harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Phi^{V^R}_{(\gamma^R, \tau^R)}(\bar{g}^R , \bar{\pi}^R) =\Phi^{V^R}_{(\gamma^R, \tau^R)}(\gamma^R, \tau^R)+ (2\psi^R + \lambda \zeta R^{-2}(\log R)^\frac{1}{2}, V^R).
\begin{cases}\overset \cdot a_0 = a_0 a_1 a_2, \overset \cdot a_1 = a_1a_2 a_3, \overset \cdot a_2 = a_2 a_3 a_4 \,-\, a_2 a_1 a_0, \overset \cdot a_3 = -a_3 a_2 a_1, \; \overset \cdot a_4 =-a_4 a_3 a_2;\end{cases}
{\cal G}(x-x',0)_{\mu\nu,\gamma\delta}\, = \,\langle h_{\mu\nu}(x,0) \,h_{\gamma\delta}(x',0) \rangle = \int d^N m |\sigma_m(0)|^2 \langle\epsilon_{\mu\nu}^{(m)}(x) \epsilon_{\gamma\delta}^{(m)}(x')\rangle \,.
K(h^{\frac{1}{2^{n}}},2)^{r_{n}}A \sharp_{\nu}B \leqslant A \nabla_{\nu}B -\sum_{k=0}^{n-1}r_{k}[ A\sharp_{\frac{m_k}{2^k}}B -2A\sharp_{\frac{2m_k+1}{2^{k+1}}}B +A\sharp_{{\frac{m_k+1}{2^k}}}B] \leqslant K(h^{\frac{1}{2^{n}}},2)^{R_{n}}A \sharp_{\nu}B,
\left|\left|\frac{x_1v_1-y_1u_1}{\mathcal{N}(q_1)}-\frac{l}{\mathcal{N}(q_2)}\right|\right|= \left|\frac{x_1'v_1-y_1'u_1}{\mathcal{N}(q_1)}-\frac{l}{\mathcal{N}(q_2)}\right|.
\begin{cases}\partial_t\tilde{u}^{N,\alpha}=\tfrac{1}{2}\partial_x^2\tilde{u}^{N,\alpha}+F_N^\alpha(\tilde{u}^N)*\eta_2^\epsilon+\sigma_\beta^\alpha\partial_x\xi^\beta*\eta^\epsilon, \tilde{u}^N(0)=\tilde{u}(0)\in(\mathcal{C}_0^{\delta-1})^d.\end{cases}
\begin{array}{rcl}2k\nabla_{U}\tilde{\tau}_{h}(U)+2k\:\textup{div}(U)\tilde{\tau}_{h}(U)+ \frac{k(a+c)}{\varphi}\textup{div}(U)\tilde{\tau}_{h}(U)-\overline{\Delta}\tilde{\tau}_{v}(U)+g(\overline{\Delta}U,U)\tilde{\tau}_{v}(U)=0. \end{array}
W^{\ast}= \frac{1}{1 2}( z + \frac{b}{z}+ 2 ) x_{0}^{6}+ \frac{1}{6}x_{3}^{6}+ \frac{1}{6}x_{4}^{6}+ \frac{1}{2}x_{5}^{2}+ c^{- 1 / 6}x_{0}x_{3}x_{4}x_{5},
\begin{aligned}I_{3}\leq \frac{\lambda c_{2}}{(c_{1}t_{1})^{2}\Gamma (2 \alpha )}\Biggl| \int_{0}^{t_{1}}\left ((t_{1}-s)^{2 \alpha -1}- (t_{2}-s)^{2 \alpha -1}\right) ds \Biggr| & \leq \frac{\lambda c_{2}}{(c_{1}t_{1})^{2}\Gamma (2 \alpha +1 )}\Biggl| t_{1}^{2 \alpha}- t_{2}^{2 \alpha}+ (t_{2}-t_{1})^{2 \alpha}\Biggr|,\end{aligned}
= \frac{1}{2}+ \frac{1}{3}- \frac{1}{3}- \frac{1}{4}+ \frac{1}{4}+ \frac{1}{5}- \frac{1}{5}- \frac{1}{6}+ \frac{1}{6}+ \frac{1}{7}- \frac{1}{7}- \frac{1}{8}
V ( r ) \; = \; ( 1 - H^{2}r^{2}- \frac{2 M}{r})^{2}\; [ \frac{1}{E^{2}}( \mu r + \frac{\Omega^{2}}{2 r})^{2}( 1 - H^{2}r^{2}- \frac{2 M}{r}) - 1 ] .
U(o, \, o\, |\, z) = \sum_{i=1}^r \frac{- (\phi_i(z)-mU(o, \, o\, |\, z))}{2m}+ \sum_{i=1}^r\frac{[(\phi_i(z)- mU(o, \, o\, |\, z))^2+4\lambda m_iz^2]^{1/2}}{2m},
G_{A,k,g}(\omega) = \sum_{i=1}^k \mu_{\theta^i\omega}(A'\cap \{\tau_A\le g\}), H_{A,k,g}(\omega) = \sum_{i=1}^k \sup_{j\ge p}\left|\mu_{\theta^i\omega}(A'\cap \sigma^{-g}\{\tau_A>j\})-\mu_{\theta^i\omega}(A')\mu_{\theta^{i+g}\omega}(\tau_A>j)\right|, K_{A,k,g}(\omega) = \sum_{i=1}^k \mu_{\theta^i\omega}(A') \, \mu_{\theta^i\omega}(\tau_A\le g).
R_{L}^{( n )}( t , t^{\prime}; \{q \}) = R_{L}^{[ n ]}( t , t^{\prime}; \{q \}) - \frac{1}{2}\sum_{m , m^{\prime}}\Phi_{m}( t ; L ; \{q \}) \hat{V}_{m m^{\prime}}^{- 1}\Phi_{m^{\prime}}( t^{\prime}; L ; \{q \})
\hat{G}_{a}^{\prime}=\frac{1}{\sqrt{n}}\hat{G}_{a}, \hspace{0.3cm}\hat{G}_{ab}^{\prime}=\frac{1}{\sqrt{n}}\hat{G}_{ab}, \hspace{0.3cm}\hat{G}_{a5}^{\prime}=\frac{1}{\sqrt{n}}\hat{G}_{a5}.
