harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{R}^{ab}=R^{ab}-K^a_c\wedge K^{cb}+\frac{1}{2}C^{ab}_{\;\cdot c}\tilde{T}^c+\frac{1}{4}\gamma^I_J \gamma^K_L\theta^a_I \theta^b_K A^J\wedge A^L+i\left(DK^{ab}-\frac{1}{4}\gamma^I_J(\theta^a_Ie^b+\theta^b_Ie^a)\wedge A^J\right).
|p_1(x)-p_2(x)| = \left|\sum_{i=1}^{n+1}(p_1(r_i)-p_2(r_i))\prod_{j\neq i}\frac{x-r_j}{r_i-r_j}\right| &\leq \sum_{i=1}^{n+1}\left|p_1(r_i)-p_2(r_i)\right|\prod_{j\neq i}\frac{|x-r_j|}{|r_i-r_j|}&< \sum_{i=1}^{n+1}\upsilon\gamma^{-n}&=n\upsilon\gamma^{-n}&=\epsilon,
D^*(r_s) \leq \bar{D}(r_s)= \frac{\sum_{i=0}^{K-1-s}\binom{K}{s+1+i}(N-1)^i N}{\binom{K-2}{s-1}+\sum_{i=0}^{K-1-s}\binom{K-1}{s+i}(N-1)^iN}
\partial_t v = d\Psi(0,0) (b, v) + \partial_t q(a,\hat{u}) &= -\nabla_{A_{\infty}}^* \nabla_{A_{\infty}}v - \nabla_{v}(J L_{u_{\infty}}\mu(u_{\infty})) - \nabla_{A_{\infty}}^* L_{u_{\infty}}b + \nabla b \times d_{A_\infty}u_{\infty}&+ d q(a_t, \hat{u}_t) [b, v]
\frac{\partial}{\partial z}H(x,z,p) = H(x,z,p)\sum_{i=1}^n \frac{1}{z + nx_ip_i}&\geq nH(x,z,p)\left( \prod_{i=1}^n \frac{1}{z + nx_ip_i}\right)^\frac{1}{n}&= nH(x,z,p)^\frac{n-1}{n},
F[\Phi,X ,\varphi,\chi] =\int \!dx\;\left(\Phi^j J^{1~j}[\varphi ]- \frac i2~\overline X^j \gamma^1 \chi^j \right)=\int dx\Bigl(\Phi^i (\sqrt{1-{\varphi}^2}\frac{\stackrel{\leftrightarrow}{\partial}}{\partial x}\varphi^i +\varepsilon^{ijk}\varphi^j\frac \partial{\partial x}\varphi^k) -\frac i2\overline X^j \gamma^1 \chi^j\Bigr)~.
\Delta = \left( \begin{array}{c c}{{\sqrt{\zeta + \rho^{2}}}}&{{0}}\\{{0}}&{{\rho}}\\{{\bar{z}_{2}}}&{{- z_{1}}}\\{{\bar{z}_{1}}}&{{z_{2}}}\\ \end{array}\right) , \quad \bar{\Delta}= \left( \begin{array}{c c c c}{{\sqrt{\zeta + \rho^{2}}}}&{{0}}&{{z_{2}}}&{{z_{1}}}\\{{0}}&{{\rho}}&{{- \bar{z}_{1}}}&{{\bar{z}_{2}}}\\ \end{array}\right) .
\rho \left( 1 + \frac{\rho}{\lambda}\right) = \frac{6 K}{k_{4}^{2}}\frac 1{a^{2}}- \frac{1 2{\cal U}_{0}}{\lambda k_{4}^{4}}\frac 1{a^{4}}- \rho + \frac{2 ( 3 H_{0}^{2}- \Lambda )}{k_{4}^{2}},
a ( \lambda ) = \rho_{g}^{2}\frac{\left( e^{\lambda}+ 1 \right)^{4}}{e^{2 \lambda}}\qquad \mathrm{~ a n d ~}\qquad b ( \lambda ) = \frac{\left( e^{\lambda}- 1 \right)^{2}}{\left( e^{\lambda}+ 1 \right)^{2}},
\int_{\Omega}\nabla \theta \cdot \mathcal{B}_{\mathrm{rot}}\theta \mathrm{d}x= -\int_{\Omega}\theta \mathrm{div}\,\, \mathcal{B}_{\mathrm{rot}}\theta \mathrm{d}x= -\int_{\Omega}\theta^{2}\mathrm{d}x.
d\alpha = \sum_{i=1}^ndf_i \wedge dx_1 \wedge \cdots \wedge \widehat{dx_i}\wedge \cdots \wedge dx_n & = \sum_{i=1}^n(-1)^{i-1}dx_1 \wedge \cdots \wedge dx_{i-1}\wedge df_i \wedge dx_{i+1}\wedge \cdots \wedge dx_n.
\varphi(\mathrm{v},\mathrm{y},\mathrm{t}, \tau) := \displaystyle\int_{0}^{\tau}\dfrac{\alpha |\mathrm{y}-\mathrm{v}|^2}{4(\mathrm{s}-\tau)^2}\textbf{exp}\Big(-\dfrac{\alpha|\mathrm{y}-\mathrm{v}|^2}{4(\mathrm{s}-\tau)}\Big)\Phi^{\textbf{e}}(\xi,\mathrm{t};\mathrm{z},\mathrm{s})d\mathrm{s}.
