}.
\mathrm{E}(X^n)=\frac{1}{\mathrm{B}(\alpha,\beta)}\frac{1}{\sigma}\int_{-\infty}^\infty x^n\left[\Phi_s\left(\frac{x-\mu}{\sigma}\right)\right]^{\alpha-1}\left[1-\Phi_s\left(\frac{x-\mu}{\sigma}\right)\right]^{\beta-1}\phi_s \left(\frac{x-\mu}{\sigma}\right) \mathrm{d}x.
\sum_{m=0}^{r-1}\frac{m p_n^2(m)}{h_n}=\frac{r-1}{2},\sum_{m=0}^{r-1}\frac{m^2 p_n^2(m)}{h_n}=\frac{h_{n+1}}{h_n}+\left(\frac{r-1}{2}\right)^2+\frac{h_{n}}{h_{n-1}},
S_{L}[{\hat{\Phi}}] \ = \{\frac{4 \pi}{L + 1}}\, \mathrm{T r}[{\frac{1}{2}}({\hat{\cal L}}_{i}{\hat{\Phi}})^{2}+{\frac{r}{2}}{\hat{\Phi}}^{2}+{\frac{\lambda}{4 !}}{\hat{\Phi}}^{4}] \ .
V_{i}^{L}= \lim_{\delta p_{i}\to 0}\frac{E_{\kappa}( \vec{p}+ \delta \vec{p}) - E_{\kappa}( \vec{p})}{\delta^{L}p_{i}}= e^{- \frac{E ( \vec{p})}{\kappa c^{2}}}\frac{\partial E_{\kappa}( \vec{p})}{\partial p_{i}}\ = \e^{- \frac{E ( \vec{p})}{\kappa c^{2}}}V_{i}\, .
A(z) = F_{n}^{N}(z)-CF_{n-1}^{N+1}(z) = \frac{1}{\sqrt{2\omega}}\int_{z}^{+\infty}d\zeta \Big( C\zeta^{N+1}\big( L_{n-1}^{N+1}(\zeta) \big)^{2}-\zeta^{N}\big( L_{n}^{N}(\zeta) \big)^{2}\Big) \exp ( -\zeta )
\begin{array}{l}{{{[}U_{a}, U^{b}{]}= S_{a}^{b}\in g^{0}}}\\{{{[}U_{a}, U_{b}{]}= K_{a b}\in g^{+ 2}}}\\{{{[}U^{a}, U^{b}{]}= K^{a b}\in g^{- 2}}}\\{{{[}S_{a}^{b}, U_{c}{]}= U_{( a b c )}\in g^{+ 1}}}\\ \end{array}
\max_{i,j}|(G^0_{\ell})_{ij}|=\max_{i,j}|(G_\ell)_{ij}+((G_\ell)\mathsf{W}_{ab}G^0_\ell)_{ij}| &\prec \max_{i,j}|(G_\ell)_{ij}|+\frac{1}{\sqrt{M}}\max_{i,j}|(G_\ell)_{ij}|\cdot \max_{i,j}|(G^0_\ell)_{ij}|\prec 1+\frac{1}{\sqrt{M}}\max_{i,j}|(G^0_\ell)_{ij}|.
\big[{\rm dist}\, \big(T f, T( E_0) \big) \big]^2 = \bigg\Vert \sum_{j = n}^\infty a_j (T) \, \langle f, v_j \rangle \, u_j \bigg\Vert^2 = \sum_{j = n}^\infty [a_{j}(T)]^{2}| \, \langle f, v_j \rangle|^2 & \leq [a_{n}(T)]^{2}\sum_{j = n}^\infty | \langle f, v_j\rangle|^2 \leq [a_{n}(T)]^{2};
\|[d(f_t)_z+\alpha(t)dh_z]^{-1}\| \leq\sum_{j=0}^\infty \alpha(t)^j\|(d(f_t)_z)^{-1}\|^{j+1}\|dh_z\|^j &\leq\sum_{j=0}^\infty \frac{\alpha(t)^j}{a^{j+1}}\leq \frac{2}{a}.
