harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal S}={\cal S}( A_{\mu}^{p}, B_{\mu \nu}^{p}) = \int{d^{4}}x \{- \frac{1}{4}{\varepsilon}^{\mu \nu \rho \sigma}G_{\mu \nu}^{p}B_{\rho \sigma}^{p}+ \frac{1}{2}A_{\mu}^{p}A^{p \mu}\},
\Phi(\vec{r},t;\vec{r}^{\prime},t)=\frac{c}{[4\pi Dc(t-t^{\prime})]^{3/2}}exp[-\frac{|\vec{r}-\vec{r}^{\prime}|^{2}}{4Dc(t-t^{\prime})}]exp[-\mu_{a}c(t-t^{\prime})]
\mu_{1}^{\Delta}= \left( \mu_{1}^{A}, m_{1}^{I}\right) , \nu_{2}^{\Delta}= \left( \nu_{2}^{A}, n_{2}^{I}\right) , \nu_{3}^{\Delta}= \left( \nu_{3}^{A}, n_{3}^{I}\right) , \omega^{\Delta}= \left( \omega^{A}, o^{I}\right) ,
H_2 =\frac{1}{6}\big(\vec{L}_0\cdot \vec{u}\big)^2+\left|\vec{r}\times \frac{\vec{u}}{|\vec{u}|}\right|^2V_{\rm Cal}= \frac{1}{6}\big( (yp_z-zp_y)+(zp_x-xp_z)+(xp_y-yp_x)\big)^2 \hphantom{H_2 =}{}+\frac{1}{3}\big((x-y)^2+(x-z)^2+(y-z)^2\big) V_{\rm Cal},
F^{( \pi )}( m , T ) = - T \sum_{a}m_{a}\int_{- \infty}^{\infty}\frac{d \beta}{2 \pi}\left[ e^{\beta}\ln \left( 1 + e^{- \epsilon_{a R}( \beta ) / T}\right) + e^{- \beta}\ln \left( 1 + e^{- \epsilon_{a L}( \beta ) / T}\right) \right]
\delta = \sum_{j=0}^{\infty}\pi_j \delta(\tilde{s}_j)> \pi_0 \delta(\tilde{s}_0) + \sum_{j=1}^{\infty}\pi_j \delta(\tilde{s}_1) &= \frac{1}{E(\tau_2 - \tau_1)}M(2p - 1) + \left[1 - \frac{1}{E(\tau_2 - \tau_1)}\right] (-2/3) > 2.
\frac{\partial}{\partial t}\frac{\delta\bar\ell}{\delta\bar\sigma^1}=-\operatorname{ad}^*_{\bar\sigma^1}\frac{\delta\bar\ell}{\delta\bar\sigma^1}+\operatorname{div}^{(-\bar\sigma_2)}\left(-\frac{\delta\bar\ell}{\delta \bar\sigma^2}\right),
=\sum_{l=0}^{|k|-1}\frac{2|k|-(2l+1)}{2k^2 h^2}\left[ \phi(x-(l+1)h)-2\phi(x-lh) + \phi(x-(l-1)h) \right] &=\frac{1}{2k^2h^2}\left( \phi(x+kh)+\phi(x+(k+1)h)+(2|k|-1)\phi(x+h)-(2|k|+1)\phi(x) \right),
\phantom{xxxxxx}L_t: \; \; (\l_0y_0 +i \l_1 y_1)-t (i\l_2y_2 + \l_3 y_3)\;=\; t(\l_0y_0 -i \l_1 y_1) -(i\l_2y_2 - \l_3 y_3)=0 \hbox{and}\phantom{xxxxxx}L'_t: \; \; (\l_0^{-1}w_0 +i \l_1^{-1}w_1)-t (i\l_2^{-1}w_2 + \l_3^{-1}w_3)\;=\; t(\l_0^{-1}w_0 -i \l_1^{-1}w_1) - (i\l_2^{-1}w_2 - \l_3^{-1}w_3) =0
H_{T}= \displaystyle \frac{1}{2}\dot{x}^{j}\dot{x}^{j}+ U ( \vec{x}) - i R_{j k}( \vec{x}) \theta_{1}^{j}\theta_{2}^{k}- ( \pi_{\theta_{a}^{j}}- \displaystyle \frac{i}{2}\theta_{a}^{j}) \lambda_{a}^{j}
K_{W_e/U}\le d_e^*K_{W_r\times_{V^r}V^e/U}= d_e^*K_{W_r\times_{V^r}V^e/W_r}+b_{e,r}^*K_{W_r/U}= \alpha_e^*K_{V^e/V^r}+b_{e,r}^*K_{W_r/U}= \alpha_e^*(1-p^{e-r})K_V +b_{e,r}^*K_{W_r/U},
\frac{1}{N^{3}}\int_{G_{N}}f(x)\sum_{i=1}^{N}M_{\epsilon,i}^{2}(x)dx\le C\left[1+\left(\frac{\mathcal{D}_{N}(f)}{N}\right)^{\frac{3}{4}}\right]\left(\epsilon^{\frac{3}{2}}+\frac{\epsilon}{N}\right).
e_{4}( k , N ) ={\frac{8 N^{4}}{C_{( 1^{k}; 0 )}}}\left[{\frac{1}{2 C_{( 1 ; 0 )}}}+ \sum_{i = 1}^{4}{\frac{1}{C_{( 1^{k}; 0 )}- 3 C_{( 1 ; 0 )}- C_{r_{i}}}}{\frac{d_{r_{i}}}{N d_{( 1^{k}; 0 )}}}\right] .