[\beta_jY_{i,\ell}^{t_s,t}-\sqrt{P_r}h_{r,j}^{t_s}d_r]\mod\Lambda = [\sqrt{P_r}h_{r,j}^{t_s}\hat{V}^{t_s,t}_{i,\ell}+(\beta_j-1)\sqrt{P_r}h_{r,j}^{t_s}\hat{V}^{t_s,t}_{i,\ell}+\beta_jn_2]\mod\Lambda &=[\sqrt{P_r}h_{r,j}^{t_s}\hat{V}_{i,\ell}^{t_s,t}+n']\mod\Lambda,
\Gamma_{1}={\frac{1}{\lambda_{1}}}{\frac{1}{\lambda_{2}}}\left({\frac{2}{\lambda_{1}+ \lambda_{2}}}\right)^{2}, \quad \Gamma_{2}={\frac{1}{2}}{\frac{1}{\lambda_{1}^{2}}}\left({\frac{2}{\lambda_{1}+ \lambda_{2}}}\right)^{2}, \quad \Gamma_{3}={\frac{1}{\lambda_{1}^{3}}}{\frac{2}{\lambda_{1}+ \lambda_{2}}}.
n^{\phi}(y,z)=2\frac{j(z)}{j(y)}\int_{y}^{z}\int_{-\infty}^{0}\phi''(\omega)\Phi(\omega,x,z)\tilde{\Phi}(-\omega,x,-y)dxd\omega =:\frac{j(z)}{j(y)}K^{\phi}(y,z)
- \ln Z_{1}= \int_{0}^{\infty}\frac{d s}{s}\left( \mathrm{T r}\, e^{s D^{2}}- \frac 1 4 \mathrm{T r}\, e^{s ( D^{2}{\bf 1}_{1 6}+ i \Gamma . F / 2 )}- \frac 1 2 \mathrm{S t r}\, e^{s \Delta_{2}}\right) ,
f^*(z_1,\ldots,z_{m})=\sum_{n=m}^{\infty}P_n(z_1,\ldots,z_{m})=\sum_{i_1\geq1,\ldots,i_m\geq1}c_{i_1,\ldots,i_m}z_1^{i_1}\cdots z_m^{i_m},
\|u\|_{D^{s,p}}^p+ \int_{\mathbb{R}^N}V|u|^p dx =\|u\|_{s,p,V_{+}}^p-\int_{\mathbb{R}^N}V_{-}|u|^p dx &\geq \|u\|_{s,p,V_{+}}^p-\|V_{-}\|_{N/(ps)}\|u\|_{p_s^{*}}^p &\geq \|u\|_{s,p,V_{+}}^p-S_{D^{s,p}}^{-1}\|V_{-}\|_{N/(ps)}\|u\|_{D^{s,p}}^p &\geq (1-S_{D^{s,p}}^{-1}\|V_{-}\|_{N/(ps)}) \|u\|_{s,p,V_{+}}^p.
\sigma_{- 2 l_{1}- j}\left( \left( A_{1}- \lambda \right)^{- 1}\right) ( x , - \xi , \lambda ) = \left( - 1 \right)^{j}\sigma_{- 2 l_{1}- j}\left( \left( A_{1}- \lambda \right)^{- 1}\right) ( x , \xi , \lambda ) .
d_{L}\alpha (s_0,\ldots, s_k) := \sum_{j = 0}^k (-1)^j X_j \alpha(s_0, \ldots, \hat s_j, \ldots, s_k) + \sum_{i < j}(-1)^{i+j}\alpha(s_i * s_j, s_0, \ldots, \hat s_i, \ldots, \hat s_j, \ldots, s_k)
\| (\nabla f)(g_2(x)-t(g_2-g_1)(x)) \|_{L_x^p}\leq \| \nabla f \|_{L^p}\| \det \nabla ((1-t)g_2(x) + tg_1(x)) \|^{1\slash p}_{L_x^{\infty}}&\leq C( \| \det\nabla g_1 \|_{L^{\infty}}+ \| \det\nabla g_2 \|_{L^{\infty}})^{1\slash p}\| \nabla f \|_{L^p}.
fg =(T^h+\widetilde{T}^h+R^h)(T^v+\widetilde{T}^v+R^v)(f,g) &=T^hT^v(f,g)+T^h\widetilde{T}^v(f,g)+T^hR^v(f,g) &\quad+\widetilde{T}^hT^v(f,g)+\widetilde{T}^h\widetilde{T}^v(f,g)+\widetilde{T}^hR^v(f,g) &\quad+R^hT^v(f,g)+R^h\widetilde{T}^v(f,g)+R^hR^v(f,g).
s=s^{\prime}[1-\frac{1}{2s^{\prime2}}[2(xx^{\prime}+yy^{\prime})-(x^{\prime2}+y^{\prime2})]+\frac{1}{8s^{\prime4}}[2(xx^{\prime}+yy^{\prime})-(x^{\prime2}+y^{\prime2})]^{2}+\cdot\cdot\cdot].
\frac{d}{ds}(\varepsilon,|\cdot|^2) =(\mathcal{L}_{\mu}\varepsilon,|\cdot|^2)-(\nabla\cdot(\varepsilon\nabla\phi_{\varepsilon}),|\cdot|^2)+\frac{{\mu}_s}{2{\mu}}(\Lambda\varepsilon,|\cdot|^2)+(\Theta_{\mu}(\varepsilon),|\cdot|^2) &+(F,|\cdot|^2)+(E,|\cdot|^2).
\mathbb{T}^{T(2,2a+1)^{(2,2b+1)}}_{\mu}(\rho^{\rm{AN}}_{u,\omega_2}) = \frac{(2b+1)}{2}\times \left( \frac{\omega_2^{2a+1}+\omega_2^{-(2a+1)}}{\omega_2^{2}-\omega_2^{-2}}\right)^2
I_{\chi}(u_{1,k}, u_{2,k}) = \int_{X}f_k = \int_{\{u_1> \phi-k\}}f_k+ \int_{\{u_1 \le \phi-k\}}f_k & =\int_{\{u_1> \phi-k\}}f+ \int_{\{u_1 \le \phi -k\}}f_k &= I_\chi(u_1,u_2)- \int_{\{u_1\le \phi -k\}}f + \int_{\{u_1 \le \phi -k\}}f_k.
