harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
p_{n}\cdot p_{k}=|p_{n}||p_{k}|cos\theta_{nk},|p_{n}|=\frac{1}{c}\sqrt{E_{n}^{2}-(m_{n}c^{2})^{2}},|p_{k}|=\frac{1}{c}\sqrt{E_{k}^{2}-(m_{k}c^{2})^{2}},
|(-\Delta_p)^s\Upsilon(x)|\le \begin{cases}\dfrac{c_1}{|x|^{\alpha(p-1)+ps}}\alpha(p-1)<N, begin{align*}10pt] \dfrac{c_2\log(|x|)}{|x|^{N+ps}}\alpha(p-1)=N, begin{align*}10pt] \dfrac{c_3}{|x|^{N+ps}}\alpha(p-1)>N, \end{cases}
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1- y^m z^n}\right)^{\frac{m^1}{n^2}}= \exp\left\{\sum_{n=1}^{\infty}\left( \frac{y - (1 + n) y^{n+1}+ n y^{n+2}}{(1-y)^2}\right) \frac{z^n}{n^2}\right\}
\Delta_{\mu, \kappa}(4\eta; -8\eta, \eta) = - 2\pi i e^{-\frac{2\pi i \eta}{\kappa}\mu^2 - 2\pi i \mu\eta}\frac{\Gamma(-6\eta; -2\kappa\eta, -8\eta)}{\Gamma(-2\eta; -2\kappa\eta, -8\eta)}\frac{\theta_0(2\mu\eta; -2\kappa\eta) (-2\kappa \eta; -2\kappa\eta)^2}{(-4\eta; -4\eta)(-2\eta+1/2; -2\eta)}.
\frac{d\mathbb{P}_{N}^{\tilde{b},\tilde{\Gamma}}}{d\mathbb{P}_{N}}=\exp\left\{\sum_{c=1}^{m}\sum_{i\in I_{N}^{c}}U_{i}^{(c)}+\sum_{\substack{1\le c_{1},\,c_{2}\le m,\,i\in I_{c_{1}}^{N},\,j\in I_{c_{2}}^{N}}}V_{ij}^{(c_{1},\,c_{2})}\right\}
\frac{\partial \beta}{\partial t}=\dot{\beta}={}\dot{\gamma}+ u^{1}\dot{X}_{1}+ \dotsb + u^{m-d-1}\dot{X}_{m-d-1}&+ \dot{u}^{m-d}X_{m-d}+ u^{m-d}\dot{X}_{m-d}+ \dotsb &+ \dot{u}^{m-1}X_{m-1}+ u^{m-1}\dot{X}_{m-1},
\hat{\mathbb{E}}_{\tau+}[\varphi(\phi_m(X^x_{\tau+t^m_0},X^x_{\tau+t^m_1},X^x_{\tau+t^m_2},\cdots,X^x_{\tau+t_m^m}))]=\hat{\mathbb{E}}[\varphi(\phi_m(X^y_{t^m_0},X^y_{t^m_1},X^y_{t^m_2},\cdots,X^y_{t_m^m}))]_{y=X^x_{\tau}}.
\tilde{m}_{0}^{\ast}: = K^{\ast - 1}m_{0}K^{\ast}= \left( \begin{array}{c c}{{0}}&{{- \kappa_{e}^{\frac{1}{2}}T \kappa_{o}^{-{\frac{1}{2}}}}}\\{{- \kappa_{o}^{-{\frac{1}{2}}}\bar{T}\kappa_{e}^{\frac{1}{2}}}}&{{0}}\\ \end{array}\right) = - \tilde{m}_{0}\, .
\rho(\xi-y) = \sum_{|\ell|<N_1}\frac{(-y)^{\ell}}{\ell!}(\partial^{\ell}\rho)(\xi)+R_{N_1}(\xi,y) \alpha_x(\rho(\xi-y)) = \sum_{|\ell|<N_1}\sum_m\frac{(-y)^{\ell}}{\ell!}(\partial^{\ell}\rho_m)(\xi)e^{ix\cdot m}\left(\prod_{j=1}^nU_j^{m_j}\right)+\alpha_x(R_{N_1}(\xi,y))
\mathfrak{I}_2u-\mathfrak{I}_1u= T\beta^{-1}(\rho(a,b)u)-\alpha^{-1}([a,b,Tu]) = T\rho(a,b)\beta^{-1}(u)-[a,b,T\beta^{-1}(u)] = \wp(a,b)u\in \mathcal{B}^1_T(V,L),
{\frac{e^{2}}{6}}c_{A}\sum_{k}{\frac{1}{\omega_{k}^{3}}}\int d^{3}x E^{2}={\frac{e^{2}}{6}}{\frac{c_{A}}{2 \pi^{2}}}l o g \left({\frac{\Lambda}{\mu}}\right) \int d^{3}x E^{2}.
\mathbb{E}\left.\left(\exp\left(i \sum_{j=1}^n \theta_j X(t_j) \right)\right| \mathcal{F}^{\sigma}\right)=\exp\left(-\frac{1}{2}\int_{\mathcal{X}\times \mathbb{R}^d}\left(\sum_{j=1}^n\theta_j g(x,t_j-s)\right)^2 \sigma^2(s)p(dx) ds\right).
R(x)=2\sum_{k=1}^{\infty}\frac{k^{\overline{k}}x^{k}}{(2\pi)^{2k}(\frac{B_{2k}}{2k})}=2\sum_{k=1}^{\infty}\frac{k^{\overline{k}}x^{k}}{(2\pi)^{2k}\beta_{2k}}.