\mathcal{F}\left( b\right) :=\left\{x\in \mathbb{R}^{n}:a_{t}^{\prime}x\leq b_{t}t\in T\right\}, \mathcal{F}^{op}\left( c,b\right) :=\arg \min \left\{c^{\prime}x:x\in \mathcal{F}\left( b\right) \right\}.
c_1(E) = - s_1(E), \; c_2(E) = s_1(E)^2 - s_2(E), \ldots, c_i(E) = - s_1(E) c_{i-1}(E) - s_2(E) c_{i-2}(E) - \cdots - s_i(E),\; \forall \; i \geqslant 3.
\sum_{j = 0}^k{\sum_{s = 0}^j{( - 1)^s \binom kj\binom js\frac{{G_{m + (n + 1)k - 2j + (n + 1)s}}}{{F_n^s}}}}= ( - 1)^{(n + 1)k}\frac{{G_m}}{{F_n^k}},n\ne 0\,,
T(\alpha[\begin{matrix}\vec{f}\\ \\ 0\\ \end{matrix}]+\beta[\begin{matrix}0\\ \\ \vec{b}\end{matrix}])=\alpha[\begin{matrix}1\\ 0\\ \vdots\\ 0\\ \epsilon_{f}\\ \end{matrix}]+\beta[\begin{matrix}\epsilon_{b}\\ 0\\ \vdots\\ 0\\ 1\end{matrix}].
\lVert \Delta^{p,*}_a(h_p^- - h_q^-) \rVert+ \lVert \Delta^{p,*}_a(h_q^- - h_s^-) \rVert = \lVert \Delta^{p,*}_a(h_p^- - h_q^-) \rVert+ \lVert \Delta^{q,*}_a(h_q^- - h_s^-) \rVert &< \epsilon_s - \epsilon_p,
W^{(1)}= \frac{a +b_1}{2}y^5+\left(- 12 a + 37 b_1 + \frac{b_2 - b}{2}\right)y^9 &+\left(138 a + 691 b_1 + 14 b_2+ 11 b + \frac{b_3 +c}{2}\right)y^{13}&+\left(- 1012 a - 19335 b_1 - 9 b_3 - 10 c - \frac{675 b_2 + 231 b + d}{2}\right)y^{17}+ \cdots.
\big(\boldsymbol{\theta}(y),\mathbf{a}(y)\big)\in \mathcal{K}(y)\,,\mathcal{A}(y)=\{r\cdot\mathbf{a}(y): r\in \mathcal{R}(y)\}\,,\min\, \mathcal{R}(y)=1{\mathrm{for\all\}}y\in \mathbb{R}^n_+\,.
\langle \mathcal L(h), \, g\rangle_{L^2_{x,v}}=\langle \mathcal L(h), \, g-\Pi_{\mathcal L}(g)\rangle_{L^2_{x,v}}=\langle \mathcal L(h^{\perp}), \, g^{\perp}\rangle_{L^2_{x,v}}\leq C^{\mathcal L}||h^{\perp}||_{\Lambda}\, ||g^{\perp}||_{\Lambda}\,,
\frac{\dot{y}}{\dot{x}}=\dot{y}:\dot{x}\equiv\frac{dy}{dt}:\frac{dx}{dt}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{dy}{dx}=\frac{d}{dx}(f(x))=Dy=f^{\prime}(x)=y^{\prime}.
e^{-\nu}(\partial_{3}+iA_{3}+i\partial_{4}-A_{4}+{1 \over 2}\partial_{3}\nu +{i \over 2}\partial_{4}\nu)\psi_{1}+e^{-\mu}(\partial_{1}+iA_{1}-i\partial_{2}+A_{2}+{1 \over 2}\partial_{1}\mu -{i \over 2}\partial_{2}\mu)\psi_{2}=0,
D(h^{N}_{\theta}) =\Pi_{+}D(h_{\theta})=H^{1}(\mathbb{R}^{2}_{+}\setminus\Gamma_{\theta}^{+}), D(h^{D}_{\theta}) =\Pi_{-}D(h_{\theta})=\{v\in H^{1}(\mathbb{R}^{2}_{+}\setminus\Gamma_{\theta}^{+}):v(\cdot,0)=0\},
U_{J_\ell}= b_\ell -\sum_{j=1}^{\ell-1}\left\lfloor\frac{b_\ell-b_j}{a_n}\right\rfloor y_j + \sum_{j=\ell+1}^{m}\left\lfloor\frac{b_\ell-b_j}{a_1}\right\rfloor y_j
\int_{t+2(r+1)}^{\infty}\frac{\lambda''(s) r}{(s-t)}ds =\int_{t+2(r+1)}^{\infty}\frac{\lambda''(s) r}{1+s-t}ds + \int_{t+2(r+1)}^{\infty}\lambda''(s) r \left(\frac{1}{s-t}-\frac{1}{1+s-t}\right) ds
\frac{1}{2N^2}\Big(\sum_{j=1}^{N}h_{j}\Big)H_{N}+ \frac{1}{2N^2}H_{N}\Big(\sum_{j=1}^{N}h_{j}\Big) & = \frac{1}{N^2}\Big(\sum_{j=1}^{N}h_{j}\Big)^2 - \frac{a_{N}}{2N^2(N-1)}\sum_{i=1}^{N}\sum_{1\leq j<k \leq N}\big(h_{i}|x_j-x_k|^{-1}+ |x_j-x_k|^{-1}h_{i}\big).
\begin{array}{l}{{\nabla^{2}\phi_{e}= - \rho_{e}( \mathbf{x}) , ~ \nabla^{2}\phi_{m}= \rho_{m}( \mathbf{x}) ,}}\\{{\nabla^{2}\mathbf{A_{m}}= - \mathbf{J_{e}}( \mathbf{x}) , ~ \nabla^{2}\mathbf{A_{e}}= \mathbf{J_{m}}( \mathbf{x})}}\\ \end{array}
\left(\phi(-\infty),\psi(-\infty)\right)=(b,0), begin{align*}0.2cm] \left(\phi(-\infty),\psi(-\infty)\right)=(1,0), begin{align*}0.2cm] \left(\phi(-\infty),\psi(-\infty)\right)=(u_1^*,u_1^*), begin{align*}0.2cm] \left(\phi(-\infty),\psi(-\infty)\right)=(u_2^*,u_2^*).