{\cal L}= -{\frac{1}{4}}E^{( 1 1 )}[{\cal R}^{( 1 1 )}+{\frac{1}{1 2}}F_{M N P Q}^{( 1 1 )}F^{( 1 1 ) \M N P Q}-{\frac{8}{1 2^{4}}}\varepsilon^{M_{1}\cdots M_{1 1}}F_{M_{1}\cdots M_{4}}F_{M_{5}\cdots M_{8}}A_{M_{9}M_{1 0}M_{1 1}}] ,
\tilde z=\Psi_2^{-1}(\hat z)= \tilde z_3+\frac{(\tilde z_3-\tilde z_1)(\tilde z_2-\tilde z_3)(\hat z_2-\hat z_1)(\hat z-\hat z_3)}{(\tilde z_2-\tilde z_1)(\hat z_2-\hat z_3)(\hat z-\hat z_1)-(\tilde z_2-\tilde z_3)(\hat z_2-\hat z_1)(\hat z-\hat z_3)}\,.
{{\cal R}^{\hat{t}\hat{\imath}}}_{\hat{t}\hat{\imath}}=-{1\over G^2}{(\ln B)'}^2=-{(d-2)^2Q^2\over 2\Delta (D-2)}\left[r^{d-2}+{1\over d-2}\sqrt{\Delta\over 2 (D-2)}|Q|\right]^{-2\left(1+{p+1\over \Delta}\right)}r^{-a^2{D-2\over \Delta}}.
\scriptstyle\sum\limits_{i=1}^N \frac{w_i e^{-\rho_I\sum\limits_{l=0}^{\infty}w_{Il}\delta_0 (-1 +{}_2F_1(-\frac{1}{\alpha_I}, l+\mu, \frac{\alpha_I-1}{\alpha_I}, -\frac{T R^{2 \alpha_d}\delta_0^{-\alpha_I}\gamma_0}{c_0 t_i \overline{g_p}\mu (1+\kappa)}))}}{e^{\frac{T \sigma^2 R^{2 \alpha_d}}{c_0 t_i \overline{g_p}}}}
p_{b\perp}\sim -\varepsilon^b\sim \frac{D a p_b}{(D-1)(a-r)}\sim \frac{D\Gamma ((D+1)/2)(\xi -\xi_{D})}{2^{D}\pi^{(D+1)/2}(a-r)^{D+1}}\left( 1-2\delta_{B0}\right) .
\{\{F,G\},H\}+ (-1)^{GH}\{\{F,H\},G\}+ (-1)^{F(G+H)}\{\{G,H\},F\}= Q\{F,G,H\}+ \{QF,G,H\}+(-1)^{FG}\{QG,F,H\}+(-1)^{H(F+G)}\{QH,F,G\}.
\mathbf{D}\triangleq\begin{pmatrix}\mathbf{D}_{11}\mathbf{D}_{21}\cdots \mathbf{D}_{N1}\mathbf{D}_{12}\mathbf{D}_{22}\cdots \mathbf{D}_{N2}\vdots \vdots \ddots \vdots \mathbf{D}_{1N}\mathbf{D}_{2N}\cdots \mathbf{D}_{NN}\end{pmatrix},
\frac{\delta{\tilde A}}{\delta x}= -(u_t+p_m)=0,\frac{\delta{\tilde A}}{\delta u}=(x_t-u)=0, \frac{\delta{\tilde A}}{\delta p}= x_m-\tilde{w}_p=(x_m-\tau)=0, \frac{\delta{\tilde A}}{\delta S}= -\tilde{w}_S-r_t=-(r_t+T)=0,\frac{\delta{\tilde A}}{\delta r}=S_t=0.
P^{l,m}=\frac{\sqrt{2}}{\delta(x,y)}\sum_{n=0}^{\log_2\tfrac{2^m}{\delta(x,y)}-1}\frac{1}{2^n}\Omega_{I^{(n)}(x,y)}(x,y) &=\frac{\sqrt{2}}{\delta(x,y)}\Biggl(\Omega(x,y)-\sum_{n=\log_2\tfrac{2^m}{\delta(x,y)}}^{\infty}\frac{1}{2^n}\Omega_{I^{(n)}(x,y)}(x,y)\Biggr) &=P_{l,m}(x,y)+Q_{l,m}(x,y),
A_\mu^a\left(x\right)=\sum\limits_{\vec{k}}\left(\sum\limits_{\sigma=1,2}A_{\vec{k},\sigma}^af_{k,\mu}^\sigma \left(x\right)+A_{\vec{k}}^{L,a}f_{k,L,\mu}\left(x\right)+B_{\vec{k}}^af_{k,S,\mu}\left(x\right) \right)+h.c.,
-\int_0^T\int_0^L\int_0^\infty 2g(s)xu\eta^t(x,s)dsdxdt\leq L^2 \int_0^T\int_0^\infty g(s)\|u\|^2dsdt+\int_0^T\|\eta^t(\cdot,s)\|^2_{L_g}dt = L^2 g_0\int_0^T\|u\|^2dt+\int_0^T\|\eta^t(\cdot,s)\|^2_{L_g}dt.