\mathrm P \left( X \in \left( t_{J-1},t_{J}\right) \right) = \mathrm P (X > F^{-1}( (J-2) \epsilon ) ) = 1- F \left(F^{-1}( (J-2) \epsilon ) \right) \leq 1- \lfloor 1/\epsilon \rfloor \epsilon < \epsilon.
u ( p^{2}, m_{1}^{2}, m_{2}^{2}) = \frac{\sqrt{p^{2}+ ( m_{1}+ m_{2})^{2}}- \sqrt{p^{2}+ ( m_{1}- m_{2})^{2}}}{\sqrt{p^{2}+ ( m_{1}+ m_{2})^{2}}+ \sqrt{p^{2}+ ( m_{1}- m_{2})^{2}}}\ .
\sqrt{\det\left(\frac{\hat{{\cal G}}_{\perp}^{(0)}}{\hat{{\cal G}}_{\perp}^{(1)}}\right)}=\frac{{\cal N}_{\perp}}{8i\sin\left(\frac{|\omega_-(\kappa^2)|\beta_T\mu}{2}\right)\sinh\left(\frac{\omega_+(\kappa^2)\beta_T\mu}{2}\right)}e^{-h_{\perp}(\beta_T\mu,\kappa^2)}
a_{N_1+N_2}(\sigma) - a_{N_1}(\sigma) - a_{N_2}(\sigma) =\log \left( \mathbb{E}_{\mu_{N_1}^{\sigma}\bigotimes \mu_{N_2}^{\sigma}}\left[ \exp \left( - \sum_{(i, j) \in I_{N_1,N_2}}M_{ij}u_i v_j \right) \right] \right).
\nabla^{\frac{H}{2}}\left(\int_0^Tg_s\delta R_s^H\right)(x) = \int_0^T(\nabla^\frac{H}{2}g_s)(x)\delta R_s^H + 2c_{H}^{B,R}\frac{\mathrm{B}\left(\frac{H}{2},1-H\right)}{\Gamma\left(\frac{H}{2}\right)^2}\int_0^Tg_s|s-x|^{H-1}\delta B_s^{\frac{H}{2}+\frac{1}{2}}.
I(u)= \frac{1}{2}\int_M |\nabla u|^2 dv_g +\frac{m_0^2}{2}\int_M u^2 dv_g -\frac{1}{p}\int_M (u^+)^{p}dv_g -\frac{\omega^2}{2}\int_M (1-q\Phi(u))u^2 dv_g~,
\begin{cases}\overline{u}_t=d_i\Delta \overline{u}-\lambda \overline{u}+g_i(t,\cdot), \Omega_T, \nabla_x \overline{u}\cdot n(x) = 0, [0,T] \times \partial \Omega, \overline{u}_0=u_{i0}(x)\ge0, x \in \Omega,\end{cases}
e_{k\ell}(c\cdot z) =e_{k_1\ell_1k_2\ell_2}\big(c\cdot k_2\ell_2\bar k_2\bar \ell_2 z_1+c\cdot k_1\ell_1\bar k_1\bar \ell_1 z_2\big) &=e_{k_1\ell_1}\big(( \bar k_2\bar \ell_2 c)\cdot z_1\big) e_{k_2\ell_2}\big((\bar k_1\bar \ell_1 c)\cdot z_2\big).
h_{e_{i}e_{j}}d e^{i}d e^{j}= \mathrm{T r}( V_{1}\cdot V_{1}) = \frac 2{( e^{2})^{2}}e_{0}^{2}\left( ( d e_{1})^{2}+ ( d e_{2})^{2}\right) \quad ( i , j = 1 , 2 )
\ell^{ss}_{(\xi, x, y)}(v) - \ell^{ss}_{(\xi, x, y')}(v) = ( y - y ' ) \, e^{- \int_x^v A (\xi, s) \, d s}\in [ e^{- \|A \|_\infty}, e^{\| A \|_\infty}] \cdot | y - y' | .
\begin{aligned}A =\big[[f_a,f_b]_3,[f_a,f_c]_3,[f_a,f_d]_3\big], B =\big[f^3_a,f_{a3},[f_b,f_c]_3,[f_b,f_d]_3\big], C =\big[f^3_a,f_{a3},f^3_b,f_{b3},[f_c,f_d]_3\big].\end{aligned}
\begin{cases}U(k)=\lefteqn{-H_{1}(\tilde{P}_{1}(k+1))^{-1}G_{\nu}(\tilde{P}_{1}(k+1))^{T},}\mathbb{U}(k)=U(k)+\tilde{U}(k)=\tilde{H}_{1}(\tilde{P}_{1}(k+1),\tilde{Q}_{1}(k+1))^{-1}\tilde{G}_{\nu}(\tilde{P}_{1}(k+1),\tilde{Q}_{1}(k+1))^{T},\end{cases}
\underset{\mathbf{x}\in \mathbb{R}^{np}, \mathbf{z}\in \mathbb{R}^{mp}}{}f(\mathbf{x}) + \frac{\rho(1-\eta)}{2}\|\mathbf{Ax}+ \mathbf{Bz}\|^2 \sqrt{\eta}(\mathbf{Ax}+ \mathbf{Bz}) = \mathbf{0}.
W ={i\over 4}u^+_iu^+_j \bar D^i_{\dot\alpha}\bar D^{j\dot\alpha}\sum^\infty_{n=1}\int du_1\ldots du_n\;{(-ig)^{n}V^{++}(u_1) \ldots V^{++}(u_n) \over (u^+u^+_1)(u^+_1u^+_2) \ldots (u^+_nu^+)}
\begin{pmatrix}Y_1 Y_2 \end{pmatrix}= Y = X R|_{\hat t\times\hat t}= \begin{pmatrix}X_1 X_2 \end{pmatrix}\begin{pmatrix}R_{11}R_{12}R_{22}\end{pmatrix}= \begin{pmatrix}X_1 R_{11}X_1 R_{12}+ X_2 R_{22}\end{pmatrix},
s_{g a u g e}= \int_{0}^{T}\sum_{l l^{\prime}}\left( \frac 1 S \frac{d S_{l}( t )}{d t}- ( r_{b}- r_{l}) \right) \frac{A_{l l^{\prime}}( t )}{2}\left( \frac 1 S \frac{d S_{l^{\prime}}( t )}{d t}- ( r_{b}- r_{l^{\prime}}) \right) d t \ .