- i M^{2}= i g^{4}C_{2}( R ) C ( R^{\prime}) \int_{q}\int_{k 5 5}{\frac{N ( k , k^{5},{\hat{k}}^{5}, q )}{( k^{2}) ( k^{2}- ( k^{5})^{2}) ( k^{2}- ({\hat{k}}^{5})^{2}) ( q^{2}) ( ( q - k )^{2})}}\ ,
Q(x^{\Lambda_2}| y) = - \log \int_{\substack{\frac{1}{K-R}\sum_{i \in \Lambda_1^{(k)}}x_i = \tilde{y}_k k \in [M] \setminus \{l \}}}\exp\left(-H(\bar{x}^{B(l)})\right)\exp \left( - Q_l (x^{\Lambda_2^{(l)}}| \bar{x}^{B(l)}) \right) \mathcal{L}(d \bar{x}^{\Lambda_1^{(l)}}),
\sum_{\substack{b\leq N/a (a,b)=1 b\equiv\psi(a)\mod{n}}}\mu^2(b)=\frac{\phi(a)N}{a^2n}\cdot\frac{1}{\zeta(2)}\cdot\prod_{p\mid an}\frac{p^2}{p^2-1}+O(2^{\omega(a)}\sqrt{N/a}).
(\tilde{Q}_i)_{j,k}=([\tilde{Q}]_{i+1,i})_{j,k}=-\sum_{h=0}^{m-1}[\tilde{v}_{i-h}\tilde{w}_{i-h}]_{j,k}\frac{\lambda_j^{m-h-1}\lambda_k^h}{\lambda_j^m-\lambda_k^m},
F^{(0)}_{p,q}(t)= \frac{t^p\left(t^{q+2}-2t^{q+1}+1\right)+t^{q}-1}{(t-1)^2}, F^{(1)}_{p,q}(t)= \frac{t^{p-2}\left(t^{q+2}-2t^{q+1}+1\right)-t^{q}+1}{t-1}= \frac{t^{q}\left(t^{p}-2t^{p-1}-1\right)+t^{p-2}-1}{t-1}, F^{(2)}_{p,q}(t)= t^{p-2}\left(t^{q}-2t^{q-1}-1\right)-t^{q-2}-1.
\zeta_{\mathcal{A}}(s_{d},\ldots,s_{1}) \mapsto \sum_{d'=0}^{d}(-1)^{s_{d'+1}+\ldots+s_{d}}\zeta(s_{d'+1},\ldots,s_{d})\zeta(s_{d'},\ldots,s_{1}) \mod \zeta(2)
\int_{|z|<R}|D\log|f|(z)|^2\,dz=\int_{|z|<R}\frac{|D|f|(z)|^2}{|f(z)|^2}\,dz=\int_{|\xi|<tR}\frac{|D|f|(\frac{\xi}{t})|^2}{|f(\frac{\xi}{t})|^2}d\frac{\xi}{t}&=\int_{|\xi|<tR}\frac{(\frac{1}{t})^{2d-2}|D|f|(\xi)|^2}{(\frac{1}{t})^{2d}|f(\xi)|^2}\frac{1}{t^2}\,d\xi=\int_{|\xi|<tR}|D\log |f| (\xi)|^2\,d\xi,\forall t>0.
x^{-1}sx \simeq_R^f zww^{-1}y^{-1}syww^{-1}z^{-1}(\; x =_G yww^{-1}z^{-1}) &\simeq_R^f zww^{-1}z^{-1}tzww^{-1}z^{-1}(\; w^{-1}y^{-1}syw\simeq_R^f w^{-1}z^{-1}tzw) &\simeq_R^f t (\; zww^{-1}z^{-1}=_G \epsilon,\; \; \epsilon\; ).
\left( - \frac{h^{2}}{8 \pi^{2}M_{1}}\bigtriangledown_{1}^{2}- \frac{h^{2}}{8 \pi^{2}M_{2}}\bigtriangledown_{2}^{2}+ V_{n}( r ) \right) \psi_{\mathrm{v i b - r o t}}= E_{t}\psi_{\mathrm{v i b - r o t}},
P_1(Zg) =(Z_{21}g_{11}+Z_{22}g_{21})(Z_{21}g_{12}+Z_{22}g_{22})-(b^2-a)(Z_{11}g_{11}+Z_{12}g_{21})(Z_{11}g_{12}+Z_{12}g_{22}) &=Z_{21}Z_{22}g_{11}g_{22}+Z_{22}Z_{21}g_{21}g_{12}-(b^2-a)(Z_{11}Z_{12}g_{11}g_{22}+Z_{12}Z_{11}g_{21}g_{12}) &=P_1(Z)g_{11}g_{22}+P_1(Z)g_{21}g_{12}=0
\sum_{\bar{\nu}^\flat \in{\mathcal Wt}(\bar{\lambda}^\flat)^\tau}{\rm tr}(\tau \, | \, V_{\bar{\lambda}^\flat, 1}(\bar{\nu}^\flat)) \, e^{\bar{\nu}^\flat}= \sum_{w \in W_0}w \Big( \prod_{\alpha \in N'_\tau(\breve{\Sigma}^\vee)^+}\frac{1}{1 - e^{-\alpha}}\, \Big) \cdot e^{w\bar{\lambda}^\flat}.