\frac{R_{21}}{-i-\omega(1,1)}u_1+\frac{R_{23}}{-i-\omega(3,3)}u_3=i(-R_{21}+R_{23})=-iR_{21}+iR_{23}=0+i=i=u_2; \frac{R_{41}}{-i-\omega(1,1)}u_1+\frac{R_{43}}{-i-\omega(3,3)}u_3=-iR_{41}+iR_{43}=i-2i=-i=u_4.
\Xi^{\lambda}(K_{a}F_{a}\otimes1\otimes E_{a}K_{a}^{-1}\otimes K_{a})=\sum_{j,m,n,o}c_{j}^{o}\pi(K_{a}F_{a})_{m}^{j}c_{n}^{m}\pi(E_{a}K_{\lambda})_{o}^{n}.
\mathcal{E}= E+\frac{a\Phi}{x+a^2}+ \frac{b\Psi}{x+b^2}\Theta = E^2(a^2\cos^2\theta +b^2\sin^2\theta)-\frac{1}{\sin^2\theta}\Phi^2 -\frac{1}{\cos^2\theta}\Psi^2 + K \mathcal{X}=\Delta \Bigl[E^2 x+(a^2-b^2)\Bigl(\frac{\Phi^2}{x+a^2}-\frac{\Psi^2}{x+b^2}\Bigr)-K\Bigr]+ r_s^2 (x+a^2)(x+b^2)\mathcal{E}^2.
\|\tilde{w}_\epsilon(t)\|_{H^{m}_{k+\ell}(\mathbb{R}^2_+)}^2\le \tilde C_8 e^{C_2 t}\int^t_0e^{-C_2 \tau}\| \tilde{w}_\epsilon(\tau)\|_{H^m_{k+\ell}(\mathbb{R}^2_+)}^2 d\tau &+\tilde C_9e^{C_2 t}\| \tilde{u}_0\|_{H^{m+1}_{k + \ell'-1}(\mathbb{R}^2_+)}^2,
\frac{1}{n!}\frac{\mathrm{d}^n}{\mathrm{d}x^n}\frac{e^x}{1+e^x}= \frac{1}{2\pi i}\oint_{\partial B(x,r)}\frac{e^z}{(1+e^z)(z-x)^{n+1}}\, \mathrm{d}z = \frac{1}{2\pi i}\int_0^{2\pi}\frac{e^{x+re^{it}}}{(1+e^{x+re^{it}}) (re^{it})^{n+1}}\cdot rie^{it}\, \mathrm{d}t
\int_{0}^{t}\|h^{n-1}(\tau, \cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau \leq t \max_{0 \leq \tau \leq t}\|\partial_{t}^{n-1}\bar{f}(\tau, \cdot)\|_{L^{2}(\Omega)}^{2}&+ C(\phi) \int_{0}^{t}\|\bar{D}^{s - 1}z(t, \cdot)\|_{L^{2}(\Omega)}^{2}\mathrm{d}\tau
\frac{1}{T}\sum_{t=2}^{T+1}\Delta_{t}\le48R_{1}\max\left\{52^{\frac{1}{p}}T^{\frac{1-p}{p}}\left(1+\log T\right)^{\frac{2}{p}}\sigma\gamma^{\frac{p-1}{p}};2\left(2LR_{1}+LR_{0}+\mu\sigma+\left\Vert g_{0}\right\Vert_{*}\right)T^{-1}\gamma\right\}=\widetilde{O}\left(T^{\frac{1-p}{p}}\right).
\zeta_1+\zeta_2+\zeta_3+\zeta_4=\delta_1\alpha-1, \ \ \zeta_1, ..., \zeta_4>0 \frac1{\theta_1}+\frac1{\theta_2}+\frac1{\theta_3}+\frac1{\theta_4}=1, \ \ \theta_1=\frac2{\delta_1}, \ \ \theta_3=\frac2{\delta_2}, \ \ 1<\theta_2, \theta_4<\infty,
{\rm Tr}\, d\Phi \, d\Phi^\dagger= \sum_i d\lambda_i^2+ \sum_{\alpha,\beta}dt_\alpha \,dt_\beta \,\left(\sum_i\lambda_i \alpha_i \right)\left(\sum_j\lambda_j \beta_j \right) \,{\rm Tr}\,E_\alpha E_\beta.
V_{\mathbb{Z}\beta}^{T_{2},-}\boxtimes_{V_{\mathbb{Z}\beta}^{+}}V_{\mathbb{Z}\beta}^{T_{2},-}=V_{\mathbb{Z}\beta}^{+}+V_{\frac{\beta}{2}+\mathbb{Z}\beta}^{-}+\sum_{1\le r\le2k-1,r\ \mbox{even}}V_{\frac{r}{4k}+\mathbb{Z}\beta}.
\mathcal A\ := \ \Big\{F\in{[m]\choose k-1}\ :\j_3\in F,\F\cap (H_1\cup H_2)=\{j_2\}\Big\}, \mathcal A'\ := \ \Big\{F\in{[m]\choose k-1}\ :\j_3\in F,\F\cap (H_1\cup H_2)=\{j_1\}\Big\}.