\mu_\omega\left(A \cap(\sigma^{-k-n}B)\right)= \int 1_A\cdot 1_B\circ \sigma^{k+n}~d\mu_\omega=\int \L_{\theta^n\omega}^{k}\left(\L^n1_A\right)\cdot 1_B~d\mu_{\theta^{n+k}\omega}.
(a_1, a_2, \cdots, a_{2m}, 1, 2, \cdots, 2n)(1, 2, \cdots, 2n) = \sum_{j=2}^{2m}(-1)^j (a_1, a_j, 1, 2, \cdots, 2n)(a_2, a_2, \cdots, \widehat{a_j}, \cdots, a_{2m}, 1, 2, \cdots, 2n),
r^{n-1}(g^m(r))'=-\int_0^r\rho^{n-3+\frac{n-2-nm}{m}}\left\{\tilde\alpha g(\rho)+\tilde\beta \rho g'(\rho)\right\}d \rho\quad\forall r>0 \Rightarrow\quad g'(r)=-\frac{g^{1-m}(r)}{m r^{n-1}}\int_0^r\rho^{n-3+\frac{n-2-nm}{m}}\left\{\tilde\alpha g(\rho)+\tilde\beta \rho g'(\rho)\right\}d\rho\quad\forall r>0.
T_n(x_1, \ldots, x_n; \hbar) = \sum_{\pi}T_1(x_{\pi (1)}) \ldots T_1(x_{\pi (n)}) \Theta (x_{\pi (1)}^0 - x_{\pi (2)}^0 ) \ldots \Theta (x_{\pi (n-1)}^0 - x_{\pi (n)}^0),
S=\frac{1}{G}\int dxdt\:\sqrt{-\bar{g}}\:e^{-2\bar{\phi}}(\bar{R}+4(\bar{\nabla}\bar{\phi})^2+4\lambda^2)-\frac{1}{2}\int dxdt\:\sqrt{-\bar{g}}\: \sum_{i=1}^N(\bar{\nabla}f_i)^2,
E_1 = \frac{1}{\mu}\,e_1,E_i= \frac{1}{a}\,e_i,~i=2,3, E_j = -\frac{c}{a\sqrt{a^2b^2-c^2}}\,e_{j-2}+ \frac{a}{\sqrt{a^2b^2-c^2}}\,e_j,~j=4,5.
I = \int dt \left(\frac 12 q'^2 - \frac{1}{2l}\varsigma^1{\varsigma^1}' - \frac{1}{2l}\varsigma^2{\varsigma^2}' - \frac{g}{2q^2}+ \frac{\sqrt g}{2lq^2}[\varsigma^1, \varsigma^2] + \frac{3}{cl^2}T_B q^2 - \frac{3}{cl^2}T_F \varsigma^1 q\right)\,.
\begin{array}{l}(Z\cdot\psi)(X_1,\cdots,X_4)=\sum\limits_{1\leq i\leq 4}[Z,X_i]\bullet \psi(X_1,\cdots,\hat{X}_i,\cdots,X_{4}) -\sum\limits_{1\leq i<j\leq 4}\psi(X_1,\cdots,[[Z,X_i],X_j]\hat{X}_j,\cdots, X_{4})\end{array}
L_{0}= \frac{M}{2}{\dot{q}}^{2}- V(q) + \sum_{\alpha=1}^{N}\{\frac{m_{\alpha}}{2}{\dot{x}_{\alpha}}^{2}-\frac{m_{\alpha}}{2}\omega_{\alpha}^{2}x_{\alpha}^{2}+{\epsilon}_{\alpha}x_{\alpha}q \},
L^{( 0 )}\rightarrow L^{( N )}= \sum_{k = 0}^{N}\int d x ( \dot{\phi_{k}}\phi_{k}^{\prime}- \phi_{k}^{2}) + \sum_{m = k + 1}^{N}\sum_{k = 0}^{N - 1}\int d x ( \dot{\phi_{k}}- \phi_{k}^{\prime}) \phi_{m}^{\prime}
\int dxE_{n}(s)(\frac{d}{ds}\psi_{n}^{*}(s))\psi_{n}(s)+\int dx\psi_{n}^{*}(s)(U^{\prime}-U)\psi_{n}(s)+\int dxE_{n}(s)\psi_{n}^{*}(s)(\frac{d}{ds}\psi_{n}(s))=
\Lambda_{\nu}^{\mu}(x,y)=\delta_{\nu}^{\mu}\,\delta^{3}(x-y)-J^{\mu\alpha}\int d^{\,3}\!\rho \;d^{\,3}\!\sigma \,\delta_{\alpha (x)}\varphi^{i}(\rho )\,J_{ij}(\rho ,\sigma )\,\delta_{\nu (y)}\varphi^{j}(\sigma )
(2^*-1)\alpha_n \sum_{i=1}^k \mu_i^{\frac{n-2}{2}}\int_\Omega U_{\mu_i,\xi_i}^{2^*-2}H(x,\xi_i) \psi^0_{\mu_j,\xi_j}dx =o\Big(\frac{\epsilon}{|\ln\epsilon|^2}\Big).
\begin{array}{ll}\mathcal{D}^{\alpha}\xi_t + k_1\gamma p\nabla\cdot (\mathcal{D}^{\alpha}v) = -k_1\mathcal{D}^{\alpha}(v\cdot\nabla\xi) - k_1\gamma\sum\limits_{|\alpha_1|>0}\mathcal{D}^{\alpha_1}\xi \nabla\cdot (\mathcal{D}^{\alpha_2}v).\end{array}
ds^{2}=(H_{1}H_{2}...H_{n})^{-4t/r}\eta_{\mu \nu}dy^{\mu}dy^{\nu}+(H_{1}H_{2}...H_{n})^{4u/r)}[\sum_{i=1}^{n}H_{i}^{-2/r}dz_{i}^{m_{i}}dz_{i}^{m_{i}}+dx^{\gamma}dx^{\gamma}],
h(x) =\frac{h(x_{n+1})-h(x_{n})}{x_{n+1}-x_n}(x-x_n)+h(x_n) &=\frac{\sqrt{2}}{2}\left(\frac{\sqrt{2}+1}{3}\right)^n x+(\sqrt{2}+1)^n\left(\frac{2-\sqrt{2}}{4}\right).