e^{t \lambda_{+}+\frac{t}{\nu}|\xi|^{2(1-\sigma)}}-1=\left(t \lambda_{+}+\frac{t}{\nu}|\xi|^{2(1-\sigma)}\right)e^{\theta(t \lambda_{+}+\frac{t}{\nu}|\xi|^{2(1-\sigma)})}
G=\nabla g=[\begin{matrix}\partial g_{x}/\partial x&\partial g_{x}/\partial y&\partial g_{x}/\partial z\\ \partial g_{y}/\partial x&\partial g_{y}/\partial y&\partial g_{y}/\partial z\\ \partial g_{z}/\partial x&\partial g_{z}/\partial y&\partial g_{z}/\partial z\end{matrix}]
1 + \left( \begin{array}{c}{{n}}\\{{2}}\\ \end{array}\right) + \left( \begin{array}{c}{{n}}\\{{4}}\\ \end{array}\right) + \ldots = 2^{n - 1}\, \, \, \, \, , \, \, \, \, \, \left( \begin{array}{c}{{n}}\\{{1}}\\ \end{array}\right) + \left( \begin{array}{c}{{n}}\\{{3}}\\ \end{array}\right) + \left( \begin{array}{c}{{n}}\\{{5}}\\ \end{array}\right) + \ldots = 2^{n - 1}\, \, \, \, \, , \, \, \, \, \,
\allowdisplaybreaks\begin{aligned}\bigvee_{j = 1}^{i-1}(-u_i + x_j) \vee (-u_{i+1}+ 1 + x_i) \leq \bigvee_{j = i+1}^{n-1}(-u_j + x_j) \vee x_n \vee (-u_n) \enspace i = 1, \dots, n-1 \bigvee_{j = 1}^n x_j \leq 0 -\infty \leq x_1 \leq x_2 \leq \dots \leq x_{n-1}\leq u_n + x_n \, .\end{aligned}
\tilde{\rho}_{B^2}(s) = \frac{\beta}{2\pi}\ \ \ \ \ \ \rho_{B^2}(s) = \frac{\beta}{2\pi}\sqrt{\frac{b^2 -s}{s-c^2}}\ \ \ \ \ \ \rho_{B}(\eta ) = \frac{\beta \eta}{\pi}\sqrt{\frac{b^2 -\eta^{2}}{\eta^{2}-c^2}}
\Pi_{1}\,=\, -\frac{1}{4\pi (-z)^{1/2}}\arctan \sqrt{-z};\, \,\, \,\, \Pi_{2}=\,\frac{1}{16\pi m\, z}\left[ 1-\left( \frac{1+z}{2 (-z)^{1/2}}\right)\arctan\sqrt{-z}\right] .
F ( x , y , k^{--}p^{-}) = - 2 C_{F}g^{2}P^{+}\psi ( y ){\frac{\sqrt{x ( 1 - x )}}{\sqrt{y ( 1 - y )}}}{\frac{( k^{--}p^{-})}{{\cal E}- \omega ( 1 - x ) - \omega ( x ) + i \epsilon}}.
\exp\left(-\sum_{n=1}^N\Big(\int_0^x\varphi_2s_n\,dt\Big)\frac{1}{(2i\lambda)^n}\right)=1+ \frac{1}{2 i\lambda}\int_0^x \varphi_1\varphi_2\,dt+ O\Big(\frac{1}{\lambda^2}\Big)
\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial v}(X_t, \dot X_t, t) = \dot \gamma_t \frac{\partial\mathcal{L}}{\partial v}(X_t, \dot X_t, t) + e^{\gamma_t}\frac{d}{dt}\nabla h(Z_t) - e^{\gamma_t}\frac{d}{dt}\nabla h(X_t).
\int d \Phi_{1}( r_{*}) d \tilde{\Phi}_{2}( r_{*}) \exp{i \int d r_{*}( \Phi_{1}( r_{*}){\cal D}_{1}\Phi_{1}( r_{*}) + \tilde{\Phi}_{2}( r_{*}){\cal D}_{1}\tilde{\Phi}_{2}( r_{*}) )}
K(w\prime_{n},w\prime_{m}^{\ast};w\prime_{r},w\prime_{s}^{\ast};t)=\sum_{\kappa>0}f_{\kappa}(w\prime_{n},w\prime_{m}^{\ast})f_{\kappa}(w\prime_{r},w\prime_{s}^{\ast})e^{iE_{\kappa}t}
D_\alpha^{\mathbf i}= \partial_\alpha^{\mathbf i}+ i{\bar\theta}^{\dot\alpha{\mathbf i}}\partial_{\alpha\dot\alpha}\ ,\qquad\quad{\bar D}_{\dot\alpha{\mathbf i}}= -{\bar\partial}_{\dot\alpha{\mathbf i}}- i{\theta}^\alpha_{\mathbf i}\partial_{\alpha\dot\alpha}.
F(x_0) =sx_0^{2/s}+tA^{1/t}x_0^{-1/t}= (s+2t) x_0^{2/s}= (s+2t) 2^{-2t/(s+2t)}A^{2/(s+2t)}& = n2^{-2t/n}A^{2/n}= n2^{s/n-1}A^{2/n}.
\mathbb{E}[\mathbf{w}'\mathbf{w}'^{\mathcal{H}}] =p\kappa^{2}_{T}\left(\mathbf{H}+\mathbf{\Delta H}\right)\left(\mathbf{H}+\mathbf{\Delta H}\right)^{\mathcal{H}}&+p\mathbf{\Delta H}\left(\mathbf{\Delta H}\right)^{\mathcal{H}}+(p\kappa^{2}_{R}m+N_{0})\mathbf{I}_{n}.
h^{\alpha\gamma}h^{\beta\delta}\partial_\alpha\cdot\partial_\beta\cdot R^P(d\phi(\partial_\gamma),d\phi(\partial_\delta))\psi= -2s^2g^{ij}\partial_t\cdot\partial_i\cdot R^P(d\phi(\partial_t),d\phi(\partial_j))\psi &+g^{ij}g^{kl}\partial_i\cdot\partial_k\cdot R^P(d\phi(\partial_j),d\phi(\partial_l))\psi.
\widehat{H}_0:=\frac{2\sigma+1}{8\sigma^2}|\xi|^4t\cos(|\xi|^2t)\mathrm{e}^{-\frac{1}{2\sigma}|\xi|^4t}\ \ \mbox{and}\ \ \widehat{H}_1:=\frac{1}{2\sigma^2}|\xi|^4t\sin(|\xi|^2t)\mathrm{e}^{-\frac{1}{2\sigma}|\xi|^4t}.
\biggl(f_1*_1(f_2f_3)\biggr)(\xi)+f_1(\xi)\biggl(f_2*_1f_3\biggr)(\xi)-\biggl(f_1*_1f_2\biggr)(\xi)f_3(\xi)-\biggl((f_1f_2)*_1f_3\biggr)(\xi)=0.