C_1(b) = \frac{I_F'(b) \varphi_F(b) - I_F(b) \varphi_F'(b)}{\psi'(b) \varphi_F(b) - \psi(b) \varphi_F'(b)}, C_2(b) = \frac{I_F'(b) \psi(b) - I_F(b) \psi'(b)}{\psi'(b) \varphi_F(b) - \psi(b) \varphi_F'(b)}.
f(U^{(0)},V^{(0)}) = F(U^{(0)},V^{(0)};\gamma)\ge F(U^{(p)},V^{(p)};\gamma) &\ge \frac{1}{2}\sum_{ij\in E_{\mathrm{ss}}}(\alpha_{ij}^{(p)})^2 + \frac{1}{2}\sum_{ik\in E_{\mathrm{sa}}}(\alpha_{ik}^{(p)})^2.
\frac{c_{(k+1,k+1)}^2}{c_{(k,k)}^2}-\frac{c_{(k,k)}^2}{c_{(k-1,k-1)}^2}=\frac{e^{8k+8}\frac{(2k+2)^4-(2k+4)^2(2k)^2}{(2k+4)^2(2k)^2(2k+2)^4}+e^{4k}\frac{p_1(k)}{p_2(k)}+\frac{p_3(k)}{p_4(k)}}{e^{8k+8}\frac{1}{(2k)^2(2k+2)^2}+e^{4k}\frac{p_5(k)}{p_6(k)}+\frac{p_7(k)}{p_8(k)}}
X_{t}^{\mathrm m_{i}}\, =\, X_{0}^{(u)}+ \int^{t}_{0}b( s, X_{s}^{\mathrm m_{i}}, u \, \delta_{\widetilde{X}^{\mathrm m_{i}}_{s}}+ ( 1- u) \mathrm m_{i,s}){\mathrm d}t +{\mathrm d}B_{t}\, ; 0 \le t \le T \, ,i \, =\, 1, 2 \, .
\begin{cases}dX^{t,\xi}_{s}=b(X_{s}^{t,\xi})ds+\sum_{i,j=1}^dh_{ij}(X_{s}^{t,\xi})d\langle B^i,B^j\rangle_s+\sum_{j=1}^{d}\sigma_j(X_{s}^{t,\xi})dB^j_s,\ \ \ \s\in [t,T], X_{t}^{t,\xi}=\xi,\end{cases}
\rho(H) \geq x^{\mathrm{T}}(\mathcal{A}x)=\frac{1}{m}\sum_{\{i_1,i_2,\ldots,i_r\}\in E(H)}\sqrt[r]{d_{i_1}d_{i_2}\cdots d_{i_r}}& \geq\Bigg(\prod_{\{i_1,i_2,\ldots,i_r\}\in E(H)}\sqrt[r]{d_{i_1}d_{i_2}\cdots d_{i_r}}\Bigg)^{\frac{1}{m}}.