{\frac{1}{2}}\sum_{i = 1 , 2}\left({\frac{S_{i}}{\Delta^{3}}}\right)^{2}\left[ \log \left({\frac{S_{i}}{\Delta^{3}}}\right) -{\frac{3}{2}}\right] - \left({\frac{S_{1}}{\Delta^{3}}}+{\frac{S_{2}}{\Delta^{3}}}\right)^{2}\log \left({\frac{\Lambda}{\Delta}}\right) + \sum_{n \geq 3}\sum_{i = 0}^{n}c_{n , i}\left({\frac{S_{1}}{\Delta^{3}}}\right)^{n - i}\left({\frac{S_{2}}{\Delta^{3}}}\right)^{i},
{\frac{1}{2 C_{\cal R}}}\sum_{\mu , \nu \in{\cal R}}\left( L^{\pm}L^{\mp}\right)_{\mu \nu}={\frac{1}{2}}p^{2}+{\frac{1}{2}}\sum_{\rho \in \Delta_{+}}g_{| \rho |}( g_{| \rho |}- 1 ) \,{\frac{| \rho |^{2}}{( \rho \cdot q )^{2}}}+{\frac{\omega^{2}}{2}}q^{2}\mp \omega ({\frac{r}{2}}+ \sum_{\rho \in \Delta_{+}}g_{| \rho |}) ,
\Phi_{|D(v)}\simeq \{(t; [x_1, \dots, x_h];[0: \dots :0: t y_{n-p+1}: \dots : t y_{n}: y_{r+p+1}: \dots :y_{n}] \in \mathbb{C}\times \mathbb{P}^{n-1}\times \mathbb{P}^{n-1}| t \sum_{a=r+1}^{r+p}x_a y_{n-p+a-r}+ \sum_{a=r+p+1}^h x_a y_a=0 \}.
\Pi ({\vec{x}}) = \sum_{n_{1}, n_{2}, n_{3}= - N}^{+ N}\frac{1}{X^{3 / 2}}{\sqrt{\kappa_{\vec{n}}}}P_{\vec{n}}e x p \left( - 2 \pi i ( \sum_{j = 1}^{3}n_{j}x_{j}) / X \right) .
\begin{pmatrix}1 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 0 1 1 0 0 1 0 0 1 1 0 -1 -1 0 0 0 1 0 0 -1\end{pmatrix}\begin{pmatrix}a_{3}b_{2}b_{3}a_{3}' b_{2}' b_{3}'\end{pmatrix}=\begin{pmatrix}1/2-a_{1}-a_{2}-b_{1}1/2-a_{1}'-a_{2}'-b_{1}' -a_{2}-a_{2}' -b_{1}-b_{1}' 0 -a_{1}+a_{1}'\end{pmatrix}.
d_i+d_{i+1}=|N(x_i)\backslash\{x_1,\ldots,x_{i}\}|+|N(x_{i+1})\backslash\{x_1,\ldots,x_{i+1}\}| =|N(x_i)\backslash\{x_1,\ldots,x_{i}\}\cup N(x_{i+1})\backslash\{x_1,\ldots,x_{i+1}\}| \leq k-i,
E ={\frac{T}{2 \pi}}\rho_0^2 \int_1^{\infty}dy \left[{\frac{y^{n/6}(y- \lambda)^{\delta/4}}{\sqrt{y^{2n/3}(y -\lambda)^{\delta}- (1 -\lambda)^{\delta}}}}-{\frac{1}{\sqrt{y}}}\right]
f_i(\delta)-f_i(0) =[f_i(\delta)-f_i(\delta')]+[f_i(\delta')-f_i(0)] &\ge f_i'(\delta')(\delta-\delta')+f_i'(0)(\delta'-0) &>f_i'(0)(\delta-\delta')+f_i'(0)(\delta'-0) &=f_i'(0)(\delta-0).
\mathbb{P}(Y^2- B_n^2 > t ) = \mathbb{P}(|Y| > \sqrt{t+ B_n^2}) & \leq \mathbb{P}(|\varepsilon| > \sqrt{t+ B_n^2}- B) & \leq \mathbb{P}(|\varepsilon| > (1/\sqrt{2})(\sqrt{t}+B_n) - B) & \leq e^{-\frac{1}{\nu}\sqrt{\frac{t}{2}}}e^{-\frac{1}{\nu}(\frac{B_n}{\sqrt{2}}-B)}\mathbb{E}e^{|\varepsilon|/\nu}.
\sum\limits_{j=2}^k W_j(x)=O\left(\frac{1}{\left(1 + |x-q_1|\right)^{\alpha}}\sum\limits_{j=2}^k \frac1{|q_1-q_j|^{N+2s-\alpha}}\right)=O\left(\frac{1}{\left(1 +|x-q_1|\right)^{\alpha}}\left(\frac{k}{r}\right)^{N+2s-\alpha}\right).
S={1\over\pi}\int d^2x\left(\partial_+\left(\kappa\rho+2e^{-2\Phi}\right)\partial_-\left(\Phi-\rho\right)+\lambda^2e^{2\left(\rho-\Phi\right)}+{1\over2}\,\partial_+f_i\,\partial_-f_i\right)\;.
-\frac{\partial L_{d}F\left(p\|q\right)}{\partial q_{j}}=\left(1-\alpha\right)\left\lbrace\left[\frac{a-1}{a-b}\left(Z_{j}\right)^{a}-\frac{b-1}{a-b}\left(Z_{j}\right)^{b}\right]-1\right\rbrace
u_{g,2}(\lambda,x_{g_2,2})=\frac{\Delta_{B_{g_2-1,2}}^N(\lambda)}{\Delta_{2}(\lambda)}c_{g_2,2}(\lambda,x_{g_2,2})+\sum\limits_{k=g-d(w)+1}^{g-1}\frac{\Delta_{A_{k,2}}^N(\lambda)}{\Delta_{2}(\lambda)}s_{g_2,2}(\lambda,x_{g_2,2}).
\frac{\partial f(0)}{\partial z}=\frac{\partial^2 f(0)}{\partial z^2}=\frac{\partial^3 f(0)}{\partial z^3}=0\frac{\partial^4 f(0)}{\partial z^4}=\frac{4\sqrt{2}}{\pi}\frac{\Gamma(3/2-\pi/4)}{\Gamma(1-\pi/4)}\left(\frac{3}{\pi}-1 \right)<0.