\phi_{3}^{( b r s t )}= \frac{1}{2}( p^{2}+ \gamma ) + \gamma \gamma^{i j}[ \partial_{j}c^{3}\mu_{i}] + ( \partial_{i}c^{i}) \mu_{3}+ \frac{3}{2}\partial_{i}c^{3}\partial_{j}c^{3}\mu_{j}\mu_{i}+ \partial_{i}( c^{i}\mu_{3}) .
a_{0,3}^{\pm}=-\frac{q^{4m-1}-(2q-1)^2q^{2m-1}+4(q-1)(q^2-q+1)}{2q(q-1)^2(q^{2m-2}-1)}\pm\frac{q^{m-1}s}{2q(q-1)^2(q^{2m-2}-1)}.
{\xi_0 \over \lambda}= 2~ i~ \xi_0 \left({1 \over 2 L}\sum_{n= \pm \frac{1}{2}, \pm \frac{3}{2}, \cdots ~}\right) \int_{- \infty}^{\infty}{d p^- \over 2 \pi}{1 \over 2 p^+_n p^- - \xi_0^2 + i \epsilon}~,
(\alpha_x|\beta_y) =\delta(x-y) (\alpha_x|\beta_{\pm \i \pi}) =0 (\alpha_{\pm \i \pi}|\beta_x) =0 (\alpha_{\pm \i \pi}|\beta_{\pm \i \pi}) =1 (\alpha_{\pm \i \pi}|\beta_{\mp \i \pi}) =0
\Lambda(T) = (1+D^2)^{-1/2}[|D|^2,T] = \frac{|D|}{(1+D^2)^{1/2}}\delta(T) + (1+D^2)^{-1/2}\delta(T)|D| = 2\alpha(T) - (1+D^2)^{-1/2}\delta^2(T) = 2\alpha(T)-\beta(T).
\left(\begin{array}{c}\zeta_1^{(2)}\zeta_2^{(2)}\zeta_1^{(1)}\end{array}\right) \circ{\rm gw}_{\infty}= \frac{1}{2}\left(\begin{array}{ccc}1 0 0 1 1 0 0 0 1 \end{array}\right)\left(\begin{array}{c}\lambda_1^{(2)}\lambda_2^{(2)}\lambda_1^{(1)}\end{array}\right).
\{A_{1}^{\pm \pm}( n_{1}, n_{2}) , A_{2}^{\pm \pm}( m_{1}, m_{2}) \}= \frac{2}{N_{k}}\, f^{\pm \pm}( n_{1}, n_{2}) \, \delta_{n_{1}, m_{1}}\, \delta_{n_{2}, m_{2}}\ \ \ ,
{\cal S}_1=\int\sqrt{-G}[-\partial_\xi\phi_{(1)}\partial_\xi\phi_{(0)}-\frac{\partial V}{\partial\phi}\mid_{(0)}\phi_{(1)}+K_\xi(-\frac{1}{2}\partial_\xi\phi_{(0)}\partial_\xi\phi_{(0)}-V(\phi_{(0)}))\xi]d^3\sigma d\xi.
\frac{\pi^2}{12}= \sum_{k \geq 0}\frac{(-1)^k}{(k+1)^2}= \sum_{k \geq 0}(-1)^k \left[ \frac{13}{(3k+1)^2}- \frac{13}{(3k+2)^2}+ \frac{4}{(3k+3)^2}\right],
\begin{matrix}\Omega_{xx}=\sum a_{\alpha\beta}x_{\alpha}x_{\beta}=0\\ \Omega_{yy}=\sum a_{\alpha\beta}y_{\alpha}y_{\beta}=0\\ \Omega_{xy}=\sum a_{\alpha\beta}x_{\alpha}y_{\beta}\\ (\alpha,\beta=1,2)\end{matrix}
a_{k,n}=\int_\frac{k-1}{n}^\frac{k}{n}6t(1-t)\ln\left(\frac{t}{1-t}\right)\mathrm{d}t &=\frac{k^2(3n-2k)}{n^3}\ln\frac{k}{n-k}-\frac{(k-1)^2(3n-2k+2)}{n^3}\ln\frac{k-1}{n-k+1}&\ +\ln\frac{n-k}{n-k+1}+\frac{1-2k}{n^2}+\frac{1}{n}\ , b_{k,n}=\int_\frac{k-1}{n}^\frac{k}{n}6t(1-t)\mathrm{d}t=\frac{3(2k-1)}{n^2}+\frac{2(-3k^2+3k-1)}{n^3}\ .
pdf_{MJ}(\gamma;\theta)d\gamma=\frac{\pi^{\frac{1}{2}}2^{\frac{1-d}{2}}}{\Gamma(\frac{d}{2})}K_{\frac{d+1}{2}}(\frac{1}{\theta})^{-1}\theta^{\frac{1-d}{2}}e^{-\frac{\gamma}{\theta}}(\gamma^{2}-1)^{\frac{d}{2}-1}\gamma d\gamma
m_{[i_{1}-1,i_{1}-1]}m_{[i_{1},i_{1}]}+\sum_{s=i_{1}}^{j_{2}-1}m_{[s,s]}m_{[s+1,s+1]}+m_{[j_{2},j_{2}]}m_{[j_{2}+1,j_{2}+1]}=(2(j_{2}-i_{1})+2)m_{[i_{2},j_{2}]}m_{[i_{1},n]}+\cdots,
\mathbf{b}_{\alpha,\rho}= \mathbf{f}_{\alpha,\rho}\boxtimes (\mathbf{f}_{\alpha,1})^{-1}, \alpha\in(0,1] , \rho \in \{0,1/2,1\}, &\mathbf{b}_{\alpha,\rho}=\mathbf{n}_{\alpha,\rho}\circledast (\mathbf{n}_{\alpha,1})^{-1}, \alpha\in(0,1] , \rho \in[0,1].