\int_0^\infty{{x^q}{e^{- px}}}{}_1{F_1}\left({a,b;\omega x}\right)dx = \frac{{\Gamma \left({q \!+\! 1}\right)}}{{{p^{q \!+\! 1}}}}{}_2{F_1}\left({a,q \!+\! 1;b;\frac{\omega}{p}}\right)
\hat{h}_{v}^{- 1}\frac{\sinh m x}{\cosh^{2 s_{0}+ 2}m x}= \int \frac{d k}{k^{2}+ m^{2}}\mid \varphi_{k}> < \varphi_{k}\mid \frac{\sinh m x}{\cosh^{2 s_{0}+ 2}m x}>
\begin{array}{l}\partial_-\partial_0=q^{-1}\partial_0\partial_-~,~~~~~\partial_0\partial_+=q^{-1}\partial_+\partial_0~, \partial_-\partial_+-\partial_+\partial_-=(q^{1/2}-q^{-1/2})\partial_0\partial_0~.\end{array}
\mathrm{div}_T\Big(h\,\tfrac{1}{2}\nabla_\xi H^2(\nabla v) \Big) \,h\,\tfrac{1}{2}\nabla_\xi H^2(\nabla v)\cdot \nu -h^2\, \nabla_T\Big[ h\,\tfrac{1}{2}\nabla_\xi H^2(\nabla v)\Big] \,\Big[\tfrac{1}{2}\nabla_\xi H^2(\nabla v)\Big]_T\cdot \nu =h\,H(\nu)\,H^2(\nabla v)\,\mathrm{tr}\,\mathcal{B}^H
|\mathbf{r}_{ij}|^{2}= |(\mathbf{a}_{1}\zeta+\mathbf{a}_{2}\zeta^{2}+...)^{2}| &= |\mathbf{a}_{1}|^{2}|\zeta|^{2}+2\mathbf{a}_{1}\cdot\mathbf{a}_{2}\zeta^{3}+... r_{ij}= |\mathbf{a}_{1}||\zeta|\Big(1+2\frac{\mathbf{a}_{1}\cdot\mathbf{a}_{2}}{|\mathbf{a}_{1}|^{2}}\zeta+... \Big)
\begin{array}{l l}{i S_{e f f}[ \sigma , A ]}{\hspace *{2 1 e m}}{= i \left\{\frac 1 2 \sigma i G_{\sigma}^{- 1}\sigma + \frac 1 2 A^{\mu}i D_{\mu \nu}^{- 1}A^{\nu}\right\}}{\hspace *{2 1 e m}}\end{array}
\Pi_{\rho_1 + \cdots + \rho_n}= \sum_{i=1}^n \langle \Pi_{\rho_1+\cdots+ \rho_n}, x^{\rho_1 + \cdots + \rho_{i-1}}\rangle \beta(\rho_1, \dots ,\rho_i) & = \sum_{i=1}^n \pi_{\rho_i + \cdots+ \rho_n}\beta(\rho_1, \dots, \rho_i).
(Y^2)_{mn}= \sum_{a=1}^x Y_{m a}Y_{a n}=\sum_{a=1}^x (\sum_{q=1}^Q S(m,a;q)) (\sum_{r=1}^Q S(a,n;r)) = \sum_{q=1}^Q \sum_{r=1}^Q \sum_{a=1}^x S(m,a;q) S(a,n;r) = \sum_{q=1}^Q \sum_{r=1}^Q x \delta (q,r) c_q (m-n) = x \sum_{q=1}^Q c_q(m-n)=x X_{mn},
\Omega_n=\frac{1}{\sqrt \pi}\left(\frac{2\pi e}{n}\right)^{\frac n2}\frac{\exp\left(\frac 16 E_0(\frac n2)\right)}{\sqrt[6]{8\Phi_0(\frac n2)}},\Omega_n=\frac{1}{\sqrt \pi}\left(\frac{2\pi e}{n}\right)^{\frac n2}\frac{\exp\left(\frac 16 E_1(\frac n2)\right)}{\sqrt[6]{8\Phi_0(\frac n2)+8\mathrm{MC}_1(\frac n2)}}.
\|\mathcal{D}_{J,1}^{m_1}\mathcal{D}_{J,2}^{m_2}\widehat{d}_{j}\|^2_{L^2(\mathcal{W}^+_J)}\cdot \|\mathcal{D}_{J,1}^{n_1}\mathcal{D}_{J,2}^{n_2}W_J\|^2_\infty \lesssim 2^{-2jm_1s}2^{-2jm_2s\alpha}2^{js(1-\alpha)}\ell_J^{2m_1-2\beta-1}\cdot 2^{-2jsn_1}2^{-2js\alpha n_2}&= 2^{-2jsd_1}2^{-2sj\alpha d_2}2^{js(1-\alpha)}\ell_J^{2m_1-2\beta-1}.