S_{SG}(\theta )=\frac{S_{0}(\theta )}{xp-p^{-1}x^{-1}}\left( \begin{array}{cccc}xp-p^{-1}x^{-1}0 0 0 0 p-p^{-1}x-x^{-1}0 0 x-x^{-1}p-p^{-1}0 0 0 0 xp-p^{-1}x^{-1}\end{array}\right) \,
\cos\phi^{(1)}=\frac{a}{L^{(1)}}(\frac{l}{4d}(a_l+a_r)+\frac{w_1}{2}),\hspace{1cm}\sin\phi^{(1)}=\frac{a}{L^{(1)}}\frac{\sqrt{3}l}{4d}(a_l-a_r),
\begin{array}{ccl}L_{a-b}-\!L_{a+b}= \displaystyle 2\;\sqrt{k+2}\;\exp\left(\!-\frac{i\pi(a^2+b^2)}{4(k+2)}\right) \sin\left(\frac{\pi ab}{2(k+2)}\right)E_{k+a+b}\ .\end{array}
\mu_1 =\frac{(n_1-n_2)\alpha+\sqrt{(n_1-n_2)^2\alpha^2+4n_1n_2\beta^2}}{2n_1\beta}\;, \mu_2 =\frac{(n_1-n_2)\alpha-\sqrt{(n_1-n_2)^2\alpha^2+4n_1n_2\beta^2}}{2n_1\beta}\;.
\mathcal{I}_n = \underbrace{\iint_{Q_T}Z \frac{B_n}{A_n}(A - A_n) [\nabla \cdot(m \nabla \varphi_n)]}_{\mathcal{I}_{n,1}}+ \underbrace{\iint_{Q_T}Z (B_n - B) [\nabla \cdot(m \nabla \varphi_n)]}_{\mathcal{I}_{n,2}}.
\ell_{n,2}^*(\beta_0, \beta_1, \eta) - \ell_{n,2}^*(0,0,1) \leq\ss_2^\tau \sum_{i=1}^n B_i - (n/2) \ss_2^\tau \Sigma_B \ss_2 \{1+o_p(1)\}+ o_p(1).
{\cal L}^{(1)}= -\partial_\mu \overline \psi_1 \gamma_{\mu\nu}\partial_\nu \psi_1-\partial_\mu \overline \psi_2 \gamma_{\mu\nu}\partial_\nu \psi_2 - m^2 \overline \psi_1 \psi_1+ m^2 \overline \psi_2 \psi_2 \quad .
S_{F}( p ) \Gamma_{\nu}( p , k ) S_{F}( p + k ) D^{\mu \nu}( k ) = \int d^{D}x \int d^{D}y \, \, e^{i k \cdot x + i p \cdot y}\langle O \vert T \left[ \psi ( 0 ) A_{\mu}( x ) \bar{\psi}( y ) \right] \vert O \rangle
n_{\alpha i}= \left( \begin{array}{c c c c c c}{{1}}&{{1}}&{{1}}&{{1}}&{{1}}&{{1}}\\{{2}}&{{2}}&{{0}}&{{0}}&{{2}}&{{2}}\\{{3}}&{{3}}&{{0}}&{{0}}&{{3}}&{{5}}\\{{2}}&{{2}}&{{1}}&{{1}}&{{2}}&{{4}}\\ \end{array}\right) .
\tilde{\zeta}_0^{NS}(t) = \left( \zeta_0(t)\,,\, \delta\!\left( \sqrt{\tfrac{2\pi}{3}}\, e^{\pi i\left( -\frac{t}{3}+ \frac{1}{6}+ \frac{k_0}{3}- \frac{1}{4}\right)}\right) \right)
\rho(UPV) =\frac{\angle UPV}{\left\vert UPV\right\vert}& =\tfrac{\arccos\left( \tfrac{(U-P)\cdot(V-P)}{\sqrt{(U-P)\cdot(U-P)}\sqrt{(V-P)\cdot(V-P)}}\right)}{\sqrt{\left( \sigma-\sqrt{(U-P)\cdot(U-P)}\right) \left( \sigma-\sqrt{(V-P)\cdot(V-P)}\right) \left(\sigma-\sqrt{(U-V)\cdot(U-V)}\right) \sigma}}
2\int_{|x|> a}f(x)^{\frac{1}{2}}dx \le \int_{|x|>a}\Psi(|x|)f(x)dx + \int_{|x|>a}\frac{1}{\Psi(|x|)}dx &\le L + \sum_{k=0}^\infty \int_{\kappa^{k}a < |x| \le \kappa^{k+1}a}\frac{1}{\Psi(|x|)}dx &\le L + \sum_{k=0}^\infty \frac{(2\kappa^{k+1}a)^d}{\Psi(\kappa^{k}a)}< \infty,
\mathbb{P}\left( \sup_{t\in I_{0}}\frac{\sqrt{t}\left(X_{o}+\sum_{j=0}^{n}2^{j/2}X_{j,0}\right)}{\sqrt{2\pi L_{2}\frac{1}{t}}}\geq \sqrt{1+\varepsilon}\right) \leq \frac{\left( L\frac{1}{\delta_{n}}\right)^{-1-\varepsilon}}{2\sqrt{\pi L_{2}\frac{1}{\delta_{n}}}}
u_{2\ell+2}=Q_1Q_0(x_1y_1)=x_2y_2^\ell-x_2^\ell y_2, u_{2\ell^2-2\ell}=Q_2 Q_0(x_1y_1)/Q_1Q_0(x_1y_1)=\sum_{k=0}^{\ell}x_2^{k(\ell-1)}y_2^{(\ell-k)(\ell-1)}, Q_0(x_1y_1) =x_2y_1-x_1y_2, Q_1(x_1y_1) =x_2^{\ell}y_1-x_1y_2^{\ell}.
a_ju_{g_j}v^*u_{g_i}^*v = u_{g_j}\sigma^j_{g^{-1}}(a_j)v^*u_{g_i}^*v = u_{g_j}v^*\sigma^i_{g^{-1}}(a_i)u_{g_i}^*v= u_{g_j}v^*u_{g_i}^*a_iv= u_{g_j}v^*u_{g_i}^*va_j
Z ( t ) ={\frac{1}{\sqrt{2 \pi}}}{\frac{1}{g \hbar^{2}t^{3 / 2}}}\exp \! \left({\frac{t v^{4}}{4 g^{2}}}\right) K_{0}\! \left({\frac{t v^{4}}{4 g^{2}}}\right) .