{\cal L}_{\mathrm{b u l k}}+{\cal L}_{\mathrm{s o u r c e}}\sim \int d^{n}\xi \; \Bigl [{\frac{1}{g_{\mathrm{I}}^{2}}}( \partial_{\xi}m_{a})^{2}+{\frac{1}{g_{\mathrm{I}}}}\sum_{s}f_{s}( m_{a}) \delta ( \xi - \xi_{s}) \Bigr ] .
C_{0,0,0}^{l_{1},l_{2},l_{3}}=\frac{(-1)^{(l_{1}+l_{2}-l_{3})/2}\sqrt{2l_{3}+1}\Delta(l_{1},l_{2},l_{3})((l_{1}+l_{2}+l_{3})/2)!}{((-l_{1}+l_{2}+l_{3})/2)!((l_{1}-l_{2}+l_{3})/2)!((l_{1}+l_{2}-l_{3})/2)!}\ ,
|\Sigma_k(a_1,\cdots,a_k)| =\sum_{i_1,\cdots,i_{k}\}\prod_{j=1}^k(\theta_{i_j+1}-\theta_{i_j}-a_j)_+ =\sum_{i_1,\cdots,i_{k}\}\prod_{j=1}^k(m_{i_j}-a_j)_+,
b(y) = y^{-1}g \Bigl( \frac{y^2 n_1^4}{N p}\Bigr) \widehat{k^*}\Bigl( \frac{M T c m p}{2 \pi^2 y n_1^2}\Bigr) e \Bigl( \frac{-T^2 c m p}{4 \pi^2 y n_1^2}\Bigr) .
e^{- 1}{\cal L}= -{\frac{1}{2}}R -{\frac{1}{4}}G_{I J}F_{\mu \nu}{}^{I}F^{\mu \nu J}-{\frac{1}{2}}g_{i j}\partial_{\mu}\phi^{i}\partial^{\mu}\phi^{j}+{\frac{e^{- 1}}{4 8}}\epsilon^{\mu \nu \rho \sigma \lambda}C_{I J K}F_{\mu \nu}^{I}F_{\rho \sigma}^{J}A_{\lambda}^{k}\ .
E_{3z^{2}-r^{2},3z^{2}-r^{2}}=[n^{2}-(l^{2}+m^{2})/2]^{2}V_{dd\sigma}+3n^{2}(l^{2}+m^{2})V_{dd\pi}+\frac{3}{4}(l^{2}+m^{2})^{2}V_{dd\delta}
u(x_j) \ge \frac{A\psi(|x_j|)}{|x_j|^{2m-2}}\frac{A^{\frac{\lambda(n-2m)}{n-\lambda(n-2m)}}\psi(|x_j|)^{\frac{(\lambda-1)(n-2m)}{n-\lambda(n-2m)}}}{|x_j|^{(n-2m)(1+b)}}&= C(m,n,\lambda) \frac{\psi(|x_j|)^{\frac{2m}{n-\lambda(n-2m)}}}{|x_j|^{n-2+(n-2m)b}}&\ge C(m,n,\lambda) \frac{\varphi(|x_j|)^{1/2}}{|x_j|^a}
L_{12}\left( \begin{array}{c}z_1 z_2 z_3 \end{array}\right) =\ \frac12\left( \begin{array}{c}-iz_2 iz_1 0 \end{array}\right) , L_{12}\left(\begin{array}{c}z^*_1 z^*_2 z^*_3 \end{array}\right)=\ \frac12 \left(\begin{array}{c}-iz^*_2 iz^*_1 0 \end{array}\right)
S=\int d^{D}x \,\sqrt{-g}\left[-\sum^{N}_{k=1}\overline{\psi}_{k}\gamma^{\mu}\nabla_{\mu}\psi_{k}+ \frac{\lambda_{0}}{2N}(\sum^{N}_{k=1}\overline{\psi}_{k}\psi_{k})^{2}\right] ,
E_{n}^{(1)}=\langle\psi^{0}|H^{\prime}|\psi^{0}\rangle=-\frac{1}{8m_{e}^{3}c^{2}}\langle\psi^{0}|p^{4}|\psi^{0}\rangle=-\frac{1}{8m_{e}^{3}c^{2}}\langle\psi^{0}|p^{2}p^{2}|\psi^{0}\rangle
{\mathcal G}_{l + \nu}^{( 0 )}( r^{\prime \prime}, r^{\prime}; E ) = \int_{0}^{\infty}d z \eta_{l + \nu}( z ; \kappa ) \Psi_{z , l + \nu}( r^{\prime \prime}; \kappa ) \Psi_{z , l + \nu}( r^{\prime}; \kappa ) .
\frac{d^+}{dt}\biggr\rvert_{t=0}f(t) = F'(\mu(A))\left[\mathbb{M}_{p,F}^{\bar{\alpha}, \mu}(A)- \frac{1}{F'(1)}\mu(A) + \frac{1}{F'(1)}V_{p}^{\mu}(A,B)\right]+ F(\mu(A))-F(\mu(B)).
T^{H_0,\ldots,H_n}_{f^{[n]}}(V_1,\ldots,V_n) =T^{H_0,\ldots,H_n}_{(fu)^{[n]}}(V_1,\ldots,V_{n-1},\tilde{V}_n)-T^{H_0,\ldots,H_{n-1}}_{f^{[n-1]}}(V_1,\ldots,V_{n-1})\tilde{V}_n.
\begin{gathered}g_{\gamma,n,a,b}\left(x\right)=V_{\gamma-1}\left(np^{-\gamma}+bp^{-\gamma}\right)\varOmega\left(|x-np^{-\gamma}-ap^{-\gamma}|_{p}p^{-\gamma+1}\right)- -V_{\gamma-1}\left(np^{-\gamma}+ap^{-\gamma}\right)\varOmega\left(|x-np^{-\gamma}-bp^{-\gamma}|_{p}p^{-\gamma+1}\right)\end{gathered}
\begin{bmatrix}A B \end{bmatrix}( X \circ Y ) \begin{bmatrix}A B \end{bmatrix}^* =\begin{bmatrix}A B \end{bmatrix}\begin{bmatrix}X_1 \circ Y_1 0 0 X_2 \circ Y_2\end{bmatrix}\begin{bmatrix}A B \end{bmatrix}^* &=A (X_1 \circ Y_1) A^* + B( X_2 \circ Y_2 ) B^*
H(x_0,s_0,x_1,s_1) = \sup \big\{s_0 \psi_0 + s_1\psi_1\, \mid\, R^*(\psi_0) + R^*(\psi_1) \leq c(x_0,x_1) \big\}&= \sup \big\{-s_0 F^*(-\phi_0) - s_1F^*(-\phi_1)\, \mid\, \phi_0 + \phi_1 \leq c(x_0,x_1) \big\}.