\int_{-h}^0 |v(x', z)|^p \, dz = \int_{-h}^0\int_{G'}\big( |v(x', z)|^p - |v(y', z)|^p \big) \delta_{\varepsilon,x'_0}(y') \, dy'dz+ (v, \delta_{\varepsilon, x_0'}|v|^{p-2}v^*)_{\Omega'}=: I_1(x') + I_2,
var_{P}\left[ \psi_{P,\mathbf{a}}\left( \mathbf{B};\mathcal{G}\right) \right] =var_{P}\left[ \psi_{P,\mathbf{a}}\left( \mathbf{G},\mathbf{B};\mathcal{G}\right) \right] +\sum_{k=0}^{p}var_{P}\left[ r_{k}\left( \overline{\mathbf{A}}_{k},\overline{\mathbf{G}}_{k},\overline{\mathbf{B}}_{k};s_{\mathbf{a,}k}^{\ast},P\right) \right].
\langle \overline{D}_x \rangle=\bigg\langle \partial_x + y^i_x\partial_{y^i}-\bigg(\Gamma_{11}^k+ (2\Gamma_{1m}^k-\delta^k_m\Gamma_{11}^1) y^m_x+ (\Gamma_{jm}^k - 2\delta^k_m \Gamma_{1j}^1 ) y^j_x y^m_x- \Gamma_{ij}^1 \, y^i_x y^j_x y^k_x\bigg) \partial_{y^k_x}\bigg\rangle
F(\infty)-F(0)=(-1)^{m+M}2(2M+1)(2M+2)\binom{2M}{k_{\rm id}-2}\binom{k_{\rm id}-2}{\frac{k_{\rm id}-2}{2}-m_{\rm id}}^{-1}\binom{k_{c}-2}{\frac{k_{c}-2}{2}-m_{c}}^{-1}.
-d_1\rho_{j}\Delta w_{\rho_{j}}= \rho_{j}w_{\rho_{j}}g\left(\rho_{j}w_{\rho_{j}}\right)\left[f\left( \rho_{j}w_{\rho_{j}}\right)-v_{\rho_{j}}\right] \ge \rho_{j}w_{\rho_{j}}g\left(\rho_{j}w_{\rho_{j}}\right)\left[f\left( \rho_{j}w_{\rho_{j}}\right)-v_\epsilon\right]
D^{j_7}_{u_hu_g}(U_7)D^{j_8}_{-v_h,-v_e}(U_8^\dagger)D^{j_9}_{-w_f,-w_b}(U_9^\dagger)D^{j_{10}}_{x_cx_g}(U_{10})D^{j_{11}}_{-y_h,-y_d}(U_{11}^\dagger)D^{j_{12}}_{z_az_e}(U_{12})
d{}^{*}{\hat{F}}^{( 4 )}+ d A^{( 3 )}\wedge{\hat{F}}^{( 4 )}= H^{( 3 )}\wedge{}^{*}J^{( 6 )}, \qquad d{}^{*}{\hat{F}}^{( 7 )}\equiv - d{\hat{F}}^{( 4 )}={}^{*}J^{( 6 )}.
D^{+\infty}_z(T) = \bigcup_{\gamma \in T}D_z^{+\infty}\cdot \gamma \mbox{\,\,\,\,\,\,and\,\,\,\,\,\,}D^{-\infty}_z(T) = \bigcup_{\gamma \in T}D_z^{-\infty}\cdot \gamma .
S_{m,q}(a,l,\vec{c}):=\sum_{\vec{b}\in (\mathbb{Z}/mq \mathbb{Z})^d}e \Big(\frac{(lq+a)(2\lambda^T A\vec{b}-k) + am F(\vec{b}) + \left< \vec{c},\vec{b}\right>}{mq}\Big).
\int_{- \infty}^{\infty}d q^{\prime}U^{- 1}( q , q^{\prime}) \phi ( q^{\prime}) = \frac{( 1 - i )}{2}\sqrt{\pi}\mathrm{e}^{\frac{i \pi a}{2}}\mathrm{s g n}( q ) \left| k q \right|^{\frac{1}{2}}J_{a}( k \left| q \right| ) .
q^{(m-n)(m-j)}\left[ \begin{array}{l}j+m m-n\end{array}\right]^{1/2}_{q^{2}}\left[ \begin{array}{l}j-n m-n\end{array}\right]^{1/2}_{q^{2}}P^{n-m,-m-n}_{j+n}(c^{*}c;q^2)c^{m-n}(a^{*})^{n+m}.