\mathbf{A}_g = \begin{bmatrix}1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 \end{bmatrix},\mathbf{A}_w = \begin{bmatrix}1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 \end{bmatrix},\mathbf{A}_a = \begin{bmatrix}0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 \end{bmatrix}.
\|\langle v \rangle_h(t)-v(t)\|_{L^p(\Omega)}^p begin{align*}3mm] \leq \begin{cases}c \big\|[w]_h(t)-w(t)\big\|_{L^2(\Omega)}^2 \textrm{for}\,\,\,p\geq 2 begin{align*}2mm]c\big\|[w]_h(t)-w(t)\big\|_{L^2(\Omega)}^2\Big(\|[w]_h(t)\|_{L^2(\Omega)}^2+\|w(t)\|_{L^2(\Omega)}^2\Big)^{\frac{2-p}{2}}\textrm{for}\,\,\,1\leq p< 2\end{cases}begin{align*}3mm] \to 0
{R^{\lambda}}_{\mu \nu \rho}= \partial_{\rho}{\Gamma_{\mu \nu}}^{\lambda}- \partial_{\nu}{\Gamma_{\mu \rho}}^{\lambda}+{\Gamma_{\mu \nu}}^{\alpha}\diamondsuit{\Gamma_{\rho \alpha}}^{\lambda}-{\Gamma_{\mu \rho}}^{\alpha}\diamondsuit{\Gamma_{\nu \alpha}}^{\lambda}.
{\cal L}_{g , 4 A}= - \frac{1}{4}\sum_{n , m , l , k \neq ( 0 , 0 , 0 )}( \sum_{j = 0}^{N}a_{j n}a_{j m}a_{j l}a_{j k}) g_{L}f^{a b c}g_{L}f^{a d e}\tilde{A}_{\mu}^{n}\tilde{A}_{\nu}^{m}\tilde{A}^{l ~ \mu}\tilde{A}^{k ~ \nu},
\begin{array}{r c l}{{{\cal A}( r , \omega ; \omega_{\bot})}}&{{=}}&{{\displaystyle \frac{1}{\pi}\int_{- \infty}^{\infty}d \omega^{\prime}\frac{\mathrm{I m}~{\cal A}( r , \omega^{\prime}; \omega_{\bot})}{\omega^{\prime}- \omega - i \epsilon}.}}\\ \end{array}
X =\{2^{i}+\lambda2^j :i,j \in \mathbb N, i,j \leq n \}, Y =\{-2^{k}-2^l :k,l \in \mathbb N, k,l \leq n \}, G =\{(2^{i}+\lambda2^j, -2^{i}-2^j) : i,j \in \mathbb N, i,j \leq n\}.
\dim(\mathcal{M}(L))\leq ~\frac{1}{2}[(m+n-r-s)(m+n-r-s-1)+2(n-s)]+(r+s)(m+n-r-s-1) -\sum_{i=2}^{l}(m+n-r-s-i) &\leq\frac{1}{2}[(m+n-r-s-1)(m+n+r+s)+2(n-s)]- \sum_{i=2}^{l}(m+n-r-s-i) &\leq \frac{1}{2}[(m + n - r - s)(m + n + r + s) + (n - m - r - 3s)]-\sum_{i=2}^{l}(m+n-r-s-i).
P_{so}(l_k) \!=\! 1\!-\!\mathbb{E}_{\Phi_J}\left\{\mathbb{E}_{\Phi_E}\left\{\prod\limits_{E_i \in \Phi_E}\left\{1\!-\!\mathbb{P}\left\{\left. \frac{P_{S_k}|h_{{S_k},{E_i}}|^2/d_{S_k,E_i}^\alpha}{\sum\limits_{J_j \in \Phi_J}P_{\bar{J}}|h_{J_j, E_i}|^2/d_{J_j, E_i}^\alpha}\!>\! \gamma_E \right|{\Phi_E,\Phi_J}\right\}\right\}\right\}\right\}.
\mu^2=\frac{5}{3}\left(\frac{B^1_{11,3}}{\mu}\right)^2,~~~4\left(\frac{B^1_{11,3}}{\mu}\right)^2=\left(\frac{B^3_{22,1}}{\mu}\right)^2+\left(\frac{B^3_{11,2}}{\mu}\right)^2+\left(\frac{B^4_{11,2}}{\mu}\right)^2.
\mathrm{d}_{\mathbb{U}}((\mathbf{x}_1, \mathbf{y}_1), (\mathbf{x}_2, \mathbf{y}_2)):= \bigg( \sum_{i=1}^{\infty}(x_{1i}-x_{2i})^2 \bigg)^{1/2}+ \sum_{i=1}^{\infty}\big| x_{1i}y_{1i}- x_{2i}y_{2i}\big|.