B_{3,4}= \mathcal{V}(1-a_3a_4^{-1}) B_{1,2,3}= \mathcal{V}(1-a_1a_2^{-1}a_3^7) B_{1,2,4}= \mathcal{V}(1-a_1a_2^{-1}a_4^7) B_{1,2,3,4}= \mathcal{V}(1-a_1a_2^{-1}a_3^7, 1-a_3a_4^{-1}).
(\vartheta\otimes 1_1)\circ(1_1\otimes \epsilon)(v_b)= (\vartheta\otimes 1)\sum\limits_{a=-n}^n(-1)^{[b]}v_b\otimes \vartheta(1) = \sum\limits_{a=-n}^n(-1)^{[b]}\epsilon(v_b\otimes v_{a})\otimes v_{-a}= \sum\limits_{a=-n}^n(-1)^{[b]}\delta_{b,-a}(-1)^{[b]}1\otimes v_{-a}= 1_1(v_b).
\beta_{n}(t)=|u_{n+1}(t)-u_{n}(t)|+|v_{n+1}(t)-v_{n}(t)|+|\rho_{n+1}(t)-\rho_{n}(t)|+|\psi_{n+1}(t)-\psi_{n}(t)|+|\phi_{n+1}(t)-\phi_{n}(t)|,
d_{R}(t):=\left\lbrace\begin{array}{ll}-\bar{\sigma}_{R}(t), t\in[\bar{t}_0,\infty)\setminus \cup_{\nu\in\mathbb{N}}\mathcal{A}_{\nu}d_{R,n}(t), t\in\mathcal{A}_{\nu},\nu\in\mathbb{N}\end{array}\right.
\int_{0}^{\infty}d \xi |\chi_{\leq 1}'(r \xi)| r |r \xi \Phi_{1}(r \xi)| \frac{|\chi_{\leq \frac{1}{4}}(\xi)|}{\log^{b-1}(\frac{1}{\xi})}\leq C r \int_{0}^{1/r}\frac{d\xi}{\log^{b-1}(\frac{1}{\xi})}&\leq \frac{C}{\log^{b-1}(r)}, r\geq\frac{t}{2}
\lim_{n\rightarrow \infty}n^{2\alpha+1}\int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'(X_s))^2 \Psi^{n,3}_s ds =\kappa_4 \int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'\sigma)^2(X_s) ds,
\sum_{l_{2}= 2 , 4 , . .}^{*}{\frac{l_{2}^{2}}{( l_{1}^{2}- l_{2}^{2})^{3}}}={\frac{1}{1 6}}\sum_{k = 1}^{*}{\frac{k^{2}}{( ( l_{1}/ 2 )^{2}- k^{2})^{3}}}.
R^{(1)}(z) = \sum_{j=0}^{m}\frac{1}{z+x_{j}}\mbox{Res}(J_{R}^{(1)}(s),s = -x_{j}), \mbox{for}z \in \mathbb{C}\setminus \bigcup_{j=0}^{m}\mathcal{D}_{-x_{j}},
\frac{\partial}{\partial a}\cosh(\ell(a,c)) =\frac{\sinh(a-c)}{2\sinh^2a}\left(2\sinh^2a-\sinh^2c-\cosh(a-c)\sinh a\sinh c\right) &>\frac{\sinh(a-c)}{2\sinh a}\left(\sinh a-\cosh(a-c)\sinh c\right) &=\frac{\sinh^2(a-c)\cosh c}{2\sinh a}>0,
\rho (r) = \begin{cases}a, r\in [a,k] \frac{1}{2}\left( \left( a^2 + \frac{\omega^2}{a^2}\right)\frac{r^2}{k^2}+ \left( a^2 + \frac{\omega^2}{a^2}\right) \frac{k^2}{r^2}+ 2\left( a^2 - \frac{\omega^2}{a^2}\right) \right)^{\frac{1}{2}}, r\in (k,b]\end{cases}
W_1(\alpha)= -\frac{1}{2}\int [123] f_\lambda\left[\frac{\tilde{\delta}(p_{3 \hat \alpha}-p_{1 \hat \alpha 2 \hat \alpha})}{p^-_{3 \hat \alpha}- p^-_{1 \hat \alpha 2 \hat \alpha}}- \frac{\tilde{\delta}(p_3-p_{12})}{p^-_3-p^-_{12}}\right]a^\dagger_{\lambda 1}a^\dagger_{\lambda 2}a_{\lambda 3}- h.c. \; .
C_{\mu}= \frac{h_{\mu}^{2}}{2}\{d - l \}C_{\mu}+ \sum_{\mu^{\prime}\not = \mu}\frac{h_{\mu^{\prime}}^{2}}{2}\{I - l \}C_{\mu^{\prime}}- \frac{h_{\varepsilon}^{2}}{2}\{I - l \}C_{\varepsilon}.