V \partial_{i}V^{- 1}= \frac{\epsilon_{i a k}\tau_{a}\hat{r}_{k}}{2 i r}( \cos \phi - 1 ) - ( \frac{\tau_{i}- \hat{r}_{i}\tau \cdot \hat{r}}{2 i r}) \sin \phi - \hat{\phi}_{i}( \frac{\tau \cdot \hat{r}}{2 i \rho}) ,
\overline{{{A^{\mu}}}}( \overline{{{q}}}) = A^{\nu}( q ) \, \overline{{{a}}}_{\nu}^{\mu}( q ) \, , \quad \left( \overline{{{A_{\mu}}}}( \overline{{{q}}}) = A_{\nu}( q ) \,{a}_{\mu}^{\nu}( q ) \, \right) ;
V(|z|_3) - V(z_{tr}) = \frac{b^2}{36}\frac{|z_{tr}|^4}{(1-|z_{tr}|^2)^2}\left( -\frac{x^2}{3}+\frac{2x^3}{27}+\ldots \right) = V(z_{tr}) \left(-\frac{x}{3}+ \frac{2x^2}{27}+\ldots \right)
\iota(f_{1},f_{2})_{v}={\zeta_{F,v}(2) \over L(1, \sigma_{v}^{\iota}\times \sigma_{v}^{\iota \vee})}\int_{F_{v}^{\times}}\iota f_{1}(y) \iota f_{2}(y){d^{\times}y\over |d|_{v}^{1/2}}.
Q_{1}( t ) = 1 - ( a_{1 1}+ a_{1 2}\, a_{2 2}\, a_{1 2}^{- 1}) \, t + ( a_{1 2}\, a_{2 2}\, a_{1 2}^{- 1}\, a_{1 1}- a_{1 2}\, a_{2 1}) \, t^{2}\ ,
\| X_s^{I, R_1}- \hat X_s \|_H^{p-1}\| \hat X_s - X_s^{I, R_2}\|_{H_\delta}(1 + \|\hat X_s\|_{H_\gamma}^c + \| X_s^{I, R_2}\|_{H_\gamma}^c ) &\leq\tfrac{p-1}{p}\| X_s^{I, R_1}- \hat X_s \|_H^p+\tfrac{1}{p}\| \hat X_s - X_s^{I, R_2}\|_{H_\delta}^p(1 + \|\hat X_s\|_{H_\gamma}^c + \| X_s^{I, R_2}\|_{H_\gamma}^c )^p
{\bf K}( \vec{\varrho}_{f}t_{f}, \vec{\varrho}_{i}t_{i}) \sim \int{\cal D}\left[ \vec{\varrho}\right]{\it e x p}\left[ \frac{i}{\hbar}\int_{t_{i}}^{t_{f}}\left[ L_{\theta}+ L_{0}\right] d t \right] ,
\begin{cases}\eta\equiv1\ \B^{(2)}, 0\leq\eta\leq1,\ |D\eta|\leq c\lambda^{(2-p_0)/(2p_0)}(\rho_2-\rho_1)^{-1}begin{align*}5pt]\zeta\equiv1\ \ \Lambda^{(2)}, 0\leq\zeta\leq1,\ |\partial_t\zeta|\leq c(\rho_2^2-\rho_1^2)^{-1}.\end{cases}
(\lambda+\mu)\frac{\partial}{\partial z}(\frac{\partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y}+\frac{\partial u_{z}}{\partial z})+\mu(\frac{\partial^{2}u_{z}}{\partial x^{2}}+\frac{\partial^{2}u_{z}}{\partial y^{2}}+\frac{\partial^{2}u_{z}}{\partial z^{2}})+F_{z}=0
(\sum_{j=1}^2\partial_{\alpha}^k f_j(\cdot,t))\circ g= \sum_{j=1}^2 \frac{\lambda_j i}{2\pi}\frac{(-1)^k k!}{(\zeta(\alpha,t)-z_j(t))^{k+1}}= \frac{\lambda i(-1)^k k!}{2\pi}\sum_{m=0}^k \frac{z_1-z_2}{(\zeta(\alpha,t)-z_1(t))^{k+1-m}(\zeta(\alpha,t)-z_2(t))^{m+1}}
G_2(x) =\sum_{d\geq0}\left(\frac{x^{d+2}}{(1-x)^{d+1}}F_T(x)+\sum_{k=1}^d\frac{x^{d+2}}{(1-x)^k}\left(F_T(x)-\frac{1}{1-x}\right) +x^{d+2}F_T(x)\left(C(x)-\frac{1}{1-x}\right)\right) =\frac{x^2}{1-2x}F_T(x)+\frac{x^3}{(1-x)(1-2x)}\left(F_T(x)-\frac{1}{1-x}\right)+\frac{x^2}{1-x}F_T(x) \left(C(x)-\frac{1}{1-x}\right)\, .