{\Gamma^{\prime}}_{\mathrm{e f f}}^{{( 2 )}}= \int \frac{d^{4}\! p}{( 2 \pi )^{4}}\, \phi ( p ) \phi ( - p ) \left( p^{2}- \frac{\hbar g^{2}}{9 6 \pi^{2}}\left( \frac{1}{\tilde{p}^{2}}- \frac{1}{\tilde{p}^{2}+ \frac{1}{\Lambda^{2}}}\right) + O ( g^{4}) \right)
g([(p_{k,l})_{k,l},w]) \coloneqq \left([(p_{k,l})_{k,l}], \left[ \prod_{l=1}^{N_1}p_{1,l}, \prod_{l=1}^{N_2}p_{2,l},\dots,\prod_{l=1}^{N_r}p_{r,l}, w \right] \right).
E(\lambda) := \Big\{z\in Q_3:\, z\mbox{is a Lebesgue point of $|Du|$ and}|Du(z)| >\lambda \Big\}, &E_h(\lambda) := \big\{z\in Q_3: h >\lambda \big\},\mbox{and}E_{\hat f}(\lambda) := \big\{z\in Q_3: \hat f >\lambda \big\}.
{S_{P}^{4}}^{\star}= \int d^{4}x \Bigl [{\frac{1}{2}}T^{\mu \nu}F_{\mu \nu}+{\frac{1}{4}}T^{\mu \nu}T_{\mu \nu}-{\frac{{\mu}^{2}}{2}}A^{\mu}A_{\mu}\Bigl ] \; ,
\begin{array}{ccc}{\rm s}_\delta(t) := \begin{cases}\sin(\sqrt{\delta}t)/\sqrt{\delta}\delta > 0 t \delta = 0 \sinh(\sqrt{-\delta}t)/\sqrt{-\delta}\delta < 0\end{cases},{\rm c}_\delta(t) := \begin{cases}\cos(\sqrt{\delta}t) \delta > 0 1 \delta = 0 \cosh(\sqrt{-\delta}t) \delta < 0\end{cases}\end{array}~.
d_{CC}(\hat{\mathbb{X}}^N_{1},\hat{\mathbb{Y}}^N_{1})=d_{CC}(\hat{\mathbb{X}}^N_{0,1},\hat{\mathbb{Y}}^N_{0,1})\lesssim \lim_{\delta\to 0}\inf_{\lambda\in \Lambda}\{|\lambda|\vee d_p(\hat{\mathbf{X}}^{\phi,\delta}_\lambda,\hat{\mathbf{Y}}^{\phi,\delta})\}=:\beta_{p-var}(\hat{\mathbf{X}},\hat{\mathbf{Y}}).
\prod_{i=0}^{N-1}S_2\left(n_i\right) = \sum_{0\leq k \leq_b n}\prod_{i=0}^{N-1}\binom{n_i}{k_i}f(n_i,k_i) = \sum_{k=0}^{n}\binom{n}{k}_b \prod_{i=0}^{N-1}f(n_i,k_i).
\sum_{i=1}^N w_iY_{k,j}(\mathbf{x}_i)Y_{\kappa,\iota}(\mathbf{x}_i)=\left\langle Y_{k,j},Y_{\kappa,\iota}\right\rangle_{L^2(\mathbb{S}^2)}=\delta_{k,\kappa}\delta_{j,\iota},
\frac{A(a,q;z)\varphi(z)-C(a,q;z)}{B(a,q;z)\varphi(z)-D(a,q;z)}=-\frac{\mathcal{A}(a,q;z)\Upsilon(\varphi(z)) +\mathcal{C}(a,q;z)}{\mathcal{B}(a,q;z)\Upsilon(\varphi(z)) +a\mathcal{D}(a,q;z)}
\pi ={\cal P}+ \frac{i}{W}\frac{\partial W}{\partial z}( \psi \overline{{\psi}}- \frac{3}{2}) , \overline{{\pi}}= \overline{{{\cal P}}}- \frac{i}{W}\frac{\partial W}{\partial \overline{{z}}}( \psi \overline{{\psi}}- \frac{1}{2}) ,
\Delta{C}_{n+1}=0, &\Delta{C}_{t}=\Big(\mu_1(t)(\beta_t-1)-\mu_0(t)(\alpha_t-1)\Big)+H(\alpha_t)-H(\beta_t)+\log\Big(\frac{1+2^{\mu_1(t)+\Delta{C}_{t+1}}}{1+2^{\mu_0(t)+\Delta{C}_{t+1}}}\Big),~t\in\{n,\ldots,0\}.
\mathbb{E}[\mathrm{sgn}^{\dag}(\mathbf{y}_n)\mathbf{F}^{\dag}\mathbf{W}_n^{\dag}\mathbf{F}\mathbf{x}]=\sum_{q=1}^Q\mathbb{E}[\mathrm{sgn}^{\dag}(y_{nq})(\mathbf{F}^{\dag}\mathbf{W}_n^{\dag}\mathbf{F}\mathbf{x})_q].