\xi_{f l a t}( c ) = \frac 1 2 \int_{0}^{\infty}d x \frac{1}{\sqrt{\cosh ( x + c )}}\rightarrow \xi_{f l a t}( 0 ) = \sqrt{\frac{8}{\pi}}\left( \Gamma ( \frac{5}{4}) \right)^{2}\simeq 1 . 3 1 1 .
\begin{cases}t -\mu^2\left(\sum\limits_{i=1}^m \cfrac{2t}{t^2 - \sigma_i^2}+ \cfrac{n-m}{t}\right) = z_o, \sigma_i + 2\mu^2 \cfrac{\sigma_i}{t^2 - \sigma_i^2}, = \sigma_i^o t>|\sigma_i|,\end{cases}
\widetilde{\rho}_{(i,j+1)}:=\widetilde{g}_{(i,j+1)}\widetilde{\rho}_{(i,j)}\widetilde{g}_{(i,j+1)}\widetilde{\rho}_{(i,j)}^{-1}, i=1,\dots, r,\j=1,\dots, a_{i}-2.
f_{Cayley}= \sum_{i_1=0}^{\tau_1}\!\cdots\!\sum_{i_{d-1}=0}^{\tau_{d-1}}\sum_{j_1=0}^{\tau_{d-1}}\!\cdots\!\sum_{j_{d-1}=0}^{\tau_1}A_{i_1,\ldots,i_{d-1},j_1,\ldots,j_{d-1}}\prod_{k=1}^{d-1}\phi_{i_k}(s_k) \prod_{k=1}^{d-1}\phi_{j_k}(t_k)
\int_{0}^{\infty}\rho^{-\frac{d}{2}-2z}J_{\frac{d}{2}-1}(|\xi|\rho)d\rho = 2^{-\frac{d}{2}-2z}|\xi|^{\frac{d}{2}-1+2z}\frac{\Gamma(-z)}{\Gamma(\frac{d}{2}+z)},
\ell_{(a,w)}F(x,z)= F(a^{-1}x,\overline{w}z \sigma(a,a^{-1})\overline{\sigma(a^{-1},x)}) = F(a^{-1}x,\overline{w}z \sigma(a,a^{-1}x)) = w \overline{z}\,\overline{\sigma(a,a^{-1}x)}f(a^{-1}x) = \overline{z}w \,\ell^\sigma_a f(x).
I [ \hat{g}, \hat{\phi}] = \frac{1}{\pi}\int d^{3}x \sqrt{- \hat{g}}\left( \frac{\hat{R}+ 2 l^{- 2}}{1 6}- \frac{1}{2}( \nabla \hat{\phi})^{2}- \frac{1}{1 6}\hat{R}\hat{\phi}^{2}+ \frac{\nu}{8 l^{2}}\hat{\phi}^{6}\right) ,
0=P_{i}=\frac{2\rho_{i}}{\rho}+\frac{h_{11i}}{h_{11}\log h_{11}}-\beta[\frac{(X,\nu)}{(\nu,E_{n+1})}]_{i}-\alpha\frac{2(\nu_{i},E_{n+1})}{(\nu,E_{n+1})^{3}},
A+1 \leq \frac{1-3a^2}{2a^4}+1 = \frac{1-3a^2+2a^4}{2a^4}= \frac{(1-2a^2)(1-a^2)}{2a^4}&=\frac 1 2 ( \frac 1{a^2}-2) ( \frac 1{a^2}-1).
\exp^{-1}W_k(x) = \exp\left[ \sum_{n=1}^{\infty}\frac{(1 - q^{-nk})x^n}n \right] W_k(x) \exp^{-1}\Rightarrow &\Rightarrow \ \exp^{-1}W_k(x) = \frac{1-\frac x{q^k}}{1-x}\cdot W_k(x) \exp^{-1}\Rightarrow &\Rightarrow \ \Phi_m \exp^{-1}W_k(x) \cdot (1-x) = \Phi_m W_k(x) \exp^{-1}\cdot \left(1 - \frac x{q^k}\right)
\eta_{w,q}(Y_{T_{k}}Y_{T_{k}^{\ast}}) =\eta_{w,q}(q^{A_{1}}Y_{T_{+}}+q^{A_{2}}Y_{T_{-}}) =q^{A'_{1}}Y_{I(T_{+})}^{-1}Y_{\Omega_{w}^{-1}(T_{+})}+q^{A'_{2}}Y_{I(T_{-})}^{-1}Y_{\Omega_{w}^{-1}(T_{-})},
B_{i}^{ii},\,\,\,\,\,\,\,\,\,\,A_{i}+B_{ij},\,\,\,\,\,\,\,\,\,\,A^{i}+B^{j},\,\,\,\,\,\,\,\,\,\,A_{i}-B^{i},\,\,\,\,\,\,\,\,\,\,A_{i}^{i}=B_{i},
a^{+}(\hat{n}+1-\lambda)_{k,\lambda}a =(\hat{n})_{k+1,\lambda}=(a^{+}a)_{k+1,\lambda}&=\sum_{m=0}^{k+1}S_{2,\lambda}(k+1,m)(a^{+})^{m}a^{m}&=\sum_{m=1}^{k+1}S_{2,\lambda}(k+1,m)(a^{+})^{m}a^{m}&=\sum_{m=0}^{k}S_{2,\lambda}(k+1,m+1)(a^{+})^{m+1}a^{m+1}.
\cos \frac{2 \pi ( k + j )}{n - 1}- \frac{1}{n}\cos \frac{2 \pi ( k + j ) n}{n - 1}= \cos \frac{2 \pi ( k^{\prime}+ j )}{n - 1}- \frac{1}{n}\cos \frac{2 \pi ( k^{\prime}+ j ) n}{n - 1}= \gamma^{K}
\ \bigcup_{n \geq 0}\tilde{f}_{i}^{n}(\mathcal{B}_{v}(\lambda) \otimes \mathcal{B}_{w}(\mu)) \setminus \{0 \}= \ (\mathcal{B}_{s_i v}(\lambda) \otimes \mathcal{B}_{w}(\mu)) \sqcup (S_{i}(\tilde{e}_i^{\max}(\mathcal{B}_v(\lambda))) \otimes (\mathcal{B}_{s_i w}(\mu) \setminus \mathcal{B}_{w}(\mu))).