\sum_{i\in I}\prod_{j\in I, j\neq i}\frac{-z_i-z_j-\beta z_i z_j}{(1+\beta z_j)(z_j-z_i)}= \begin{cases}1 |I| \displaystyle{1 -\prod_{i\in I}\frac{1}{1+\beta z_i}}|I| \end{cases}
\delta_B ( f^{(2)}u^3 \left[ \psi^2 \right] ) + \delta_F ( f^{(4)}u^2 \left[ \psi^4 \right] ) + u^i u \epsilon \dot{N}^{i(2)}\left[ \psi^2 \right] \psi+ \frac{i}{2}u^2 \psi \dot{M}^{(2)}\left[ \psi^2 \right] \epsilon=0 ~,
\begin{cases}x_{1}^{\left(t+1\right)}=\;\delta_{1}x_{2}^{\left(t\right)}y^{\left(t\right)}\medskip x_{2}^{\left(t+1\right)}=\;\gamma_{2}x_{1}^{\left(t\right)}y^{\left(t\right)}\medskip y^{\left(t+1\right)}=\;\bigl(\gamma x_{1}^{\left(t\right)}+\delta x_{2}^{\left(t\right)}\bigr)y^{\left(t\right)}.\end{cases}
\psi_{\scriptscriptstyle{l}}( k , r ) = \frac{e^{- i \pi W / 2}\, e^{i \pi | l |}}{2 \sqrt 2}\, \left( H_{\scriptscriptstyle{W}}^{( - )}( k r ) + e^{i \pi ( W - | l | )}\, e^{2 i \delta_{l}( k )}\, H_{\scriptscriptstyle{W}}^{( + )}( k r ) \right) \, ,
p^*\left(\prod_{f\in I}X_f\right)=\prod_{i=1}^n A_{i'}=\prod_{i=1}^n\frac{\lambda_{i-a}\Delta_{\{i-a,i-a+1,\dots, i-1\}}}{\Delta_{\{i-a+1,i-a+2,\dots, i\}}}= \prod_{i=1}^n \lambda_i.
S_g = \int_Cd^4\eta\big{[}\frac{1}{8\pi G_0}\Lambda - \frac{1}{16\pi G_0}(R + K^2) + \alpha_0R^2 + \beta_0RK^2\big{]}+ \oint_{\partial C}d^3\eta\big{[}\gamma_0RK + \delta_0K^3\big{]}
\mathcal{H}^s_{\infty}(E_n^{(1)})\ll\begin{cases}\left(\frac{\delta}{3^n}\right)^s \tilde{\delta}\leq \frac{1}{3^n}, 3^n\cdot \delta\cdot \left(\frac{\delta}{3^n}\right)^s \tilde{\delta}> \frac{1}{3^n}.\end{cases}
\begin{aligned}\int_{B_{2}}\vert \nabla w \vert^{2}dx \leqslant \frac{1}{12}\int_{\mathbb{R}^{N}}(\alpha u+ \beta v)w dx+CK^{2}r_{0}^{1-N}+ CKr_{0}^{\frac{1-N}{2}}\Vert f \Vert_{2}+ CK + Cr_{0}\Vert f \Vert_{2}^{2}+ C\sqrt{K}\Vert g_{1}\Vert_{2}+ C\sqrt{K}\Vert g_{2}\Vert_{2}+\int_{B_{2}}w^{2}dx,\end{aligned}
Z ( s ) = \sum_{n_{1}, n_{2}}\left( \tilde{n}_{1}^{2}+ \frac{1}{| \tau |^{2}}\tilde{n}_{2}^{2}\right)^{- s}= \frac{1}{\Gamma ( s )}\sum_{n_{1}, n_{2}}\int_{0}^{\infty}\frac{d t}{t}t^{s}e^{- ( \tilde{n}_{1}^{2}+ \frac{1}{| \tau |^{2}}\tilde{n}_{2}^{2}) t}\equiv \frac{1}{\Gamma ( s )}J ( s ) ,
G_{\mu \nu}( q ) = \frac{- i}{q^{2}+ i \epsilon}\left[ g_{\mu \nu}- \frac{q \cdot N^{\ast}( q_{\mu}N_{\nu}+ q_{\nu}N_{\mu})}{q \cdot N q \cdot N^{\ast}+ i \epsilon}+ \frac{N^{2}( q \cdot N^{\ast})^{2}q_{\mu}q_{\nu}}{( q \cdot N q \cdot N^{\ast}+ i \epsilon )^{2}}\right] .