\frac{d^mG_0}{dv^m}=v^3\frac{d^m\hat{B}_0}{dv^m}+(3m-1)v^2\frac{d^{m-1}\hat{B}_0}{dv^{m-1}}+(3m-2)(m-1)v\frac{d^{m-2}\hat{B}_0}{dv^{m-2}}&\qquad+(m-1)^2(m-2)\frac{d^{m-3}\hat{B}_0}{dv^{m-3}}.
\Upsilon_{\alpha \beta \gamma}^{g_{1}g_{2}}\: = \: \left( \Upsilon_{\alpha \beta \gamma}^{g_{2}}\right) \, \left( g_{2}^{*}\Upsilon_{\alpha \beta \gamma}^{g_{1}}\right) \, \left( \Omega_{\alpha \beta}^{g_{1}, g_{2}}\right) \, \left( \Omega_{\beta \gamma}^{g_{1}, g_{2}}\right) \, \left( \Omega_{\gamma \alpha}^{g_{1}, g_{2}}\right)
( \Phi^{\prime}{}^{n}, \Phi_{m}^{\prime}{}^{*}) = \delta_{m}^{n}\; , \qquad ( \Phi^{\prime}{}^{n}, \Phi^{\prime}{}^{m}) = ( \Phi_{n}^{\prime}{}^{*}, \Phi_{m}^{\prime}{}^{*}) = 0 \; ,
(g-1)(x_2) = s_1s_2(x_2)-x_2 &= s_1(x_2 + \alpha_{2}(x_2) \cdot v_2) - x_2 &= (\alpha_1(x_2)+\alpha_1(v_2)\alpha_2(x_2)) \cdot v_1 + \alpha_2(x_2) \cdot v_2
\dot{\bar{x}}^*(t) = \bar{A}^* \bar{x}^*(t) + \bar{B}^* u(t) \dot{\bar{x}}_{r}(t) = - \varepsilon \bar{x}_r(t) + \displaystyle \sum_{j=1}^{n_{\rm in}}u_j(t) \bar{N}_j^* \bar{x}^*(t) + \bar{H}^* ( \bar{x}^*(t) \otimes \bar{x}^*(t) ) \bar{y}(t) = \sqrt[4]{p''}\; \bar{x}_r(t)
E_n^{A_1...A_n}(x_1,...,x_n;\Gamma)=\sum_{m_1...m_n}\int dxE_{n|m_1...m_n}^{A_1...A_n}(x,\Gamma(x),\partial_\mu\Gamma(x),...)\prod_{k=1}^n(\partial_\mu)^{m_k}\delta(x-x_k),
F_{1}=\begin{bmatrix}\lambda_{m+1}& \lambda_{m+2}& \ddots & \lambda_{m+r}& \Re \lambda_{m+r+1}\Im \lambda_{m+r+1}& -\Im \lambda_{m+r+1}\Re \lambda_{m+r+1}& \ddots & \Re \lambda_{n-1}\Im \lambda_{n-1}& -\Im \lambda_{n-1}\Re \lambda_{n-1}\end{bmatrix},
\frac{1}{(2\pi i)^{\frac{n}{2}}}\int_{\delta^{n0}}res\left(\frac{\omega_\beta}{f_n}\right)=0 \Longrightarrow \frac{1}{(2\pi i)^{\frac{n}{2}}}\int_{\delta^{n0}}res\left(\frac{\omega_\beta}{f_n^2}\right)=0.
\beta\,\tfrac{L_X^{(\alpha)}(s+1,t,z)}{L_X^{(\alpha)}(s+1,t+1,z)}+\alpha\,\tfrac{L_Y^{(\beta)}(t+1,s,z)}{L_Y^{(\beta)}(t+1,s+1,z)}=\beta\,\tfrac{L_U^{(\alpha)}(t+1,s,z)}{L_U^{(\alpha)}(t+1,s+1,z)}+\alpha\,\tfrac{L_V^{(\beta)}(s+1,t,z)}{L_Y^{(\beta)}(s+1,t+1,z)}.
\bar \nabla_M{\cal H}^{MPQ}-\bar \nabla_M h_N^{~P}\bar H^{MNQ}+ \bar \nabla_M h_N^{~Q}\bar H^{MNP}-(\bar \nabla_M \hat h^M_{~N}) \bar H^{NPQ}-h^M_{~N}(\bar \nabla_M \bar H^{NPQ})=0.
S \ : \ \ \ \tau \ \rightarrow \ \tilde{\tau}= - \frac{1}{\tau}\ \ \ , \ \ \A_{\bar{z}}^{+ +}( 0 , 0 ) \ \rightarrow \ \tilde{A}_{\bar{z}}^{+ +}( 0 , 0 ) = - \bar{\tau}A_{\bar{z}}^{+ +}( 0 , 0 ) \ \ \ ,
\frac{T_{m,m-1}}{T_{m+1,m}}=\frac{T_{m,m}}{T_{m+1,m}}=\frac{T_{m+1,m}}{T_{m+2,m}}=\frac{T_{m+1,m}}{T_{m+2,m+1}(1-T_{m+1,m})^2}.
\limsup_{\tau \to 0, R \to \infty, N \to \infty}\sup_{C \in \mathcal{A}}d_{\mathcal{P}(\Omega \times{\rm Config})}( \overline{\mathbf{P}}_{N}(C), \overline{\mathbf{P}}_{N}(\mathcal{R}C)) - \frac{1}{|I|}\sum_{i \in I}d_{\rm Config}(\theta_{x_{i}}\cdot C, \theta_{x_{i}}\cdot \mathcal{R}C ) =0,
c_{2i+1}= (-1)^{i}\frac{2^{s}q+1}{2^{s-1}q+1+i}{2^{s-1}q+1+i \choose 2i+1}2^{2i}= (-1)^{i}\frac{2^{s}q+1}{2i+1}{2^{s-1}q+i \choose 2i}2^{2i},
\begin{array}{ll}ds^{2}= -\left(1-\frac{r_{H}^{7-p}+\alpha}{r^{7-p}}\right)dt^{2}+\left(1+\frac{\alpha}{r^{7-p}}\right)dx^{\scriptscriptstyle{10}^{\scriptstyle{2}}}\mp2\frac{\sqrt{\alpha \left(r_{H}^{7-p}+\alpha \right)}}{r^{7-p}}dtdx^{10}+\frac{dr^{2}}{1-\left(\frac{r_{H}}{r}\right)^{7-p}}+r^{2}d\Omega_{8-p}^{2}+dy^{s}dy_{s},\end{array}
\,\Theta \, :=\, ( X_{0}^{(u)}, \widetilde{X}_{0}^{(u)}) \, =\, (X_{0}^{(u)}) \otimes ( \widetilde{X}_{0}^{(u)}) \, =\, \theta^{\otimes 2}\, , \theta \, :=\, (X_{0}^{(u)}) \, \equiv\, ( \widetilde{X}_{0}^{(u)})\,.