{\bf V}_{i}^{( i n t )}( t_{1}){\bf V}_{j}^{( i n t )}( t_{2}) = q^{2 ({\bf e}_{i}{\bf e}_{j})}{\bf V}_{j}^{( i n t )}( t_{2}){\bf V}_{i}^{( i n t )}( t_{1}) , \qquad t_{1}> t_{2}
[\begin{matrix}t\\u\\v\end{matrix}]=\frac{1}{-l_{ab}\cdot(p_{01}\times p_{02})}[\begin{matrix}(p_{01}\times p_{02})\cdot(l_{a}-p_{0})\\ (p_{02}\times-l_{ab})\cdot(l_{a}-p_{0})\\ (-l_{ab}\times p_{01})\cdot(l_{a}-p_{0})\end{matrix}],
{\cal L}= - \frac{1}{4}F_{\mu \nu}^{a}F^{a \mu \nu}\ \ , \ \F_{\mu \nu}^{a}= \partial_{\mu}A_{\nu}^{a}- \partial_{\nu}A_{\mu}^{a}- g f^{a b c}A_{\mu}^{b}A_{\nu}^{c}\ ,
\begin{array}{r c l}<r|\hat{U}^{-1}(\theta_0)|A_{\bar{z}}(z,\bar{z})> = <r|A_{\bar{z}}(z,\bar{z})>\ \ \ , <r|\hat{U}^{-1}(k_1,k_2)|A_{\bar{z}}(z,\bar{z})> = <r|A_{\bar{z}}(z,\bar{z})>\ \ \ ,\end{array}
\lambda_{k+1}^{\ast}\left( G\right) =\lambda_{k+1}^{\ast}\left(B-\left( B-A\right) \right) \geq\lambda_{k+1}^{\ast}\left( B\right)-\lambda_{1}^{\ast}\left( B-A\right) & \geq\lambda_{k+1}^{\ast}\left( B\right) -\lambda_{1}^{\ast}\left(I_{2nt}\right) & =\frac{2nt}{2\sqrt{k}}-1.
\frac{2}{x_1-x_2}-\left(\frac{1}{x_1-\zeta}+\frac{1}{x_1-\overline{\zeta}}\right)=0,\frac{2}{x_2-x_1}-\left(\frac{1}{x_2-\zeta}+\frac{1}{x_2-\overline{\zeta}}\right)=0.
S_{self}= -\frac{e^2c}{2}\overline{\sum}_{n,m,r,s}\int_{-\infty}^{\infty}dt \int_{-\infty}^{\infty}dx_1\int_{-\infty}^{\infty}dt^{\prime}\int_{-\infty}^{\infty}dx_1^{\prime}[ j_{nm}^{\mu}(x_1) D^c(x_1-x_1^{\prime};t-t^{\prime})j_{rs,{\mu}}(x_1^{\prime})
t\cdot\{f,g\}_2 + t^2\mathbb{A}_2 = \Phi(\widetilde{f})\Phi(\widetilde{g}) -\Phi(\widetilde{g})\Phi(\widetilde{f}) + t^2\mathbb{A}_2 = \Phi(\widetilde{f}\widetilde{g}- \widetilde{g}\widetilde{f}+ t^2\mathbb{A}_1) = t\cdot\{f,g\}_1 + t^2\mathbb{A}_2
{\tilde \tau}={\tilde \mu}_j({\tilde t},{\tilde x},{\tilde \xi}),{\tilde b}_{ji}({\tilde t},{\tilde x},{\tilde \xi})=b_{ji}({\tilde t}-\epsilon |{\tilde x}|^2,{\tilde x},{\tilde \xi}+2\epsilon{\tilde \mu}_j({\tilde t},{\tilde x},{\tilde \xi}){\tilde x})=0.
\langle a_{+}( w ) c{\bar{c}}V_{n}^{0}( 0 ) c{\bar{c}}V_{m}^{1}( 1 ) c{\bar{c}}V_{r}^{\alpha}( \infty ) \prod_{i = 1}^{L}\int V_{n_{i}}^{0}\prod_{j = 1}^{M}\int V_{m_{i}}^{1}{\frac{1}{R !}}( Q_{-})^{R}\rangle
\frac{1}{T}\sum_{t=2}^{T+1}\Delta_{t}\le\frac{2R_{1}^{2}}{\eta}=48R_{1}\max\left\{26^{\frac{1}{p}}T^{\frac{1-p}{p}}\sigma\gamma^{\frac{p-1}{p}};2\left(2LR_{1}+LR_{0}+\mu\sigma+\left\Vert g_{0}\right\Vert_{*}\right)T^{-1}\gamma\right\}
I_a^{(q,r)}(x) := \frac{W^{(q,r)}_a(x)}{W^{(q)}(-a)}- r \overline{W}^{(q+r)}(x), J_a^{(q,r)}(x,\theta ) :=Z^{(q,r)}_a(x,\theta ) - r Z^{(q)}(-a, \theta ) \overline{W}^{(q+r)}(x), J_a^{(q,r)}(x) := J_a^{(q,r)}(x,0) = Z^{(q,r)}_a(x) - r Z^{(q)}(-a) \overline{W}^{(q+r)}(x).