\rho_{W}\left( e^{h}\right) =\exp\left(\sum_{k=1}^{\infty}kb_{k}h_{k}\right) =\exp\left( \dfrac{1}{2\pi i}\int_{\left\vert z\right\vert =r_{W}}h^{\prime}\left( z\right) \log\left(1+\varphi_{W}\left( z\right) \right) \right) dz
M_f S_0 =M_f \sum_{\sigma\in W_f}q^{\ell(w_0^f)-\ell(\sigma)}H_\sigma\sum_{\tau\in W^f}q^{\ell(^fw_0)-\ell(\tau)}H_\tau &=M_f \sum_{\sigma\in W_f}q^{\ell(w_0^f)-2\ell(\sigma)}\sum_{\tau\in W^f}q^{\ell(^fw_0)-\ell(\tau)}H_\tau &=[|W_f|]M_f \sum_{\tau\in W^f}q^{\ell(^fw_0)-\ell(\tau)}H_\tau &=[|W_f|]\sum_{\tau\in W^f}q^{\ell(^fw_0)-\ell(\tau)}M_{f\cdot \tau}.
I = \frac{\Gamma(\beta+2) \, \Gamma(\gamma)}{\Gamma(\beta+\gamma+1)}\, (1-x)^{-\beta}\, \int_0^1 v^{\beta+\gamma}\, (1-xv)^{-(\gamma-\beta)}\, dv & = \frac{\sqrt{\pi}\, \Gamma(\beta+2)}{\Gamma(\beta+\gamma+2)}\, (1-x)^{-\beta}\,{}_2 F_1(\gamma-\beta,\beta+\gamma+1;\beta+\gamma+2;x),
\kappa \cos^{2}{( \frac{\pi}{2}- \delta )}\approx 1 + \frac{1}{2 \kappa^{2}\cos^{4}{( \frac{\pi}{2}- \delta )}}- \frac{c_{1}\sin{( 2 \kappa \sin{t})}+ c_{2}\cos{( 2 \kappa \sin{t})}}{\kappa^{2}\cos{( \frac{\pi}{2}- \delta )}}.
2(\cos(2\overline{v}_{1}(t,R))-1)\partial_{R}\overline{v}_{1}(t,R) - 2(\cos(2\overline{v}_{2}(t,R))-1)\partial_{R}\overline{v}_{2}(t,R) &=2\left(\cos(2\overline{v}_{1}(t,R))-1\right)\left(\partial_{R}(\overline{v}_{1}-\overline{v}_{2})\right)+2\partial_{R}\overline{v}_{2}\left(\cos(2\overline{v}_{1})-\cos(2\overline{v}_{2})\right)
F^{\Delta_1\Delta_2\Delta_3}_{x_1x_2x_3}=\frac{1}{|x_{12}|^{\Delta_1+\Delta_2-\Delta_3}\,|x_{23}|^{\Delta_2+\Delta_3-\Delta_1}\,|x_{31}|^{\Delta_3+\Delta_1-\Delta_2}}.
A \, | k \rangle = \sum_{l = 0}^{[k/2]}a_{k,l}(g) | k - 2 l \rangle \,,\;\;D \, | k \rangle = \sum_{l = 0}^{[k/2]}d_{k,l}(g) | k - 2 l \rangle \,, & C \, | k \rangle = \sum_{l = 0}^{[\frac{k-1}{2}]}c_{k,l}(g) | k - 1 - 2 l \rangle \,,