D^n_x(1)=\left\lbrace (y^1,y^2,...,y^n)\in T_xM^n:F(x,y)<1\right\rbrace, y= y^{\epsilon}z^i_{\epsilon}\textnormal{and}z=\left( z^i_{\epsilon}\right)=\left( \frac{\partial \varphi^i}{\partial x^{\epsilon}}\right).
\{J_{\varphi}^{a}( \varphi ) , ~ J^{b c}( \varphi^{\prime}) \}= \left( f^{a b d}J^{d c}( \varphi ) + f^{a c d}J^{b d}( \varphi ) \right) \delta ( \varphi - \varphi^{\prime}) .
\| (\alpha_\omega)^s(e^{(n)}) - e^{(n)}\| = \frac{1}{|K_n|}\left\| \sum_{g\in K_n}(\alpha_\omega)^{sg}(e) - \sum_{g\in K_n}(\alpha_\omega)^g(e) \right\| &\leq \frac{1}{|K_n|}|sK_n\triangle K_n| < \frac{1}{n}.
\tilde{\eta}_p(t)=\sum_{k=1}^{p+1}\frac{(-1)^{n-k}\,(p+1)!\,p!}{(p+1-k)!\,k!\,(k-1)!}\int_0^t ds_1\int_0^{s_1}ds_2\cdots\int_0^{s_{p-k}}u^{k-1}(x)\breve\eta_p(x)\,dx.
M_\xi-\omega N \coloneqq \begin{bmatrix}0 A_\xi - \imath\omega I_n \! B A_\xi^\mathsf{H}+ \imath\omega I_n 0 C^{\mathsf{H}}B^\mathsf{H}C D_\xi^{\mathsf{H}}+D_\xi \end{bmatrix},
- 2 \bar{\phi}= - 2 \phi + \frac{1}{2}\psi_{1}- 3 \psi_{2}+ 2 \psi \ , \f_{1}= - \frac{\sqrt{2}}{3}\phi + \frac{\sqrt{2}}{3}\psi_{1}+ \frac{4 \sqrt{2}}{3}\psi
[\mathsf{L}_1 , \mathsf{u}_2 ] = \mathsf{u_1u_2}[\mathsf{B}_{12}^{-1}- \mathsf{I \otimes I}, p(\mathsf{T}_2)_1 ] \mathsf{u}_{1}^{-1}+i\hbar\mathsf{u_1u_2}\mathsf{(T_2 \otimes T_2)}\mathsf{B}^{-1}_{12}\mathsf{u}^{-1}_1
1\{\bar g(\theta_L)\le c(\theta_L)\}\Phi\Big(\frac{t+R}{\varsigma}\Big)=1\{\bar g(\theta_L)< c(\theta_L)\}\Phi\Big(\frac{\bar g(\theta_L)- c(\theta_L)+s_L(\theta_L)R}{\varsigma s_L(\theta_L)}\Big),
|A| \ge q^{ug}\left (1 - \dfrac{1}{q}\right )^{N}- \left(\sqrt{q}- 1 \right)^{(u - 1)g + 1}q^{g}& \ge q^{ug}\left (1 - \dfrac{1}{q}\right )^{N}- \sqrt{q}^{(u - 1)g + 1}q^{g}& = q^{ug}\left (1 - \dfrac{1}{q}\right )^{N}- q^{\frac{(u + 1)g + 1}{2}}.
\theta^1_q= dq^1, \theta^2_q= dq^2\theta_x = dx, \theta_p^1 =dp^1 - q^1 dx, \theta_p^2= dy^2 - q^2 dx, \theta^1_y = dy^1 - p^1dx, \theta^2_y = dy^2 -p^2dx, \theta_z = dz - p^2dp^1 +q^2dy^1 - q^1 dy^2 +(p^2q^1 - p^1q^2)dx.
\sum_{k=1}^{N/2}\P\left( \mathfrak{E}_{B,N,N',y},\, d W_k\leq -(k^{\frac{9}{10}}+B^{9/5})/10 \right) \ll_dN^{-3/2}\sum_{k=1}^{\infty}e^{-\frac{k^{0.2}B^{1.2}}{400 d^2}}\leq \frac{\epsilon}{N^{\frac{3}{2}}}
\frac{\overline{IA}\cdot\overline{IA}}{\overline{CA}\cdot\overline{AB}}+\frac{\overline{IB}\cdot\overline{IB}}{\overline{AB}\cdot\overline{BC}}+\frac{\overline{IC}\cdot\overline{IC}}{\overline{BC}\cdot\overline{CA}}=1
\Phi^{\Delta , J}( t , \bar{t}, \psi ; \tau , \bar{\tau}, \phi ) = \sum_{n = 0}^{\infty}\sum_{m = 0}^{2 J}a_{m n}t_{0 n}^{\Delta}( t , \bar{t}, \psi ) t_{0 m}^{J}( \tau , \bar{\tau}, \phi ) = e^{i ( \Delta \psi - J \phi )}\varphi ( t , \bar{t}, \tau , \bar{\tau}) \, .