\left| u_{k}\right| \rightarrow{\frac{2^{\nu - 3 / 2}}{\sqrt{2 k}}}{\frac{\Gamma \left( \nu \right)}{\Gamma \left( 3 / 2 \right)}}\left| C_{+}\left( k \right) + C_{-}\left( k \right) \right| \left({\frac{y}{1 - \epsilon}}\right)^{1 / 2 - \nu}.
P_f^n\Big( \Big| \sum_{i=1}^n K\Big( \frac{X_i-x}{h_x}\Big)- E\Big[K\Big( \frac{X_i-x}{h_x}\Big)\Big] \Big|\geq 2\gamma \|K\|_{\infty}\log n + 2\|K\|_2\sqrt{\gamma G_hf(x) nh_x \log n}\Big) \leq 2n^{-\gamma}.
S_{\mathrm{a n o m a l o u s}}={\frac{1}{T_{H}}}\int_{\Sigma}e^{- \phi}\{\varrho_{L}-{\cal L}_{E}\}d^{3}r ={\frac{k}{\hbar}}\int_{\Omega}\{\varrho_{L}-{\cal L}_{E}\}\sqrt{g_{E}}d^{4}x .
R_{3,c}^{\mathrm{PNC}}=\min_i \log_2\left(1+\frac{P_{3,c}^{\mathrm{PNC}}(g_{r1},g_{r2}) g_{ri}}{1+P_{3,p}^{\mathrm{PNC}}(g_{r1},g_{r2})g_{ri}}\right) &=\log_2\left(1+\frac{P_{3,c}^{\mathrm{PNC}}(g_{r1},g_{r2}) \min( g_{r1},g_{r2})}{1+P_{3,p}^{\mathrm{PNC}}(g_{r1},g_{r2})\min(g_{r1},g_{r2})}\right),
R_P^*= R_P^*{}^{b}_{a}-R_P^*{}^{b i}_{a}D_{i}+R_P^*{}^{b ij}_{a}D_{i}D_{j}+ \cdots +(-1)^r R_P^*{}^{b i_1\cdots i_r}_{a}D_{i_1}\cdots D_{i_r}
x_j\, K_n(x,N)= &p_j(N-|n|)\,K_n+\sum_i C_{ij}K_{n+e_i}+p_jN\sum_k A_{jk}n_k\,K_{n-e_k}&-p_j\sum_kA_{jk}(|n|-1)n_kK_{n-e_k}+\sum_{i,k}C_{ij}A_{jk}n_k\,K_{n-e_k+e_i}\ .
\overline{\alpha}(z) =\prod_{i=1}^{(s-1)/2}\left(\left(1-\frac{z}{a_i}\right)\left(1-\frac{z}{n-a_i}\right)\right) &=\prod_{i=1}^{(s-1)/2}\frac{1}{a_i(n-a_i)}\prod_{i=1}^{(s-1)/2}\left(-\Big(a_i-\frac{n}{2}\Big)^2+\frac{x^2}{4}\right),
|I_1| \leq (n+2-\beta)|z|\int_{|z|\leq|x-z|<\frac{1}{\beta}<\infty}\frac{1}{|x-z|^{n+1-\beta}}\,dx+C\sum_{k=2}^\infty \int_{|z|\leq|x-z|<\frac{1}{\beta}<\infty}\frac{|z|^k}{k!|x-z|^{n+k-\beta}}\,dx &\leq C\frac{|z|^\beta}{1-\beta}\leq C\frac{\beta^{-\beta}}{1-\beta}.
\frac{1}{d_j (A)}= \underset{{\mathcal{M}\subset \mathbb{C}^{2n}}\atop{\dim \mathcal{M}=j}}{\max}\,\,\,\underset{{x \in \mathcal{M}}\atop{\langle x, Ax \rangle = 1}}{\min}\,\,\, \langle x, i J x \rangle,
\mathcal{F}_2 = \{\lambda \mid \partial_{\lambda}\Delta(\lambda)=0, partial_{\lambda}^2 \Delta(\lambda)\neq 0\}, \mathcal{F}_1 = \{\lambda \mid \partial_{\lambda}^2 \Delta(\lambda)= 0,\ \partial_{\lambda}^3 \Delta(\lambda)\neq 0\}.
\begin{cases}\underbar r_1^{k+1}-\epsilon \le u(x,t;t_0,u_0,v_0) \le \bar r_1^{k+1}+\epsilon\cr\underbar r_2^{k+1}-\epsilon \le v(x,t;t_0,u_0,v_0) \le \bar r_2^{k+1}+\epsilon\end{cases}
\left(^{V^{n}}{\! a}_{1}\right)^{a}= a_{1}^{b}\: \left( V^{n}[ \vec{\varphi}\, ] \right)^{b a}+ v_{1}^{a}= a_{1}^{a}- \frac{2 \pi}{L}\, \hat{a}_{1}^{a}\; .
d\omega^{(k)}(X) =\sum_{i=1}^{k+1}(-1)^{i-1}X_i(\omega^{(k)}(X[i]))+\sum_{1\le i<j\le k+1}(-1)^{i+j}\omega^{(k)}([X_i,X_j]\wedge X[i,j]) &=\sum_{i=1}^{k+1}(-1)^{i-1}X_i(\eta(X[i],\textbf{1}))+\sum_{1\le i<j\le k+1}(-1)^{i+j}\eta([X_i,X_j], X[i,j]).
\beta_{n+1}\left( \alpha,r\right) -\beta_{n+1}\left( \alpha,r-s\right)=\left( n+1\right) \sum_{k=0}^{n}S_{2}\left( n,k,r\mid\alpha\right)\frac{\left( -1\right)^{k}\left\langle s\right\rangle_{k+1}}{k+1}.