\forall \; n_0 \; \forall \; \alpha \; \exists \; \gamma \; \gamma(0) =n_0 \; \wedge \; \forall \; n \; \gamma(S(n))=\alpha(\gamma(n)) [\forall \; n \; \exists \; m \; \varphi(n,m)]\Rightarrow \exists \; \gamma \; [\forall \; n \; \varphi(n, \gamma(n))] & \forall \; \alpha \; \forall \; n \; \forall \; m \; (\alpha(n)=m \vee \alpha(n)\neq m)
\Gamma_0:=\displaystyle \inf_{w\in \Gamma\setminus\{0\}}\left\{\displaystyle \int_{-1}^1 (\alpha +\beta-{2}+2w_1^2+2w_2^2)(\alpha+\beta+2w_1^2+2w_2^2)|w_1|^\alpha|w_2|^\beta e^{w^2_1+w^2_2}dx\right. -\left.(2-p-q)\lambda \int_{-1}^1f(x)|w_1(x, 0)|^p|w_2(x,0)|^q dx \right \}>0,
B_{\alpha^{2}}(\alpha^{2}(x),y\ast z)= B(\alpha^{4}(x),\alpha(y)\alpha(z))=B(\alpha^{3}(x),\alpha^{2}(y)\alpha^{2}(z))=B(\alpha(x)\alpha(y),\alpha^{4}(z)) = (-1)^{|x||z|+|y||z|}B(\alpha^{4}(z),x\ast y)=B_{\alpha^{2}}(x\ast y,\alpha^{2}(z)),
(1\; 2)=\left(\begin{array}{ccc|c}1 1 -1 0 0 1 0 1 0 \hline I_{n-2}\end{array}\right)\mbox{\; and \;}(k\;\; k+1)=\left(\begin{array}{c|cc|c}I_k \hline 0 1 & 1 0 \hline I_{n-k-1}\end{array}\right)
\dot{x}{}^{a}= \kappa^{- 1}\bar{\pi}{}^{A}\pi^{A^{\prime}}, \, \, \, \, \acute{x}{}^{a}= i \zeta ( \dot{\bar{\pi}}{}^{A}\pi^{A^{\prime}}- \bar{\pi}{}^{A}\dot{\pi}{}^{A^{\prime}}) .
A\rho=\sum_{i,j}\rho_{ij}A|i\rangle\langle j|\rightarrow\sum_{i,j}\rho_{ij}(A|i\rangle)\otimes|j\rangle=\sum_{i,j}\rho_{ij}(A\otimes I)(|i\rangle\otimes|j\rangle)=(A\otimes I)|\rho\rangle\rangle=L(A)[\rho],
\widetilde{I}\left(R,c\right)=\sup\left\{\int_{1}^{R}\frac{1-f^{2}}{r}dr+4\int_{1}^{R}\frac{\left( 1-f^{2}\right)^{2}}{r}dr:\int_{1}^{R}J\left(1-f^{2}\right)rdr\leq c\right\}.
(-1)^kH(e_{k,k}\otimes\beta_{i+1,i})=H(e_{k+1,k}\otimes\beta_{i,i}) =\begin{cases}\beta_{i+k+1,i+k}, i+k+1\leq r 0, ,\end{cases}H(e_{k,k}\otimes\beta_{i,i}) =\begin{cases}\beta_{i+k,i+k}, i+k\leq r-1 0, .\end{cases}
B(\bar u_N,\bar u_N) =\sum_{m,j=1}^N t^{-\mu_m-\mu_j}B(\xi_m,\xi_j) &=\sum_{n=1}^{N}\frac1{t^{\mu_n}}\Big( \sum_{\stackrel{1\le m,j\le N,}{\mu_m+\mu_j=\mu_n}}B(\xi_m,\xi_j)\Big)+\frac1{t^{\mu_{N+1}}}\sum_{\stackrel{1\le m,j\le N,}{\mu_m+\mu_j=\mu_{N+1}}}B(\xi_m,\xi_j)+h_{N+1,2},
\left\vert A_{kp}\left( z\right) \right\vert \leq \exp \{\left( \underline{\tau}_{M}(A_{kp})+\varepsilon \right) r^{\mu (A_{kp})}\}=\exp \{\left(\underline{\tau}_{M}(A_{kp})+\varepsilon \right) r^{\mu (A_{l0})}\}.
\begin{array}{l}{\frac{2 M}{( D - 3 - p ) A_{D - 2 - p}}= \frac{D - 2 - p}{D - 3 - p}r_{H}^{D - 3 - p}+ ( D - 2 ) \delta \alpha ,}\\{\left( \frac{Q_{0}}{( D - 3 - p ) A_{D - 2 - p}}\right)^{2}= ( D - 2 ) \delta \alpha \left( \alpha + r_{H}^{D - 3 - p}\right) ,}\\ \end{array}
- \frac{i}{\hbar}\frac{q^{2}}{1 + q^{2}}\; \Bigl [ \; \frac{\{c , \; \dot{\bar{c}}\}_{q}}{q^{4}}+ [ \; b , e \; ]_{q}\; \Bigr ] \; \dot{c}\; ( p^{2}- m^{2}) ,
{\mathcal{W_{FB}}}=\frac{1}{N}\sum\limits_{k,l}\left[{\bf{\chi}}_{k}^{\dag}({\bf{s}}_l\otimes{\bf{p}}_l^{\dag})\phi_k+\phi^{\dag}_{k}({\bf{p}}_l\otimes{\bf{s}}_l^{\dag})\chi_k\right]
S_{N+1,1}=S_{N,1}+\sum_{j=1}^{N}\|\mu_j\|\frac{\|\mu_{N+1}\|^2}{(r_j+\dots+r_{N+1})^2}&\le S_{N,1}+\|\mu_{N+1}\|^2\sum_{j=1}^{N}\frac{r_j}{(r_j+\dots+r_{N+1})^2}.
q_r^{\varepsilon,\delta,h}\partial_t r = q_r^{\varepsilon,\delta,h}\Delta r + q_{rr}^{\varepsilon,\delta,h}|\nabla r|^2 -q_{rr}^{\varepsilon,\delta,h}&= q_r^{\varepsilon,\delta,h}\Delta r + q_{r}^{\varepsilon,\delta,h}\frac{g^{\delta,h}_q}{\varepsilon}(|\nabla r|^2 -1).