I ( L ) = \frac{1}{2 \left( \frac{n - 1}{2}\right) \!}\left( \frac{i}{8 \pi}\right)^{\frac{n - 1}{2}}\int_{S_{\infty}^{n - 1}}\mathrm{t r}[ U ( x ) ( d U ( x ) )^{n - 1}] ,
= \sum_{p\in\eta\cap a}\delta_{w,y}\varepsilon(p;\eta,a)\varepsilon(w;\eta,\gamma)\left|\eta_z^p a_p\eta_p^w\right|\left|\gamma\right| &+\sum_{p\in\eta\cap a}\delta_{z,x}\varepsilon(p;\eta,a)\varepsilon(z;\eta,\gamma)\left|\eta_z^p a_p\eta_p^w\right|\left|\gamma\right|.
TA_1(\zeta_1)\dots A_k(\zeta_k)\exp\left(-i\int_{\Gamma_{z,z'}}d\zeta\,B(\zeta) \right)\Psi = \overline{U(B;z,\zeta_1)}A_1(\zeta_1)\dots A_k(\zeta_k)\overline{U(B;\zeta_k,z')}\Psi.
\int_{\Sigma_{x}}\int_{\Sigma_{y}}\{y^{a}({\bf x}) , p_{b}^{n}({\bf y}) \}_{P B}f ({\bf x}) f ({\bf y}) d{\bf x}d{\bf y}= - \int_{\Sigma}\{y^{a}, p_{b}\}f ({\bf x}) d{\bf x},
\int_{S}(f_{U, \ell}^{1/2}- f_{L, \ell}^{1/2})^2 \leq \sum_{j=1}^k \int_{S_j}\frac{e^{-x}}{1 - e^{-a}}\bigl\{f_{U,\ell_j}^{(j)}(x)^{1/2}- f_{L, \ell_j}^{(j)}(x)^{1/2}\bigr\}^2 \, dx \leq \sum_{j=1}^k \frac{e^{-\alpha_j}}{1 - e^{-a}}\epsilon_j^2 \leq \epsilon^2,
\langle T(z,\bar{z})T(0,0)\rangle =\frac{c/2}{z^{4}}\qquad \langle \bar{T}(z,\bar{z})\bar{T}(0,0)\rangle =\frac{c/2}{\bar{z}^{4}}\qquad \langle T\bar{T}\rangle =0
\langle \overline{v}, w'\rangle=\!\sum_{i\in{\rm gs}(\overline{x})}\rho_{\lambda}'(F_i(\overline{x}_{J_i}))\langle \nabla\!F_i(\overline{x}_{J_i}),w_{J_i}'\rangle\!+\!\sum_{i\in\overline{{\rm gs}}(\overline{x})}\overline{\eta}_i\langle\overline{\zeta}_i,w_{J_i}'\rangle \quad\ \forall w'\in\mathbb{R}^n.
K \left( x', x'', N, (\tau''-\tau')\right) = 2 i N\int_{\tau'}^{\tau''}d t \int_{\Sigma(\tilde{x}^0)}d^3{\bf\tilde{x}}\hspace{0.25em}K \left( x', \tilde{x}, N, t\right)\frac{\partial}{\partial x^0}K \left( x, x'', N, (\tau''-\tau'-t ) \right) \vline_{x=\tilde{x}}.
\rho^{n+2}\int_{B_{\rho/2}(y)}|\nabla u|^2\,dx \leq C\delta \rho^{n+2}\int_{B_{\rho}(y)}|\nabla u|^2\,dx+ C\delta^{-2-\frac{3n}{2}}\left(\int_{B_{\rho}(y)}|u|\,dx\right)^2 & \leq C \delta \rho^{n+2}\int_{B_{\rho}(y)}|\nabla u|^2\,dx+C\delta^{-2-\frac{3n}{2}}\left(\int_{B_{1}}|u|\,dx \right)^2.
\frac{{n-i\choose z_a}}{{n-i\choose z_b}}\ge\frac{(n-i-z_b)}{z_b+1}= \frac{(n-2k+z_a)}{z_b+1}\ge \frac{(n-2k+z_b+1)}{z_b+1}\ge \frac{(n-k-2)}{k-2}.
\mathrm{e}^{u_{1}}={\frac{\prod_{r = 1}^{n}| z - z_{r}|^{2}+ \left| \sum_{k = 0}^{n - 1}q_{k}z^{k}\right|^{2}}{\prod_{r = 1}^{n}( | z - z_{r}|^{2}+ \mu )}},
\left. \phantom{{A\over B}}{\epsilon^{i_1\ldots i_n}\phi_{i_1}d\phi_{i_2}\cdots d\phi_{i_n}}\right|_\Sigma =\left.{{\overline{\Delta}}\over \Delta}{\epsilon^{j_1\ldots j_n}\overline{\phi}_{j_1}d\overline{\phi}_{j_2}\cdots d\overline{\phi}_{j_n}}\right|_\Sigma
F(t) = \prod_{j=1}^{\frac{n}{2}}(t - \lambda_{2j+1}) + \frac{4}{n + 3}\sum_{j=1}^{\frac{n}{2}}\sin^{2}\left[\frac{(2j + 1) \pi}{n + 3}\right] (\lambda_{2j+1}- \lambda_{1}) \prod_{\substack{m=1 m \neq j}}^{\frac{n}{2}}(t - \lambda_{2m+1}).
\frac{|c_{j}(t+dt)|^{2}-|c_{j}(t)|^{2}}{|c_{j}(t)|^{2}}\approx\frac{dt}{a_{jj}}\sum_{n}\frac{2}{\hbar}Im(a_{nj}V_{jn})-2Re(a_{nj}\dot{R}.d_{jn})
\psi_y = \frac{\partial_Y C}{1 - \partial_YC}= \partial_Y C\sum_{j = 0}^\infty (\partial_YC)^j \psi_z = \frac{\partial_Z C}{1 - \partial_YC}= \partial_Z C\sum_{j = 0}^\infty (\partial_YC)^j.