\frac{\nu f_X(\nu t)-g_X(t)}{u_X(t)-v_X(\nu t)}= \varphi_X(t;\nu) \left( \frac{f_X(t)-g_X(t)}{u_X(t)-v_X(t)}+ \frac{\nu f_X(\nu t) -f_X(t)}{u_X(t)-v_X(t)}\right).
P_{j j + 1}w_{\cdots \alpha_{j + 1}\alpha_{j}\cdots}( \cdots , z_{j + 1}, z_{j}, \cdots ) = R_{j j + 1}( z_{j}/ z_{j + 1}) w_{\cdots \alpha_{j}\alpha_{j + 1}\cdots}( \cdots , z_{j}, z_{j + 1}, \cdots ) .
\kappa_m \left( \sum_{i=1}^n c_i X_\rho(t_i) \right) = \kappa_m\left(\rho^{1/2}\sum_{i=1}^n c_i W B(t_i)\right) + \kappa_m\left((1-\rho)^{1/2}\sum_{i=1}^n c_i B'(t_i)\right).
Ric(W,W)=-\frac{1}{2}\sum_{i}|[W,\hat{Y_i}]|^2-\frac{1}{2}\sum_{i}\langle[W,[W,\hat{Y_i}]],\hat{Y_i}\rangle+\frac{1}{2}\sum_{i<j}\langle[\hat{Y_i},\hat{Y_j}],W\rangle^2.
Y ( m_{k}; \vartheta , \phi ) = e^{i m_{n}\phi}\prod_{k = 1}^{n}\left( \sin \theta_{k}\right)^{| m_{k}|}C_{m_{k - 1}- | m_{k}|}^{| m_{k}| + n / 2 - k / 2}\left( \sin \theta_{k}\right) .
\sum_{\ell=0}^{N-1}\big|\mathcal{D}_{N,\ell}^{h,2,1}\big|\le \frac{C(1+|x|_{L^p})^K}{\tau^{2\kappa}}\bigl(t_{N-1}^{-\frac12+\kappa}\Delta t |h|_{L^q}+t_{N-1}^{\frac12+\kappa-\beta}\big|(-A)^{-\beta}h\big|_{L^q}\bigr).
(a^{+})^{k+1}a^{k+1}=\sum_{m=0}^{k+1}S_{1,\lambda}(k+1,m)(a^{+}a)_{m,\lambda}&=\sum_{m=1}^{k+1}S_{1,\lambda}(k+1,m)(\hat{n})_{m,\lambda}&=\sum_{m=0}^{k}S_{1,\lambda}(k+1,m+1)(\hat{n})_{m+1,\lambda},(k \ge 0).
\lim_{n \to +\infty}\lambda_1^{1/{p_n}}(B_{\rho_n}^{\pi/4};{p_n}) = \frac{1}{r_2}\lim_{n \to +\infty}\lambda_1^{1/{p_n}}(B_1^{\pi/4}\setminus \overline{B_{\rho_n}^{\pi/4}};{p_n}) = \frac{1}{r_2}.
\displaystyle{\liminf_{\varrho \to 0^+}\frac{\overline{J}_p(u,v; Q(x_0,\varrho))}{\varrho^N}= \liminf_{\varrho \to 0^+}\overline{J}_p(u_\varrho, v_\varrho; Q)\geq \overline{J}_p(u_0, v(x_0);Q).}
\begin{aligned}\mathcal{J}(y(u),u) \leq \mathcal{J}^\oplus(\alpha,\beta;\eta,\boldsymbol{\tau},v) \, \forall \, \eta \in V, \, \boldsymbol{\tau}\in H(,\Omega), \, v \in U_{}, \, \alpha, \beta > 0, \end{aligned}
\left[ \bigcup_{d_1 + d_2 = d}\mathcal{Z}_{d_1,d_2}\right]^\textnormal{vir}= \sum_{d_1+d_2=d}\Delta^! \left[ \overline{\mathcal{M}}_{g_1,n_1+1}(X,d_1) \right]^\textnormal{vir}\otimes \left[ \overline{\mathcal{M}}_{g_2,n_2+1}(X,d_2) \right]^\textnormal{vir}
\Phi_{1}^{-}\Phi_{1}^{+}= \Phi_{2}^{-}\Phi_{2}^{+}= \frac{\alpha}{\alpha^{2}- \beta^{2}}, \Phi_{1}^{-}\Phi_{2}^{+}= \Phi_{2}^{-}\Phi_{1}^{+}= \frac{\beta}{\alpha^{2}- \beta^{2}}.
G_{6 6}\sim 1 + \frac{2 0}{3 !}g \int \frac{d^{3}k}{( 2 \pi )^{3}}\frac{d^{3}p}{( 2 \pi )^{3}}\, \frac{1}{k^{2}}\frac{1}{p^{2}}\frac{1}{( p + k )^{2}}~ ,
\psi((T^{P\cap R}_{\lambda_{R}^+})^{-(n + k)})\sigma(E^P_{o_{-,P}}(w(\lambda_R^+)^k)) = \psi((T^{P\cap R}_{\lambda_{R}^+})^{-(n + k)}E^{P\cap R}_{o_{-,P\cap R}}(w(\lambda_R^+)^k)) & = (\psi E^{P\cap R}_{o_{-,P\cap R}}(w))((T^{P\cap R}_{\lambda_{R}^+})^{-n}).
A \in{\mathcal A}\mbox{is}\mu\mbox{-a.e.}{\mathcal L}\mbox{-invariant}\ \ \ \Leftrightarrow \ \ \ \chi_A(x) = ({\mathcal L}\varphi) (x) = P(x, A) \ \ \mbox{for}\mu\mbox{-a.e.}x \in X.