\widehat{C^{a/q,2}_{\lambda}}(\xi ) = \widehat{M_{\psi_{Q/2}, q}}(\xi ) \cdot \sum_{\ell \in \mathbb Z^{d}}\widetilde\psi_{q}(\xi - \ell /q) (1- \widetilde\psi_{\lambda Q/N}(\xi - \ell /q) )\widetilde{d \sigma_{\lambda}}(\xi - \ell /q) &:= \widehat{M_{\psi_{q/2}, q}}(\xi ) \cdot \widehat{C^{a/q,3}_{\lambda}}(\xi ).
\sum_{r=1}^{n-1}g_{rk}\Omega_{X,r}=\sum_{r=1}^{n-1}a_k \zeta^{rk}(n x_r^2-2x_r\sum_{p=1}^{n-1}x_p) &=\sum_{r=1}^{n-1}(n-2)a_k \zeta^{rk}x_r^2- \sum_{1\leq p<r \leq n-1}2(\zeta^{rk}+\zeta^{pk})a_k x_r x_p .
\dfrac{h(w;\lambda)}{\varGamma((r-p-q)w+1-a-b)}= C_3 \cdot \tilde{d}^w \cdot \dfrac{\prod_{i\in \bar{I}_p}\varGamma\left(w+\frac{i+a}{p}\right) \prod_{i \in \bar{I}_q}\varGamma\left(w+\frac{i+b}{q}\right)}{\prod_{j \in J_p}\varGamma\left(w+\frac{j-a}{r-p}\right) \prod_{j \in J_q}\varGamma\left(w+\frac{j-b}{r-q}\right)}.
\tilde{Q}_g = - \frac{1}{(2 \pi)^3}\int \frac{d^3 k}{2 k^0}\bigl[ \chi(\vec{k})^*k_\nu (\vec{k}) a^\nu (\vec{k}) +\chi (\vec{k}) k_\nu (\vec{k}) a^\nu (\vec{k})^K \bigr]
E_2(t,\eta) &=\sum_{K=K_1-K_2+K_3}|\eta_K|^2|\eta_{K_1}|^2|\eta_{K_2}|^2|\eta_{K_3}|^2\Big(\frac{1}{|\eta_{K}|^2}-\frac{1}{|\eta_{K_1}|^2}+\frac{1}{|\eta_{K_2}|^2}-\frac{1}{|\eta_{K_3}|^2}\Big)\Big|\frac{\sin(\frac{1}{2}t\Delta\omega_{KK_1K_2K_3})}{\frac{1}{2}\Delta\omega_{KK_1K_2K_3}}\Big|^2 &\quad+o(t^2L^2\log(L)+tL^4)
\|\hat Q_rQ_{\geq r_0}\|_2^2=\sum_{s\geq r_0}\|\hat Q_rQ_s\|_2^2 \leq \sum_{s\geq r_0}\frac{4\|\hat Q_rE Q_s\|_2^2}{(\mu_r-\mu_s)^2}\leq \sum_{s\geq r_0}\frac{4\|\hat Q_rE Q_s\|_2^2}{(\mu_r-\mu_{r_0})^2}=\frac{4\|\hat Q_rEQ_{\geq r_0}\|_2^2}{(\mu_r-\mu_{r_0})^2},
\frac{\langle \varphi^{2}\rangle_{0}}{M_{5}^{3}}\sim \frac{C_{0}^{2}\lambda_{0}H^{2}}{M_{5}^{3}}\left( \frac{p}{H}\right)^{\frac{3 \Delta + 8}{\Delta + 2}}\sim \left. \frac{\langle \varphi_{4}^{2}\rangle}{M_{p , e f f}^{2}}\right\vert_{t = t_{p}}.
(q^{-n+2j})^k{(q^{2j+n+1}s^2/a^2;q)_k \over (q^{4j+1}s^2/a^2;q)_k}=(q^{2j}s)^{n-2j}{(q/a^2;q)_{n-2j}\over (q^{4j+1}s^2/a^2;q)_{n-2j}}.
\forall~j\in\mathcal J\,,a_j\in\Gamma{\rm and}\overline{D(a_j,\ell)}\subset\Omega\,, &N={\rm Card}\,\mathcal J=\sum_{i=1}^kN_i\quad{\rm and}|\Gamma|=\sum_{i=1}^k|\Gamma_i|\,, &|\Gamma|\left(1+\frac{\eta}4\right)^{-1}-C\ell\leq N\times 2\ell\leq |\Gamma|\left(1+\frac{\eta}4\right)^{-1}\,.
\lambda_g \theta_f \theta_\tau \delta_h =\lambda_g\left(\sum_{u \in G}\eta(u) \rho_u \delta_h\right)=\sum_{u \in G}\eta(u) \lambda_g\rho_u\delta_h &=\left(\sum_{u \in G}\eta(u) \rho_u\right) \lambda_g\delta_h =\theta_f \theta_\tau\lambda_g \delta_h, \forall g \in G.
F_{k\ell}(v,y)-F_{k\ell}(v_{0},0) &=-\left(\tilde{a}^{k\ell}(y)-\tilde{a}^{k\ell}(0)\right)J(v_{0})-\tilde{a}^{k\ell}(y)\left(J(v)-J(v_{0})\right) &\lesssim c_\ast(y_n^2+y_{n+1}^2)+ \delta_0 (y_n^2+y_{n+1}^2).