\mathrm{L i}_{3}( i / \sqrt 3 ) = \frac{1}{8}\sum_{n > 0}\frac{( - 1 / 3 )^{n}}{n^{3}}+ \frac{i}{\sqrt 3}\sum_{n \ge 0}\frac{( - 1 / 3 )^{n}}{( 2 n + 1 )^{3}}
\gamma_0 = \frac{1}{q!}, \gamma_1 = -\, \frac{1}{(q-2)!}\bigg\{\frac{5}{3}\bigg\}, \gamma_2 = \frac{1}{2!(q-2)!}\bigg\{\frac{25}{9}(q-2)_2 + \frac{61}{6}(q-2) + 3 \bigg\}, \gamma_3 = -\, \frac{1}{3!(q-2)!}\bigg\{\frac{125}{27}(q-2)_4 + \frac{305}{6}(q-2)_3 + \frac{681}{5}(q-2)_2 + 73 (q-2) + 2 \bigg\}.
\eta(0) = 0, \eta(t) \mbox{is non-decreasing and}\int_0^T \zeta_0(t) \, \eta(dt) = 0, &{\tilde{\eta}}(0) = 0,{\tilde{\eta}}(t) \mbox{is non-decreasing and}\int_0^T{\tilde{\zeta}}_0(t) \,{\tilde{\eta}}(dt) = 0 .
{\cal T}_{S}= \left[ \begin{array}{c c}{1}&{\qquad 0}\\{}&{}\\{1}&{\qquad 1}\\ \end{array}\right] \qquad{\cal S}_{S}= q \gamma_{q}\left[ \begin{array}{c c}{0}&{\qquad - 1}\\{}&{}\\{1}&{\qquad 0}\\ \end{array}\right]
\begin{aligned}(I-\psi_a(H^a))e^{-itH}f \sim (I-\psi_a(H^a))\Phi_a(x_a/t)\sum_{d\le a}\tilde P^d_{M_d}e^{-itH}f &=\Phi_a(x_a/t)(I-\psi_a(H^a))\sum_{d\le a}\tilde P^d_{M_d}e^{-itH}f.\end{aligned}
\sqrt{F}/F =\langle d_1^8\rangle F\times\langle q(3,1)^4\rangle F\times\langle q(2,1)^2\rangle F\times\langle q(2,3)^2\rangle F\times &\times\langle q(1,1)\rangle F\times\langle q(1,3)\rangle F\times\langle q(1,5)\rangle F\times\langle q(1,7)\rangle F.
\vcenter t o . 7 6 i n{\special{e m : g r a p h 4 . p c x}\vss \mathrm{t}o 1 i n{\llap{p_{1}, i}\hss p_{3}, k}\vss \mathrm{t}o 1 i n{\llap{p_{2}, j}\hss p_{4}, l}\vss}
L_{_{\mathrm{G F + F P}}}^{1}= - \frac{i}{4}\mathrm{T r}\left( \delta_{1}( \chi^{\mu \nu}[ \lambda F_{\mu \nu}- \frac{1}{2}S_{\mu \nu \alpha \beta}F^{\alpha \beta}+ \frac{1}{2}a B_{\mu \nu}] ) \right) .
\begin{pmatrix}A 0 0 1\end{pmatrix}:=\left(\prod_{2\leq k\leq N-1}A(k)(\phi_k,\psi_k)\right)e^{\lambda_3\phi_{N-1}},B:=F_{N-1}(\phi_{N},\ldots,\phi_{\frac{N(N-1)}{2}},\psi_{N},\ldots,\psi_{\frac{N(N-1)}{2}},\omega_1,\ldots,\omega_{N-2}).
{\cal F}^{( 4 )}( \vec{X}) = \sum_{k = - \infty}^{+ \infty}\sum_{n = - \infty}^{+ \infty}{\cal F}_{k , n}( | \phi | , \omega ) \exp \left( i k \left( \sigma + n \theta + \frac{\theta \omega}{2 \pi}\right) \right)
u_j(x) = \int_{x-r}^{x+r}\prod_{\stackrel{i=1}{i\neq j}}^n(\eta_i(y)-\eta_i(x))(-\partial \eta_j(x))\sin^{[n]}(\tilde \Psi(y)-\tilde\Psi(x))\ \d y, q(x) = \int_{x-r}^{x+r}\prod_{i=1}^n(\eta_i(y)-\eta_i(x))\sin^{[n+1]}(\tilde \Psi(y)-\tilde \Psi(x))(-\partial \tilde \Psi(x)) \ \d y
\ \int_{B(x,r)}(|\nabla u|^{p-2}\nabla u-|\nabla v|^{p-2}\nabla v)\cdot(\nabla u-\nabla v)\,d\mu = \int_{B(x,r)}|\nabla u|^{p-2}\nabla u\cdot (\nabla u-\nabla v)-|\nabla v|^{p-2}\nabla v\cdot (\nabla u-\nabla v)\,d\mu = \int_{B(x,r)}|\nabla u|^{p-2}\nabla u\cdot (\nabla u-\nabla v)\,d\mu.
P_{\ \nu}^{\mu}= -{\frac{1}{4}}R_{\ \lambda}^{\mu}R_{\ \nu}^{\lambda}+{\frac{1}{6}}R R_{\ \nu}^{\mu}+{\frac{1}{8}}\delta_{\nu}^{\mu}R_{\ \beta}^{\alpha}R_{\ \alpha}^{\beta}-{\frac{1}{1 6}}\delta_{\ \nu}^{\mu}R^{2}\ .
\tilde{\alpha}_{- m}^{( A )}\rightarrow \frac{1}{\sqrt{m}}\tilde{\alpha}_{- m}^{( A )}, \tilde{A}_{- r}\rightarrow \frac{1}{\sqrt{r}\gamma}\tilde{A}_{- r}, \tilde{B}_{- s}\rightarrow \frac{1}{\sqrt{s}\gamma}\tilde{B}_{- s}.
i \frac{\partial}{\partial t}\psi ( \vec x, t) \left|_{\vec x}\right.= i \left[ \frac{\dot a (t)}{a (t)}\right] \ \vec r \cdot \left( \frac{\partial f}{\partial \vec r}\left|_t \right. \right)+ i \frac{\partial f}{\partial t}\left|_{\vec r}\right. .