F \left( a , b , c \right) = g_{c}^{- 1}\left( g_{c}\left( a \right) * g_{c}\left( b \right) \right) = r^{- 1}\left( r \left( a \right) * \left[ r \left( c \right) \right]^{- 1}* r \left( b \right) \right) ,
D_m(\lambda,c)= \left(\begin{array}{cccc}1+\lambda_1+c_1^2 1+c_1c_2 ... 1+c_1c_m 1 +c_2c_1 1+\lambda_2+c_2^2 ... 1+c_2c_m & ... 1 +c_mc_1 1+c_mc_2 ... 1+\lambda_m+c_m^2\end{array}\right)
\begin{array}{ll}\frac{1}{2\lambda^2}\{h(\nabla_{\tilde{X}}^N \tilde{Z}, \tilde{Y}) + h(\nabla_{\tilde{Y}}^N \tilde{Z}, \tilde{X})\}+ Ric(X,Y) + \frac{\mu}{\lambda^2}h(\tilde{X},\tilde{Y}) = 0.\end{array}
\Phi = \sum_{\sigma = \pm}\int_{0}^{\infty}d \Omega \sum_{\{T , \lambda \}}\sum_{A}\left( u_{\Omega \{T , \lambda \}A}^{( \sigma )}b_{\Omega \{T , \lambda \}}^{( \sigma ) A}+ b_{\Omega \{T , \lambda \}}^{( \sigma ) A \dagger}u_{\Omega \{T , \lambda \}A}^{( \sigma ) *}\right) \ .
{\bf E}\bigl[\beta_{2, 2}(\tau, b)^k\bigr] =\prod\limits_{l=0}^{k-1}\Bigl[\frac{\Gamma\bigl((l+b_0+b_1)/\tau\bigr)\,\Gamma\bigl((l+b_0+b_2)/\tau\bigr)}{\Gamma\bigl((l+b_0)/\tau\bigr)\,\Gamma\bigl((l+b_0+b_1+b_2)/\tau\bigr)}\Bigr].
\sum_{x\in B_r(x_0)}\!\!\!V(x)\,v^2_+(x,T)\mu(x) \leq \frac{c(T,\lambda,r)}{(r_1)^2}\sum_{x\in G}\exp\Big(-\frac{\mathbf{d}^\alpha(x)}{\lambda}\Big)\chi_{\{r< \mathbf{d}(x)\le r_1\}}(x)\mu(x),
m-s+1-n_{1}=\underbrace{n_{1}+\cdots+n_{1}}_{M_{1}-1\;\mathrm{times}}+\underbrace{n_{2}+\cdots+n_{2}}_{M_{2}\;\mathrm{times}}+\cdots+\underbrace{n_{\ell}+\cdots+ n_{\ell}}_{M_{\ell}\;\mathrm{times}}
\begin{array}{c c}{{\phi_{\omega}^{E}\mid_{0}= \phi_{\omega}^{L}\mid_{2 \pi}\hspace{1 0 m m}\phi_{\omega}^{E}\mid_{\theta}= \phi_{\omega}^{L}\mid_{\theta}}}\\{{\partial_{\sigma}\phi_{\omega}^{E}\mid_{0}= - \partial_{\sigma}\phi_{\omega}^{L}\mid_{2 \pi}\hspace{1 0 m m}\partial_{\sigma}\phi_{\omega}^{E}\mid_{\theta}= - \partial_{\sigma}\phi_{\omega}^{L}\mid_{\theta}}}\\ \end{array}
\begin{array}{cc}\Big(\hat{C}^{-1}(\hat{C}C\hat{C}^{-1}-I)\underline{v}+ \hat{C}^{-1}\hat{C}\Big(Dp - U(J^{(r)}X) - \mathbf{1}\frac{\beta}{2}\Big) \Big)_i =\Big( (C\hat{C}^{-1}-\hat{C}^{-1})\underline{v}+ Dp - U(J^{(r)}X) - \mathbf{1}\frac{\beta}{2}\Big)_i\geq 0. \end{array}
-\sum_{j=1}^d \mu^{(j)}(p)e_j = \left(-\frac{\partial}{\partial t}+A\right)\Phi_1(t, p) = \left(-\frac{\partial}{\partial t}+\sum^d_{j=1}\mu^{(j)}(p)\frac{\partial}{\partial p_j}+\frac{1}{2}\sum^d_{j=1}\frac{\partial^2}{\partial p_j^2}\right)\Phi_1(t, p),
\langle E^{2}\rangle-\langle E\rangle^{2}=kT^{2}\frac{\partial\langle E\rangle}{\partial T}+kT\mu_{1}\frac{\partial\langle E\rangle}{\partial\mu_{1}}+kT\mu_{2}\frac{\partial\langle E\rangle}{\partial\mu_{2}}+...,
n = (n_- + n_+) = \frac{<N>}{V \Lambda^2}= \Bigg[(1+\beta')^\frac{1+\beta'}{2}(1-\beta')^\frac{1-\beta'}{2}4 \pi \beta\frac{e^{2\gamma_E}}{8}\frac{N_c}{N_c-1}2^{\frac{8(N_c-1)^2}{\beta N_c^2}}\Bigg]^\frac{\beta N_c^2}{8(N_c-1)^2 -\beta N_c^2}\;.
\sum_{k=N}^n k^{\underline{N}}\binom{n}{k}x^k (1 - x)^{n-k}= \sum_{k=N}^n \frac{k!}{(k-N)!}\frac{n!}{k!(n-k)!}x^k (1-x)^{n-k}& = n^{\underline{N}}x^N \sum_{k=N}^n \frac{(n-N)!}{(k-N)!((n-N)-(k-N))!}x^{k-N}(1-x)^{(n-N)-(k-N)}& = n^{\underline{N}}x^N \sum_{k=0}^{n-N}\binom{n-N}{k}x^k (1-x)^{(n-N)-k},
A(t,k,r)=\sum_{j=0}^{\min (k,t)}(-1)^j\binom{k}{j}\binom{j+t}{j}\binom{r-j-1}{r-t-1}=\sum_{j=0}^{\min (k,t)}\binom{k}{j}\binom{-t-1}{j}(-1)^{t-j}\binom{-r+t}{t-j},