\hat{\Delta}_{e q u}( x - y ) = \int d^{\, 3}u \, d^{\, 3}v \sum_{\tau = \pm}\hat{B}_{L}^{( \tau )}({\bf x}-{\bf u}) \, \hat{\Delta}_{d i a g}^{( \tau )}({\bf u}-{\bf v}; x_{0}- y_{0}) \, \hat{B}_{R}^{( \tau )}({\bf v}-{\bf y}) ,
Q(2k,0) = \sum_{i=1}^k\sum_{j=0}^{i-1}\left( x_{j,i-1}P(2k-i+j,0) - y_{j,i-1}Q(2k-i+j,0)\right), Q(2k+1,0) = \sum_{i=1}^k\sum_{j=0}^{i-1}\left( x_{j,i-1}P(2k+1-i+j,0) -y_{j,i-1}Q(2k+1-i+j,0) \right) &\ \ +\frac12\sum_{i=0}^k\left( x_{i,k}P(k+i,0) - y_{i,k}Q(k+i,0) \right).
\overset{I}{T_1}\overset{J}{T_2}= \overset{I \otimes J}{T}\!\!\!_{12}, \:\: 1_{H^*}= \overset{\mathbb{C}}{T}, \:\: \Delta(\overset{I}{T^{\, a}_{\, b}}) = \sum_i\overset{I}{T^{\, a}_{\, i}}\otimes \overset{I}{T^{\, i}_{\, b}}, \:\: \varepsilon(\overset{I}{T}) = I_{\dim(I)}, \:\: S(\overset{I}{T}) = \overset{I}{T}{^{-1}}
\left(\frac{p+1}{2}\right)^2\sum_{\substack{k=k_1-k_2+\cdots-k_{p-1}+k_{2p-1}|k_1|\gg k_2^* |\Phi(k_1-k_2+\cdots+k_p, k_1\cdots, k_p)|\gtrsim |k_1|}}\frac{\widehat{(W_tu_0)}_{k_1}}{\Phi(k_1-k_2+\cdots+k_p, k_1\cdots, k_p)}\prod_{\substack{j=3 odd}}^{2p-1}\widehat{v}_{k_i}\prod_{\substack{l=2 even}}^{2p-1}\overline{\widehat{v}_{k_l}},
r_{s,p}=\frac{\overline{\beta'(b_s)}(\overline{a}_1-\overline{b}_s)r_{s,1}+\overline{\beta'(b_1)}(\overline{b}_1-\overline{a}_1)r_{1,1}+\overline{\beta'(b_1)}(\overline{a}_p-\overline{b}_1)r_{1,p}}{\overline{\beta'(b_s)}(\overline{a}_p-\overline{b}_s)}
\int_{E_{\alpha}}R^{\mu\nu\lambda\rho}R_{\mu\nu\lambda\rho}=\alpha\int_{E}\bar{R}^{\mu\nu\lambda\rho}\bar{R}_{\mu\nu\lambda\rho}+8\pi(1-\alpha)\int_{\Sigma}\bar{R}_{abab}+O((1-\alpha)^2)~~~,
e''_{2, t}(0) = \omega \left( \left( t\Sigma^t \right)^2 \right) - \left(\omega \left( t \Sigma^t \right) \right)^2 &= \int_0^t \int_0^t \omega ( \sigma_s \sigma_u ) - \omega ( \sigma_s ) \omega ( \sigma_u ) \d s \d u &= \int_0^t \int_0^t \omega((\sigma_s - \omega (\sigma_s)) (\sigma_u - \omega (\sigma_u ) ) ) \mathrm{d}s\mathrm{d}u .
\vartheta_{00}(x)^{1/6}\vartheta_{01}(x)^{-1/3}[\frac{\vartheta_{00}(x)^{4}-\vartheta_{01}(x)^{4}}{16x}]^{1/24}=\sum_{n=0}^{\infty}Q(n)x^{n}=\prod_{k=1}^{\infty}\frac{1}{1-x^{2k-1}}
\liminf_{t\to\infty}\frac 1t \int_0^t (a_{\max}N+c_{\max})I(\bar t)d\bar t\geq\liminf_{t\to\infty}\frac 1t \int_0^t (a(\xi_{\bar t})N+c(\xi_{\bar t}))I(\bar t)d\bar t \geq \liminf_{t\to\infty}\frac 1t \int_0^t c(\xi_{\bar t})(N-S(\bar t))d\bar t \geq \liminf_{t\to\infty}\frac 1t \int_0^t c_{\min}(N-S(\bar t))d\bar t.
|F(x,\sigma,\tau)-(\sigma-\tau)x(1-x)| =\big|\tfrac{(\sigma-\tau)x(1-x)}{1+\sigma x + \tau(1-x)}-(\sigma-\tau)x(1-x)\big| \leq|\sigma-\tau||1-\sigma x-\tau(1-x)+\tilde r (x,\sigma,\tau)-1| \leq2|\sigma-\tau|\tfrac{|\sigma|+|\tau|}{1+\sigma\wedge\tau\wedge0}\,.
\mathbb{E}_{t_{n}}^{\sigma,X^{n}}\left[ \int_{t_{n}}^{t_{n+1}}Z_{t}^{t_{n},X^{n}}dB_{t}\Delta B_{n+1}\right] =\mathbb{E}_{t_{n}}^{\sigma,X^{n}}\left[ Y_{t_{n+1}}^{t_{n+1},X^{n+1}}\Delta B_{n+1}\right] +R_{z_{1}}^{n,\sigma}+R_{z_{2}}^{n,\sigma},
\boldsymbol{\mathcal{P}}\left(-n+ W(x,n)\right) \times \boldsymbol{\mathcal{Q}}\left(\#(x,n)\right) \to \mathbb{R}^{\Gamma \cap \left(-n+\mathrm{Int}_{r_0+N\sqrt{k}}W(x,n)\right)}(p,q) \mapsto \mathbb{G}(p,q)|_{\Gamma \cap \left(-n+\mathrm{Int}_{r_0+N\sqrt{k}}W(x,n)\right)}