C(\Psi,\varphi_0,J,f)=\sup_{\substack{\xi\in A^2(\{\Psi<0\},e^{-\varphi_0})^*\setminus\{0\}\xi|_{A^2(\{\Psi<0\},e^{-\varphi_0})\cap J}\equiv 0}}\frac{|\xi\cdot f|^2}{K^{\varphi_0}_{\xi,\Psi,\lambda}(0)}.
k_{i+1}^{\pm}(u+\frac{1}{2}hi)^{-1}E_{j}(v)k_{i+1}^{\pm}(u+\frac{1}{2}hi)=E_{j}(v), k_{i}^{\pm}(u+\frac{1}{2}hi)E_{j}(v)k_{i}^{\pm}(u+\frac{1}{2}hi)^{-1}=E_{j}(v),
{\cal H}={\frac{u}{\cal R}}\sqrt{\left({\frac{u}{\cal R}}\right)^{2}{P_{u}}^{2}+ \vec{L}^{2}+ \alpha^{\prime}m^{2}L^{2}g_{s}^{\frac{- 2}{D - 3}}e^{- 2{\frac{D - 4}{D - 3}}\phi}}- q A_{t},
\Sigma_{3}= \sum \limits_{\delta_1, \delta_2 = \pm 1}\sum \limits_{D_1, D_2}\frac{S(\delta_1 m_1, \delta_2 m_2, n_1, n_2, D_1, D_2)}{D_1 D_2}\mathcal{J}_{\delta_1, \delta_2}\left(\frac{\sqrt{m_1 n_2 D_1}}{D_2},\frac{\sqrt{m_2 n_1 D_2}}{D_1}\right),
G^{\prime}(t)=\int\limits_{\mathbb{Q}_{p}}u_{t}(x,t)w(x)dx=-\gamma\int\limits_{\mathbb{Q}_{p}}(\boldsymbol{D}_{x}^{\alpha}u)(x,t)w(x)dx +\int\limits_{\mathbb{Q}_{p}}F(u(x,t))w(x)dx+\int\limits_{\mathbb{Q}_{p}}(\boldsymbol{D}_{x}^{\alpha_{1}}u^{3})(x,t)w(x)dx.
v_q(a_{n,k}) \geq \min_{m>k,\,lm\leq n}\left(f(l,m,n)+v_q(a_{n-lm,m-1})\right) &\geq \min_{m>k,\,lm\leq n}\left(f(l,m,n)+\frac{m-2}{m}\binom{n-lm}{2}\right) &=\min_{m>k,\,lm\leq n}\left(\frac{m-2}{m}\binom{n}{2}+\binom{l}{2}\right) &=\frac{k-1}{k+1}\binom{n}{2}.
\psi_{X,a}(\alpha,\alpha) = (-1)^{\frac{k(k-1)}{2}}\cdot\int_X \alpha\cup\overline{\alpha}\cup a^{n-k}&=(-1)^{\frac{k(k-1)}{2}}\cdot\int_{\tilde{X}}\nu^*\alpha\cup\overline{\nu^*\alpha}\cup (\nu^*a)^{n-k}.
\begin{array}{r c l}{{{\cal L}}}&{{=}}&{{\displaystyle ( \partial_{\mu}\Phi )^{\dagger}\star \partial^{\mu}\Phi + \mu^{2}\Phi^{\dagger}\star \Phi - \lambda \Phi^{\dagger}\star \Phi \star \Phi^{\dagger}\star \Phi .}}\\ \end{array}
H^{2\theta}(\mathbb{R}^2\times \Sigma^2) =\{u\in \mathcal S'(\mathbb{R}^2\times \Sigma^2); \sum_{j,k}\int_{\mathbb{R}^2}(|\xi_0|^2+|\eta_0|^2+\lambda_j^2+\lambda_k^2)^{2\theta}|\mathcal Fu_{jk}(\xi_0, \eta_0)|^2d\xi_0d\eta_0<\infty\},
U_{\rm s}\;=\;{4\over 3n_{\rm s}(k^2)^2}\int_0^kd\zeta\, \rho_{\rm s}(\zeta)(k^2-\zeta^2)-{1\over 3}-{2\over 3}{d_{\rm s}(k^2)-p_{\rm s}(k^2)\over n_{\rm s}(k^2)^2}\;.
(w-w_0)v^2\bar g(\nabla^NH,\tau)= -\frac{1}{\varepsilon}\bar g(\nabla^N((w-w_0)v),\tau) &-{\eta}wv\left(1-\frac{1}{v^2}\right)-(w-w_0)v^2g(\nabla^NP,\tau) \leq -\frac{1}{\varepsilon}\bar g(\nabla^N((w-w_0)v),\tau) &-(w-w_0)v\left({\eta}\left(1-\frac{1}{v^2}\right)-2c^2-\frac{|A|^2}{2}\right) &-w_0v{\eta}\left(1-\frac{1}{v^2}\right),
(1-q_1^{-1}q_3)E^{(1)}_{0|1,l}=\sum_{j\ge0}q_1^{j-l}\Bigl(E_{0,l-j}E_{1,j}-(q_1+q_3)E_{0,l-j-1}E_{1,j+1}+q_1q_3 E_{0,l-j-2}E_{1,j+2}\Bigr) &+\sum_{j\ge1}q_1^{-j-l}\Bigl(q_1q_3 E_{1,-j}E_{0,j+l}-(q_1+q_3)E_{1,-j+1}E_{0,l+j-1}+ E_{1,-j+2}E_{0,j+l-2}\Bigr)\,,
(1-e q^{n-1}) w_n+ (1-ed q^{n-1}) w_{n+1}- \frac{\gamma}{\alpha}(1-d q^{n-1}) w_n -\frac{\gamma}{\alpha}(1-q^{n-1})w_{n-1}= \left( \frac{1}{\rho_a}-\frac{\gamma}{\alpha(1-\rho_a)}\right) w_n.
\begin{array}{lll}{p_\delta}^*= b_{\delta}\cdot z_{\delta}& 0_{\delta}\le z_{\delta}\le u_{\delta}, & 1_\delta \cdot z_{\delta}=1, & \tilde{A}_\delta z_{\delta}\le \tilde{c}_\delta, & \displaystyle{\sum_{I\le \delta}( A_0 \frac{I}{\delta}) z_{I,\delta}}\le b_0.\end{array}
0\, \longrightarrow\, p_{1*}{\mathcal Q}_{d-1}(A,\,B) \, \longrightarrow\, p_{1*}{\mathcal Q}_d(A,\,B)\,=\,{\mathcal K}_d(A,\,B)\,\stackrel{p_0}{\longrightarrow}\, p_{1*}\left( \frac{\mathcal{J}_d}{\mathcal{J}_{d-1}}\right)\, \longrightarrow\, 0\, .
f(N\*x)=f(N\*x_1, \ldots, N\*x_n)=\frac{1}{(2\pi)^{n/2}\*N^{n}}\*\sum_{k\in\mathbb{Z}^n}\* \hat{f}(k_1\*N^{-1}, \ldots, k_n\*N^{-1})e^{i\*k\cdot x},
\psi|q_i|\rightarrow0,\;\;\nabla_{q_i}\psi\cdot\frac{q_i}{|q_i|}\rightarrow0,\;\;\;\;|q_i|\rightarrow\infty,\;\;\;\;(x,t)\in\mathbb{R}^3\times(0,T],\;\;i=1,\ldots,K.
\langle \bar{W}^{4}| W^{4}| W^{4}| \bar{W}^{4}\rangle = \frac{1}{x_{1 3}^{8}x_{4 2}^{8}}\left[ a_{2}+ \frac{b_{4}}{s}+ \frac{c_{1}}{s^{2}}+ \frac{b_{1}}{s^{3}}+ \frac{a_{1}}{s^{4}}\right] ,
Y_{i_{1}j_{1}, \ldots , i_{p}j_{p}}^{k_{1}l_{1}, \ldots , k_{q}l_{q}}= \bar{a}_{i_{1}\ldots i_{p}}\bar{a}_{j_{1}\ldots j_{p}}b^{k_{1}\ldots k_{q}}b^{l_{1}\ldots l_{q}}\mathrm{- t r a c e s}
\|B_1\|_{\infty}\dots \|B_{k+1}\|_{\infty}=\prod_{l\in L^c}\|C_r^{\nu_l+1}\|_{\infty}\leq \|C_r\|_{\infty}^{\sum_{l\in L^c}(\nu_l+1)}=\|C_r\|_{\infty}^{k},
S_k^{+}(0)={\rm VSASM}(k), S_k^{+}(1)={\rm VSASM}(k+1),{\partial \over \partial q}S_k^{+}(q) \vert_{q=0}= 2k S_{k}^{+}(0),{\rm Coef\/f}_{q^{k}}(S_k^{+}(q) )={\rm CSTCPP}(k+1), S_k^{-}(0)={\rm CSTCPP}(k), S_k^{-}(1)={\rm CSTCPP}(k+1),{\partial \over \partial q}S_k^{-}(q) \vert_{q=0}= (2k-1) S_{k}^{-}(0),{\rm Coef\/f}_{q^{k}}(S_k^{-}(q) )={\rm VSASM}(k).
\Psi_1(0)+(1-z)\Psi_2(0)={}_2F_1\bigg[\begin{matrix}-a \frac12 &1\end{matrix}\bigg|\,z\bigg]+(1-z)^{a+1}{}_2F_1\bigg[\begin{matrix}-a \frac12 &1\end{matrix}\bigg|\,\frac{z}{z-1}\bigg] = 2\bigg(1-\frac z2\bigg)^{a+1}\cdot{}_2F_1\bigg[\begin{matrix}-\frac12a \frac12-\frac12a &1\end{matrix}\bigg|\,\frac{z^2}{(z-2)^2}\bigg]=(2-z)\cdot\Phi(0).
\frac{d\langle T\rangle}{dt}=\frac{d\langle\Psi|T|\Psi\rangle}{dt}=\langle\frac{\partial\Psi}{\partial t}|T|\Psi\rangle+\langle\Psi|\frac{\partial T}{\partial t}|\Psi\rangle+\langle\Psi|T|\frac{\partial\Psi}{\partial t}\rangle
\widehat G_{1}\sum_{i=1}^N\widehat\Gamma_i\widehat\Sigma_{ii}^{-1}\bigl(\widehat BM_{z z}^{i j}\widehat B{}'-\widehat\Sigma_{zz}^{i j}\bigr)\widehat\Sigma_{j j}^{-1}I_{K+1}^1=0,
f_1=f_1(s,z)=e^{4s}(1-z)^{2}(1+z)^{2}+e^{-4s}+2z^2, &f_2=f_2(s,z)=e^{2s}(1+z)^{2}+e^{-2s}, &f_3=f_3(s,z)=e^{2s}(1-z)^{2}+e^{-2s}.
\begin{array}{ll}\!\! \!\! \log B_0(x_0,u(x_0),Du(x_0))-\log B_0 (x_0,u_0(x_0),Du_0(x_0)) = \!\! \!\! D_u(\log B_0)(x_0,\tilde u(x_0), Du(x_0))(u-u_0) (x_0) \!\! \!\! + D_{p_k}(\log B_0)(x_0,u_0(x_0),\tilde p(x_0))D_k(u-u_0)(x_0),\end{array}